Undergraduate Courses
course descriptions for Mathematics Lower & Upper Division, and PIC Classes
mathematics courses
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Preparation: three years of high school mathematics. Requisite: successful completion of Mathematics Diagnostic Test at UCLA. Function concept. Linear and polynomial functions and their graphs, applications to optimization. Inverse, exponential, and logarithmic functions. Trigonometric functions. P/NP or letter grading.
Textbook(s)
D. Lippman & M. Rasmussen. Precalculus. An investigation of functions.
Available online at www.opentextbookstore.com/precalc
Outline update: P. Greene, 8/13
Schedule of Lectures
Lecture  Section  Topics 

1  1.1  
2  1.2  Domain and Range 
3  1.3  Rates of Change and Behavior of Graphs 
4  1.4, 1.5  Composition of Functions , Transformation of Functions 
5  1.5, 1.6  Transformation of Functions , Inverse functions 
6  2.1, 2.2  Linear Functions and Graphs of Linear Functions 
7  2.2, 2.3  Graphs of Linear Functions , Modeling with Linear Functions * 
8  2.5  Absolute Value Functions 
9  3.1  Power Functions and Polynomials 
10  3.2  Quadratic Functions 
11  3.3  Graphs of Polynomial Functions 
12  3.4  Rational Functions 
13  3.5  Inverse and Radical Functions 
14  4.1  Exponential Functions 
15  4.2  Graphs of Exponential Functions 
16  4.3  Logarithmic Functions 
17  4.4  Logarithm Properties 
18  4.5  Graphs of Logarithmic Functions 
19  5.1 and 5.2  Circles and Angles 
20  5.2, 5.3  Angles, Points in Circles using Sine and Cosine 
21  5.4  The other Trigonometric Functions 
22  6.1, 6.2 (Tangent only)  Sinusoidal Graphs and the Graph of Tangent 
23  6.3  Inverse Trig Functions 
24  6.4  Solving Trig Equations 
25  7.2  Addition and Subtraction Identities ( Omit product to sum—sum to product identities ** 
26  7.3  Double Angle and Half Angle identities 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Preparation: three and onehalf years of high school mathematics (including trigonometry). Requisite: successful completion of Mathematics Diagnostic Test (score of 48 or better) or course 1 at UCLA with a grade of C or better. Not open for credit to students with credit in another calculus sequence. Modeling with functions, limits and derivatives, decisions and optimization in biology, derivative rules and tools. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 24 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in singlevariable and multivariable calculus. This includes some material on ordinary differential equations such as those governing populationgrowth models. The course also covers some material on calculusbased probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.
The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.
Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center at MS 3974.
Textbook(s)
S. J. Schreiber, Calculus for the Life Sciences, Wiley.
Outline update: P. Greene, 11/15
Schedule of Lectures
Lecture  Section  Topics 

1  Intro  Preview of Modeling and Calculus 
2  1.1  Real Numbers and Functions 
3  1.2  Data Fitting with Linear and Periodic Functions 
4  1.3  Power Functions and Scaling Laws 
5  1.4  Exponential Growth 
6  1.5  Function Building 
7  1.6  Inverse Functions and Logarithms 
8  1.7  Sequences and Difference Equations 
9  CatchUp, Review  
10  2.1  Rates of Change and Tangent Lines 
11  2.2  Limits (no formal definition) 
12  2.3  Limit Laws and Continuity 
13  2.4  Asymptotes and Infinity 
14  2.5  Sequential Limits 
15  2.6  Derivatives at a Point 
16  2.7  The Derivative as a Function 
17  CatchUp, Review  
18  3.1  Derivative of Polynomial and Exponential Functions 
19  3.2  Product And Quotient Rule (no proof required) 
20  3.3  Chain Rule and Implicit Differentiation 
21  3.4  Derivatives of Trigonometric Functions 
22  3.5  Linear Approximation 
23  3.6  Higher Derivatives 
24  3.7  L’Hopital’s Rule 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 3A with grade of C or better. Not open for credit to students with credit for course 31B. Applications of differentiation, integration, differential equations, linear models in biology, phase lines and classifying equilibrium values, bifurcations. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in singlevariable and multivariable calculus. This includes some material on ordinary differential equations such as those governing populationgrowth models. The course also covers some material on calculusbased probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.
The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.
Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center at MS 3974.
Textbook(s)
S. J. Schreiber, Calculus for the Life Sciences, Wiley.
Outline update: P.Greene, 11/15
Schedule of Lectures
Lecture  Section  Topics 

1  4.1  Graphing Using Calculus 
2  4.2  Extreme Values 
3  4.3  Optimization in Biology 
4  4.4  Decision and Optimization 
5  4.5  Linearization and Difference Equations 
6  CatchUp, Review  
7  5.1  Antiderivatives 
8  5.2  Accumulated Change and Area under a Curve 
9  5.3  The Definite Integral 
10  5.4  The Fundamental Theorem 
11  5.5  Substitution 
12  5.6  Integration by Parts and Partial Fractions 
13  5.8  Application of Integration 
14  CatchUp, Review  
15  6.1  Introduction to Differential Equations 
16  6.2  Solutions and Separable Equations 
17  6.3  Linear Models in Biology 
18  6.4  Slope Fields and Euler’s Method 
19  6.5  Phase Lines and Classifying Equilibria 
20  6.6  Bifurcation Preview of Modeling and Calculus 
21  CatchUp  
22  Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 3B with grade of C or better. Multivariable modeling, matrices and vectors, eigenvalues and eigenvectors, linear and nonlinear systems of differential equations, probabilistic applications of integration. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 25 lectures. The remaining classroom meetings are for leeway, reviews and two midterm exams. These are scheduled by the individual instructor.
Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in singlevariable and multivariable calculus. This includes some material on ordinary differential equations such as those governing populationgrowth models. The course also covers some material on calculusbased probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.
The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.
Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center at MS 3974.
Textbook(s)
S. J. Schreiber, Calculus for the Life Sciences, Wiley.
Outline update: P. Greene, 3/16
Schedule of Lectures
Lecture  Section  Topics 

13  7.1, 7.2  Histograms, PDFs and CDFs 
46  7.3,7.4  Mean and Variance 
79  7.5  Life tables 
1012  CatchUp, Review  
1315  8.1  Multivariate Modeling 
1618  8.2  Matrices and Vectors 
1921  8.3  Eigenvalues and Eigenvectors 
2224  8.4  Systems of Linear Differential Equations 
2527  8.5  Nonlinear systems 
2830  Catch up, Review 
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Introductory number theory course for freshmen and sophomores. Topics include prime number theory and cryptographic applications, factorization theory (in integers and Gaussian integers), Pythagorean triples, Fermat descent (for sums of squares and Fermat quartic), Pell’s equation, and Diophantine approximation. P/NP or letter grading
Textbook(s)
J. Silverman, A Friendly Introduction to Number Theory (4th edition)
Schedule of Lectures
Lecture  Section  Topics 

1  2, 3  Parametrization of Pythagoran numbers; note points where unproved assumptions made. 
2  4, 5.1  Statement of Fermat. 
3  6  Minimal positive elt. of {ax+by} is gcd(a,b). 
4  7  Fundamental Theorem of Arithmetic. 
5  8  Congruences. 
6  9, 10  Fermat’s Little Theorem. 
7  11  Chinese Remainder Theorem 
8  12, 13  Prime numbers. 
9  Review  
10  Midterm #1  
11  14  Mersenne Primes. 
12  16  Powers mod m and squaring 
13  17, 18  Roots mod m. 
14  19  Primality testing. 
15  Powers mod p and primitive roots: show lcm of orders of set of generators = p1; existence of element of order = lcm.  
16  23  Squares mod p. 
17  24  Square roots and quadratic reciprocity: case of 1. 
18  25  Quadratic reciprocity. 
19  26  Primes congruent to 1 mod 4 are squares (descent). 
20  27  Integers that are sums of two squares. 
21  Review  
22  Midterm #2  
23  28  Fermat Quartic descent. 
24  33  Gaussian integers: basic properties. 
25  34  Gaussian integers have unique factorization. 
26  34  Application to representation numbers for sums of two squares. 
27  31  Diophantine Approximation. 
28  32  Pell’s equation. 
29  Review 
General Course Outline
Course Description
(5) Lecture, three hours; discussion, one hour. Preparation: at least three and onehalf years of high school mathematics (including some coordinate geometry and trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 with a grade of C or better. Differential calculus and applications; introduction to integration. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.
In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information.
The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in onevariable calculus than that provided by Math 3.
Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Most students entering the 313233 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.
The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.
Singlevariable calculus is traditionally treated at many universities as a threequarter or twosemester course. Thus Math 31AB does not cover all of the topics included in the traditional singlevariable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center.
Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.
Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.
Textbook(s)
J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO
(a) Limits should be presented very informally with an emphasis on working with their properties: the “Limit Laws”.
(b) Section 6.2 should be restricted to the topic of average value.
Outline update: 3/15 R. Brown
Schedule of Lectures
Lecture  Section  Topics 

1  Introduction  
2  2.34  Limit Laws, Limits and Continuity (a) 
3  2.5  Evaluating Limits Algebraically 
4  2.62.7  Trigonometric Limits, Limits at Infinity 
5  2.8  Intermediate Value Theorem 
6  3.1  Definition of the Derivative 
7  3.2  The Derivative as a Function 
8  3.3  Product and Quotient Rules 
9  3.56  Higher Derivatives, Trig Functions 
10  3.7  The Chain Rule 
11  3.8  Implicit Differentiation 
12  3.9  Related Rates 
13  Midterm 1 (2.38, 3.13,3.57)  
14  4.12  Linear Approximation, Extreme Values 
15  4.2  Extreme Values continued 
16  4.3  Mean Value Theorem 
17  4.4  The Shape of a Graph 
18  4.5  Graph Sketching 
19  4.6  Applied Optimization 
20  4.7, 5.1  Newton’s Method, Area 
21  5.2  The Definite Integral 
22  Midterm 2 (3.89; 4.15)  
23  5.3  The Indefinite Integral 
24  5.4  Fundamental Theorem I 
25  5.5  Fundamental Theorem II 
26  5.7  The Substitution Method 
27  6.12  Areas Between Curves, Average Value (b) 
28  6.3  Volumes of Revolution 
29  6.4  Method of Cylindrical Shells 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour; laboratory, one hour. Preparation: at least three and onehalf years of high school mathematics (including some coordinate geometry and trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 with grade of C or better. Not open for credit to students with credit for course 31A. Intended for students who still need to review precalculus material (laboratory) while starting calculus. Differential calculus and applications; introduction to integration. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.
In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information.
The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in onevariable calculus than that provided by Math 3.
Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Most students entering the 313233 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.
The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.
Singlevariable calculus is traditionally treated at many universities as a threequarter or twosemester course. Thus Math 31AB does not cover all of the topics included in the traditional singlevariable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center.
Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.
Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.
Textbook(s)
J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO
ALEKS by McGrawHill Education, UCLA Calculus Preparation
(a) Limits should be presented very informally with an emphasis on working with their properties: the “Limit Laws”.
(b) Section 6.2 should be restricted to the topic of average value.
Outline update: 3/15 R. Brown
Schedule of Lectures
Lecture  Section  Topics 

1  Introduction  
2  2.34  Limit Laws, Limits and Continuity (a) 
3  2.5  Evaluating Limits Algebraically 
4  2.62.7  Trigonometric Limits, Limits at Infinity 
5  2.8  Intermediate Value Theorem 
6  3.1  Definition of the Derivative 
7  3.2  The Derivative as a Function 
8  3.3  Product and Quotient Rules 
9  3.56  Higher Derivatives, Trig Functions 
10  3.7  The Chain Rule 
11  3.8  Implicit Differentiation 
12  3.9  Related Rates 
13  Midterm 1 (2.38, 3.13,3.57)  
14  4.12  Linear Approximation, Extreme Values 
15  4.2  Extreme Values continued 
16  4.3  Mean Value Theorem 
17  4.4  The Shape of a Graph 
18  4.5  Graph Sketching 
19  4.6  Applied Optimization 
20  4.7, 5.1  Newton’s Method, Area 
21  5.2  The Definite Integral 
22  Midterm 2 (3.89; 4.15)  
23  5.3  The Indefinite Integral 
24  5.4  Fundamental Theorem I 
25  5.5  Fundamental Theorem II 
26  5.7  The Substitution Method 
27  6.12  Areas Between Curves, Average Value (b) 
28  6.3  Volumes of Revolution 
29  6.4  Method of Cylindrical Shells 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 31A with a grade of C or better. Not open for credit to students with credit for course 3B. Transcendental functions; methods and applications of integration; sequences and series. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.
In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information
The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in onevariable calculus than that provided by Math 3.
Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C or better.
Most students entering the 313233 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.
The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.
Singlevariable calculus is traditionally treated at many universities as a threequarter or twosemester course. Thus Math 31AB does not cover all of the topics included in the traditional singlevariable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.
Ample tutoring support is available for students in the course, including the walkin tutoring service of the Student Mathematics Center.
Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.
Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.
Textbook(s)
J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO
(a) The inverse trigonometric functions can be limited to the sine, cosine and tangent and the hyperbolic functions to the sine and cosine.
(b) The amount of time devoted to techniques of integration should be determined by the instructor
(c ) The topic of improper integrals is closely related to that of sequences and series, so it makes sense to postpone it until just before the chapter devoted to those subjects
(d) Although the formal definition of the limit is not included in Math 31A, the corresponding topic in the setting of infinite sequences is appropriate for 31B.
Outline update: 3/15 R. Brown
Schedule of Lectures
Lecture  Section  Topics 

1  Introduction  
2  7.1  Derivative of Exponential Function 
3  7.2  Inverse Functions 
4  7.3  Logarithms and their Derivatives 
5  7.3  Logarithms and their Derivatives (cont’d) 
6  7.7  L’Hopital’s Rule 
7  7.8 9  Inverse Trig and Hyperbolic functions (a) 
8  8.1  Integration by Parts 
9  8.1  Integration by Parts (cont’d) 
10  8.5  Method of Partial Fractions (b) 
11  8.9  Numerical Integration 
12  9.1  Arc Length and Surface Area 
13  9.4  Taylor Polynomials 
14  Midterm 1 (7.1 3; 7.7 9; 8.1; 8.5)  
15  8.7  Improper Integrals � 
16  8.7  Improper Integrals � (cont’d) 
17  11.1  Sequences (d) 
18  11.1  Sequences (d) (cont’d) 
19  11.2  Summing an Infinite Series 
20  11.3  Series with Positive Terms 
21  11.3  Series with Positive Terms (cont’d) 
22  Midterm 2 (8.7; 8.9; 9.1; 9.4; 11.1)  
23  11.4  Absolute and Conditional Convergence 
24  11.5  Ratio and Root Tests 
25  11.6  Power Series 
26  11.6  Power Series (cont’d) 
27  11.7  Taylor Series 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Enforced requisite for course 32AH: course 31A with grade of B or better. Honors sequence parallel to courses 32A. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.
The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.
The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in threedimensional space, and in the case of Stokes’ theorem the domain is a surface in threedimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.
Textbook(s)
G. Folland, Advanced Calculus, Pearson.
Outline update: O.Radko, 7/16
Schedule of Lectures
Lecture  Section  Topics 

1  1.1  Notation, Functions, Vectors 
2  1.1  Dot and Cross Products 
3  1.2  Open and closed subsets of Rn 
4  1.3  Limits of functions, Continuity 
5  2.1  Derivative in one variable as linear approximation 
6  2.1  Vector valued functions and their derivative 
7  2.2  Differentiating functions Rn R: partial derivates 
8  2.2  Partial derivatives, continuously differentiable functions 
9  A.2  Matrices as linear transformations 
10  2.10  Derivatives Rn Rm; Jacobian 
11  2.3/2.10  Chain rule 
12  2.6  Higher differentials, Schwarz Lemma 
13  2.7  Taylor polynomial 
14  1.4  Limits of sequences 
15  1.5  Suprema, Completeness, BolzanoWeierstrass 
16  1.6  Compactness, Extreme value Theorem 
17  2.1  Mean Value Theorem 
18  A.8  Crashcourse Eigenvalues 
19  2.8  Critical points, Hessian 
20  2.9  Lagrange Multipliers 
21  A.4/A.7  Invertible Matrices and Determinants 
22  3.4  Inverse function Theorem 
23  3.1  Implicit function Theorem 
24  3.2  Curves 
25  3.3  Surfaces 
26  3.3  Immersions, Submersions, Submanifolds 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 31A with a grade of C or better. Introduction to differential calculus of several variables, vector field theory. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.
The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.
The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in threedimensional space, and in the case of Stokes’ theorem the domain is a surface in threedimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.
Textbook(s)
J. Rogawski, Calculus: Late Transcendentals Multivariable, Fourth Edition, W. H. Freeman
1) Some problems may refer to polar coordinates. One only need inform the students that x = r cos q and y = r sin q. Polar coordinates are done in detail in 32B in order to help with areas, double integrals, etc.
2) The first two of Kepler’s Laws should be done if at all possible.
3) There are two lectures on limits and continuity, in order to introduce the concepts of open, closed sets, etc.
Outline update: R. Brown, 9/14
Schedule of Lectures
Lecture  Section  Topics 

1  13.1  Vectors in the Plane 
2  13.2  Vectors in Three Dimensions 
3  13.3  Dot Product 
4  13.4  Cross Product 
5  13.5  Planes in ThreeSpace 
6  12.1  Parametric equations 
7  14.1  VectorValued Functions 
8  14.2  Calculus of VectorValued Functions 
9  14.3,4  ArcLength and Speed; Curvature 
10  14.5,6  Motion in ThreeSpace; Planetary Motion 
11  15.1  Functions of Two or More Variables 
12  13.6  A Survey of Quadric Surfaces 
13  15.2  Limits and Continuity 
14  15.2  Limits and Continuity 
15  15.3  Partial Derivatives 
16  15.3  Partial Derivatives 
17  15.4  Differentiability and Tangent Planes 
18  15.4  Differentiability and Tangent Planes 
19  15.5  Gradient and Directional Derivatives 
20  15.5  Gradient and Directional Derivatives 
21  15.6  Chain Rule 
22  15.6  Chain Rule 
23  15.7  Optimization 
24  15.7  Optimization 
25  15.8  Lagrange Multipliers 
26  15.8  Lagrange Multipliers 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: courses 31B & 32A with a grade of C or better. Introduction to integral calculus of several variables, line and surface integrals. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.
The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.
The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in threedimensional space, and in the case of Stokes’ theorem the domain is a surface in threedimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.
Textbook(s)
J. Rogawski, Calculus: Late Transcendentals Multivariable, Fourth Edition, W. H. Freeman
1) The section on polar coordinates should be used to emphasize areas inside polar curves, as a preview of polar double integrals and cylindrical coordinates, and not arcane polar coordinate curves.
2) The sections on Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are extremely important. Time must be left to cover these sections in detail.
Outline update: R. Brown, 8/12
Schedule of Lectures
Lecture  Section  Topics 

1  16.1  Integration in two Variables 
2  16.1  Integration in two Variables 
3  16.2  More General Regions 
4  16.3  Triple Integrals 
5  12.3  Polar Coordinates 
6  16.4  Integration in Polar Coordinates 
7  16.4  Integration in Polar Coordinates 
8  16.5  Applications of Multiple Integrals 
9  16.6  Change of Variables 
10  16.6  Change of Variables 
11  17.1  Vector Fields 
12  17.1  Vector Fields 
13  17.2  Line Integrals 
14  17.2  Line Integrals 
15  17.3  Conservative Vector Fields 
16  17.3  Conservative Vector Fields 
17  17.4  Parametrized Surface 
18  17.4  Parametrized Surface 
19  17.5  Surface Integrals 
20  18.1  Green’s Theorem 
21  18.1  Green’s Theorem 
22  18.2  Strokes’ Theorem 
23  18.2  Stokes’ Theorem 
24  18.2,3  Stokes’ Theorem, Divergence Theorem 
25  18.3  The Divergence Theorem 
26  18.3  The Divergence Theorem 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Enforced requisite for 32BH: courses 31B and 32A, with grades of B or better. Honors sequence parallel to courses 32B. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.
The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.
The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in threedimensional space, and in the case of Stokes’ theorem the domain is a surface in threedimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.
Textbook(s)
G. Folland, Advanced Calculus, Pearson.
Outline update: O.Radko, 7/16
Schedule of Lectures
Lecture  Section  Topics 

1  4.1  Integration on the Line 
2  4.2  Integration in Higher Dimensions 
3  4.2  Integration in Higher Dimensions 
4  4.3  Multiple Integrals and Iterated Integrals 
5  4.3  Multiple Integrals and Iterated Integrals 
6  4.4  Change of Variables for Multiple Integrals 
7  4.4  Change of Variables for Multiple Integrals 
8  4.5  Functions Defined by Integrals 
9  4.8  Lebesgue Measure and the Lebesgue Integral 
10  5.1  Arc Length and Line Integrals 
11  5.1  Arc Length and Line Integrals 
12  5.2  Greens Theorem 
13  5.3  Surface Area and Surface Integrals 
14  5.4  Vector Derivatives 
15  5.5  Divergence Theorem 
16  5.6  Some Applications to Physics 
17  5.6  Some Applications to Physics 
18  5.6  Some Applications to Physics 
19  5.7  Stokes Theorem 
20  5.8  Integrating Vector Derivatives 
21  5.8  Integrating Vector Derivatives 
22  5.9  Higher Dimensions and Differential Forms 
23  5.9  Higher Dimensions and Differential Forms 
24  5.9  Higher Dimensions and Differential Forms 
25  5.9  Higher Dimensions and Differential Forms 
26  5.9  Higher Dimensions and Differential Forms 
Course Description
(Same as C&S Bio M32.) Lecture, three hours; discussion, one hour. Requisite: Life Sciences 30A, Life Sciences 30B. Not open to students with credit for 31A, 31B, 32A, or 32B. Designed for life sciences students who have taken Life Science 30B. Methods and results of single and multivariable calculus essential for quantitative training in biology. Limits, differentiation (single and several variables), optimization, integration and methods of integration, Taylor polynomials and applications to approximation, Taylor and other power series, vector valued functions, gradients, and Lagrange multipliers. P/NP or letter grading.
Course Objectives
This course further develops the principles and computations of calculus so that life sciences students who have completed two quarters of “Mathematics for Life Scientists” (Life Science 30A and Life Sciences 30B) will be able to continue, as needed, with Mathematics 33A (Linear Algebra, Mathematics 33B (Differential Equations), or later upper division courses. In addition to covering central topics of single and multivariate calculus, it aims at developing traditional paper and pencil computational skills. Concrete objectives include: computing limits of functions, computing derivatives, using derivatives in one and several variable settings to solve (constrained) optimization problems, and evaluating and using definite and indefinite integrals.
Textbook
Neuhauser, Claudia and Roper, Marcus. Calculus for Biology and Medicine. 4^{th} ed., Pearson, 2018.
Outline update: W. Conley 10/20
General Course Outline/Schedule of Lectures
Week  Topics 
1  Intro to course. Brief review of limit concept, derivative concept, & limit definition of derivative. More review of limits/derivatives. Review of trig, log, exponential functions. & the product and quotient rules. Review of the Chain Rule. Implicit differentiation and related rates. Applications. More applications… 
2  Intro to singlevariable optimization. Applications. Optimization, critical points, first derivative test. Second derivative test. Applications. 
3  Review of vector basics. Dot product. Orthogonal projections. Intro to optimization with multiple variables. Review of partial derivatives. Directional derivatives. The gradient, and computing directional derivatives. The gradient as a vector field. 
4  Optimization of multivariable functions. The second derivative test. Optimization with a constraint, via Lagrange multipliers. Midterm 1 
5  Applications of multivariable optimization. Review of antiderivatives, indefinite integrals, and the definite integral. Review of the definite integral, area under a curve, and FTC. 
6  Applications of definite and indefinite integrals. Integration formulas for some simple functions. Intro to integration by substitution. Integration by substitution. 
7  Integration by parts. Applications of integration by part s. 
8  Midterm 2 Intro to sequences and series. Convergence. Examples. Taylor series. Ratio test for convergence. Radius of convergence. 
9  Taylor series of some famous functions. Taylor polynomials as approximations. Applications of Taylor polynomials. 
10  Fourier series for periodic functions. Examples. Computing Fourier series coefficients. Relation to linear algebra. Application to oscillatory dynamical systems. Applications of Fourier series. 
Grades:
Scheme 1:
Discussion: 5%
Homework: 15%
Midterm 1: 20%
Midterm 2: 20%
Final exam 40%
Scheme 2:
Discussion: 5%
Homework: 15%
Max. of Midterms 1 & 2: 30%
Final exam 50%
Core Competencies
Students will acquire a strong paper and pencil problemsolving ability of the traditional type with respect to problems of single and multivariable calculus. They will be able to recognize, state and use key theorems of single and multivariable calculus.
Learning outcomes:
Testable outcomes include:
 computing explicit limits and derivatives of functions in one and several variable settings to solve (constrained) optimization problems
 evaluating and using definite and indefinite integrals
 computing Taylor series and solving approximation problems using Taylor polynomials
 computing tangent lines and tangent planes
(Problems posed may involve rational, trigonometric, exponential and logarithm functions in any of the above.)
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 3B, 31B or 32A with a grade of C or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, leastsquares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in ndimensional Euclidean space. Abstract vector spaces are not covered; they are treated in Math 115A.
Students in the course should have covered the following topics in previous high school and college mathematics courses:
 solving linear systems of equations,
 matrices, matrix multiplication,
 twobytwo and threebythree determinants,
 complex numbers,
 complex polynomials, the fundamental theorem of algebra.
This background material is reviewed in the course, though briefly.
Textbook(s)
O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall.
Since the syllabus includes some important material for engineers at the end of the course (Chapter 8), the pacing of lectures is particularly important. Some time can be saved by synopsising the properties of determinants and leaving the details to the students. The students are already familiar with twobytwo and threebythree determinants.
Most of the students are already familiar with matrix multiplication.
The ad hoc definition of “linear transformation” in Section 2.1 should be replaced by the correct definition, which can then be related to the definition given in the textbook.
Chapter 4 and Section 5.5 are generally not covered.
The QR decomposition in Section 5.2 is important for the engineers.
Most students will have seen the polar form of complex numbers given in Section 7.5 (in high school), but most students will not have seen the exponential form (Euler’s formula) in previous courses.
Positivedefinite matrices (Section 8.2) and the singularvalue decomposition (Section 8.3) are very important for the engineers.
Outline update: T. Gamelin, 9/14
Schedule of Lectures
Lecture  Section  Topics 

12  Chapter 1 (1.13)  Systems of linear equations, associated matrix equations, row reduction of a matrix, GaussJordan elimination 
36  Chapter 2 (2.14)  Linear transformations, invertible matrices, matrix algebra 
710  Chapter 3 (3.14)  Subspaces of Rn, linear independence, row space, column space, bases, dimension, kernel and image of linear transformations, ranknullity theorem, coordinates 
1115  Chapter 5 (5.14)  Orthogonality, orthonormal bases, orthogonal projections, orthogonal transformations, orthogonal matrices, GramSchmidt process, QRfactorization, least squares methods 
1619  Chapter 6 (6.13)  Determinants 
2023  Chapter 7 (7.15)  Eigenvalues, eigenvectors, diagonalization of matrices, applications to discrete dynamical systems, 
2426  Chapter 8 (8.13)  Diagonalization of symmetric matrices, applications to quadratic forms, SVD (singularvalue decomposition) 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Enforced requisite: course 3B or 31B or 32A with grade of B or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, leastsquares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. Honors course parallel to course 33A. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.
The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in ndimensional Euclidean space. Abstract vector spaces are not covered; they are treated in Math 115A.
Students in the course should have covered the following topics in previous high school and college mathematics courses:
 solving linear systems of equations,
 matrices, matrix multiplication,
 twobytwo and threebythree determinants,
 complex numbers,
 complex polynomials, the fundamental theorem of algebra.
This background material is reviewed in the course, though briefly.
The topics in linear algebra that are covered in Math 33A include:  systems of linear equations, associated matrix equations,
 row reduction of a matrix,
 linear transformations,
 invertible matrices,
 subspaces, linear independence, bases, dimension,
 row space, column space, ranknullity theorem,
 determinants,
 orthogonality, orthonormal bases,
 orthogonal matrices,
 GramSchmidt process, QR factorization,
 leastsquares approximation, normal equations,
 eigenvalues, eigenvectors, similarity, diagonalization,
 applications to discrete dynamical systems,
 diagonalization of symmetric matrices,
 applications to quadratic forms, singular value decomposition.
Textbook(s)
O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall. Check Schedule of classes for most current textbook.
Since the syllabus includes some important material for engineers at the end of the course (Chapter 8), the pacing of lectures is particularly important. Some time can be saved by synopsising the properties of determinants and leaving the details to the students. The students are already familiar with twobytwo and threebythree determinants.
Most of the students are already familiar with matrix multiplication.
The ad hoc definition of “linear transformation” in Section 2.1 should be replaced by the correct definition, which can then be related to the definition given in the textbook.
Chapter 4 and Section 5.5 are generally not covered.
The QR decomposition in Section 5.2 is important for the engineers.
Most students will have seen the polar form of complex numbers given in Section 7.5 (in high school), but most students will not have seen the exponential form (Euler’s formula) in previous courses.
Positivedefinite matrices (Section 8.2) and the singularvalue decomposition (Section 8.3) are very important for the engineers.
Outline update: T. Gamelin, 3/04
Schedule of Lectures
Lecture  Section  Topics 

12  Chapter 1 (1.13)  Linear systems, GaussJordan elimination 
36  Chapter 2 (2.14)  Linear transformations, inverses, matrix algebra 
710  Chapter 3 (3.14)  Subspaces of Rn, linear independence, bases, dimension, kernel and image of linear transformations, coordinates 
1115  Chapter 5 (5.14)  Orthogonality, orthonormal bases, orthogonal projections, orthogonal transformations, orthogonal matrices, GramSchmidt process, QRfactorization, least squares methods 
1619  Chapter 6 (6.13)  Determinants 
2023  Chapter 7 (7.15)  Eigenvalues, eigenvectors, diagonalization of matrices 
2426  Chapter 8 (8.13)  Symmetric matrices, SVD (singularvalue decomposition) 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 31B with a grade of C or better. Highly recommended: course 33A. Firstorder, linear differential equations; secondorder, linear differential equations with constant coefficients; power series solutions; linear systems. P/NP or letter grading.
Course Information:
In addition, two hour exams should be given. These exams are usually given in the fourth and eighth week; the exact time they are scheduled is up to the instructor. 24 of the 26 lectures are specified.
The course Math 33B has evolved over the years. At one time it was a course in infinite series, including power series solutions of differential equations. In the Fall of 1997 the infinite series course was renumbered as Math 31C, in hopes that students would take it earlier, but by the Fall of 1998 the course was back at the end of the calculus sequence with its original label Math 33B.
In 2004, the courses Math 33A and 33B were reorganized. The differential equations portion of Math 33A was moved to Math 33B, so that Math 33A is now a course in linear algebra and Math 33B is now a course in differential equations. The topics currently treated in Math 33B are as follows:
Introduction to first order differential equations
 second order linear differential equations with constant coefficients
 power series solutions of second order differential equations
 linear systems of differential equations
Textbook(s)
Polking, Differential Equations, 2nd Ed., Prentice Hall.
Footnotes
1. On page 22 of the Polking text the author has a section on a ‘numerical solver”. He writes “We assume that each of our readers has access to a computer.” He also adds We assume that you have access to a solver [computer and software] that will draw direction fields, provide numerical solutions?, and plot solutions.” The author goes into detail on the vibrating spring example, pages 137140. You might wish to put this off until 4.4 when he returns to the topic.
3. Math 33B does not have math 33A, linear algebra, as a prerequisite. This was a concession to the Chemistry Department. You will have to give a short, fast explanation of eigenvalues and eigenvectors.
4. All eiganvalue possibilities are discussed in this section.
Outline update: 9/14
Schedule of Lectures
Lecture  Section  Topics 

1  2.11  Examples, Direction Fields 
2  2.2  Separable equations 
3  2.4  Linear Equations, x’ (t) = a (t) x (t) + f (t) 
4  2.5  Mixing Problems 
5  2.6  Exact Differential Equations 
6  2.6  Continuation of Previous Lecture 
7  2.7  Existence and Uniqueness 
8  2.9  Autonomous Equations and Stability 
9  4.12  Existence and Uniqueness, Linear Dependence, The Wronskian 
10  4.3  Second Order Constant Coefficient Equations 
11  4.3  Continuation 
12  4.4  Harmonic Motion — Unforced 
13  4.5  Undetermined Coefficients 
14  4.6  Variation of Parameters 
15  9.13  Linear Systems with Constant Coefficients 
16  9.24  2 x 2 systems 
17  9.2  Continuation 
18  9.2  Continuation 
19  9.3  Phase Plane Portraits 
20  9.4  The TraceDeterminant Plane 
21  9.5  HigherDimensional Systems 
22  9.5  Continuation 
23  9.6  The Exponential of a Matrix 
24  9.6  Continuation 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: Math 31AB; 32AB; 33A; One of Statistics 1015, Statistics 20; PIC 10A. This course gives an introduction to datadriven mathematical modeling and to combining data analysis with mechanistic modeling of phenomena from various applications. Topics include model formulation, data visualization, nondimensionalization and orderofmagnitude physics, introduction to discrete and continuous dynamical systems, and introduction to discrete and continuous stochastic models. Examples drawn from many fields and practice problems from Mathematical Contest in Modeling. P/NP or Letter grading.
Course Information:
Students will learn the basic principles of mathematical modeling and data visualization. The focus will be on mechanistic models, but in a datadriven and problemdriven way. They will get handson practice with problems from the Mathematical Contest in Modeling, including an indepth exploration through a final project.
The grade will be determined based on homework, quizzes, a midterm, a final project (done in groups, with both written and oral components), and class participation.
Textbook(s)
Required:
(MS) “A Course in Mathematical Modeling”, by Douglas D. Mooney and Randall J. Swift
(Tufte) “The Visual Display of Quantitative Information” (2nd edition), by Edward R. Tufte
Important Supplementary Booklets:
(BFG) “Math Modeling & Getting Started”, by K. M. Bliss, K. R. Fowler, and B. J. Galluzzo (a free booklet from the Society for Industrial and Applied Mathematics)
(BGKL) “Math Modeling: Computing & Communicating”, by K. M. Bliss, B. J. Galluzzo, K. R. Kavanagh, & R. Levy (a free booklet from the Society for Industrial and Applied Mathematics)
Supplementary material through past Mathematical Contest in Modeling questions and handouts on specific topics.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  MS 0, BFG p.144  Introduction and Basic Principles of Modeling 
Week 2  Tufte 13, BGKL 3  Visualization of Data 
Week 3  MS 1  Discrete Dynamical Systems 
Week 4  MS 2  Discrete Stochastic Models 
Week 5  MS 3, BFG: Appendix B  Stages, States, and Classes 
Week 6  MS 5  Continuous Dynamical Systems 
Week 7  BGKL 45, Handouts  Continuous Dynamical Systems (continued) and Related Topics 
Week 8  MS 6  Continuous Stochastic Models 
Week 9  Mathematical Contest in Modeling (MCM): Practice  
Week 10  Mathematical Contest in Modeling: Student Discussions and Projects 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Not open for credit to students with credit for course 180 or 184. Discrete structures commonly used in computer science and mathematics, including sets and relations, permutations and combinations, graphs and trees, induction. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for two midterm exams and review. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction.
Math 61 has two goals. One goal is the introduction of certain basic mathematical concepts, such as equivalence relations, graphs, and trees. The other goal is to introduce nonmathematicians to abstraction and rigor in mathematics. Finite graphs are wellsuited to this purpose. Exercises asking for simple proofs are assigned where appropriate.
Roughly half of the students in Math 61 are engineering students in Computer Science. Of the remaining students, many are in business and economics. Relatively few (about one out of ten) have declared as Mathematics majors.
Math 61 is offered each quarter. Recent enrollment statistics are given in the following table.
Textbook(s)
R. Johnsonbaugh, Discrete Mathematics (8th Edition) , PrenticeHall.
Outline update: I. Neeman 7/12
Schedule of Lectures
Lecture  Section  Topics 

1  2.4  Mathematical induction 
2  1.1, 3.1  Sets, functions 
3  3.2  Sequences and strings 
4  3.3  Relations 
5  3.4, 5  Equivalence relations, matrices of relations 
6  6.1  Basic counting principles 
7  6.2  Permutations and combinations 
8  6.3  Generalized permutations and combinations 
9  6.7  Binomial coefficients 
10  6.8  Pigeonhole principle 
11  7.1  Recurrence relations 
1213  7.2  Solving recurrence relations (including material in exercises 4046) 
14  8.1  Examples of graphs 
15  8.23  Paths and cycles 
16  8.4  Shortestpath algorithm 
17  8.5  Representation of graphs 
18  8.6  Isomorphisms of graphs 
19  8.7  Planar graphs 
20  9.1  Examples of trees 
21  9.2  More trees 
22  9.34  Minimal spanning trees 
23  9.5  Binary trees 
2425  7.3, 9.7  Decision trees, sorting (including merge sort from 7.3) 
26  9.8  Isomorphic trees 
General Course Outline
Course Description
(4) Lecutre, 3 hours; Discussion, 1 hour. Requisites: courses 31A, 31B. Introduction to probability through applications and examples. Topics include laws of large numbers, statistics, chance trees, conditional probability, Bayes? rule, continuous and discrete random variables, jointly distributed random variables, multivariate normal and conditional distributions. In depth discussion of betting schemes in gambling, occurrence of rare events, coincidences and statistical predictions. P/NP or letter grading.
Course Information:
The course introduces a list of standard probabilistic problems and analyzes them in detail within the formalism of probability as a mathematical discipline. At the end of the course, the students will be able to demonstrate their understanding of the foundations and basic facts of probability as a mathematical discipline and apply them to resolve questions with probabilistic content.
Textbook(s)
Tijms, H. Understanding Probability, Chance Rules in Everyday Life, 3rd Edition. Cambridge University Press, 2012
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Ch. 12  Laws of large numbers and simulation 
Week 2  Ch. 34  Probability in everyday life and rare events 
Week 3  Ch. 56  Probability and statistics, chance trees 
Week 4  Ch. 7  Foundations of probability 
Week 5  Ch. 8  Conditional probability and Bayes? Rule 
Week 6  Ch. 9  Discrete random variables 
Week 7  Ch. 10  Continuous random variables 
Week 8  Ch. 11  Jointly distributed random variables 
Week 9  Ch. 12  Multivariate normals 
Week 10  Ch. 13  Conditional distributions 
General Course Outline
Course Description
(3) Seminar, two hours; fieldwork (classroom observation and participation), two hours. Introduce students to K12 mathematics activity in the United States. Cultivate interest in teaching through exploration of the sequences of mathematical content and habits of mind taught in these grades. Analyze sequences of topics in the current California State Standards in Mathematics (CCSSM), the mathematical structures that underlie these sequences and cognitive aspects of learning mathematics. Experience with professional mathematician?s habits of mind outlined in the California Standards for Mathematical Practice (including proof and mathematical modeling) and effective strategies for teaching mathematics to diverse student groups. Fieldwork in local mathematics classrooms arranged by Cal Teach program. P/NP (undergraduates) or S/U (graduates) grading.
Textbook(s)
National Research Counci How Students Learn: Mathematics in the Classroom. Washington, DC: The National Academies Press (https://doi.org/10.17226/11101), 2005.
Other reading materials to be provided
Online Resources:
National Governors Association & Council of Chief State School Officers Common Core State Standards for Mathematics (http://www.corestandards.org/Math/), 2010.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Grades 13: Length (CCSSM 1.MD.2, 2.MD.3, 3.MD.4)  
Week 2  Grades 35: Area & Volume Defined (CCSSM 3.MD.5 – 7, 5.MD.3 – 5) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 1 in the classroom?  
Week 3  Grades 68: Deriving Area and Volume Formulas (CCSSM 6.G., 7.G.4, 8.G.9) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 2 in the classroom?  
Week 4  Grades 912: Areas and Volumes of Irregular Regions and Solids (CCSSM G.GMD.1) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 3 in the classroom?  
Week 5  Grades K2: Decomposing Shapes (CCSSM K.G.6, 1.G.3, 2.G.3) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 4 in the classroom?  
Week 6  Grades 35: Defining Fraction as a Number (CCSSM 3.NF.1 & 2) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 5 in the classroom?  
Week 7  Student Presentation of Performance Tasks Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 6 in the classroom?  
Week 8  Grades 35: Multiplying Fractions (CCSSM 5.NF.4) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 7 in the classroom?  
Week 9  Grades 67: Ratios and Proportional Relationships (CCSSM 6.RP.3, 7.RP.2) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 8 in the classroom?  
Week 10  Grades 812: Linear and Other Functions (CCSSM 6.EE.9, 8.EE.5, 8.F.3, F.IF.1) 
General Course Outline
Course Description
(3) (Formerly numbered Mathematics 71SL.) Seminar, two hours; fieldwork (classroom observation and participation), two hours. Facilitate development of professional mathematical and pedagogical understandings required to teach California?s K5 mathematics curriculum. Exploration of K5 mathematics, practice effective teaching strategies for all learners, and discuss current research and standards in math education. Fieldwork in local mathematics classrooms (observation and presenting lesson plan) arranged by Cal Teach program. P/NP (undergraduates) or S/U (graduates) grading.
Textbook(s)
Berlinghoff & Gouvea Math Through The Ages: A Gentle History for Teachers and Others. Oxton Publishers & MAA, 2015.
Other reading materials to be provided
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Grades K2: Connecting Counting to Cardinality (CCSSM K.CC.4)  
Week 2  Grades K2: The Base Ten System (CCSSM 1.NBT.2, 2.NBT.1) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 1 in the classroom?  
Week 3  Grades K2: The Addition & Subtraction Algorithm (CCSSM 2.NBT.9, 3.NBT.2) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 2 in the classroom?  
Week 4  Grades 35: Adding and Subtracting Fractions (CCSSM 4.NF.3) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 3 in the classroom?  
Week 5  Grades 35: Relating Area to Multiplication (CCSSM 3.MD.57) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 4 in the classroom?  
Week 6  Grades 35: The Multiplication Algorithm for Whole Numbers (CCSSM 4.NBT.5, 5.NBT.5) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 5 in the classroom?  
Week 7  Grades 35: Student Presentations of Lesson Plans Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 6 in the classroom?  
Week 8  Grades 35: Multiplying Fractions (CCSSM 5.NF.4) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 7 in the classroom?  
Week 9  Grades 35: Dividing Fractions (CCSSM 5.NF.7) Fieldwork Prompt: In what ways (if any) did you observe students engaging in CCSS SMP 8 in the classroom?  
Week 10  Grades 35: Decimals & Decimal Operations (CCSSM 4.NF.57, 5.NBT.7, 6.NS.3) 
General Course Outline
Course Description
(1) Tutorial, three hours. Limited to students in College Honors Program. Designed as adjunct to lowerdivision lecture course. Individual study with lecture course instructor to explore topics in greater depth through supplemental readings, papers, or other activities. May be repeated for maximum of 4 units. Individual honors contract required. Honors content noted on transcript. Letter grading.
General Course Outline
Course Description
(4) (Formerly numbered 192.) Lecture, three hours. Requisite: course 31B with grade of C or better. Problemsolving techniques and mathematical topics useful as preparation for Putnam Examination and similar competitions. Continued fractions, inequalities, modular arithmetic, closed form evaluation of sums and products, problems in geometry, rational functions and polynomials, other nonroutine problems. Participants expected to take Putnam Examination. P/NP grading.
Math 100 is a course in problem solving. The problems are more varied and unexpected than in a typical undergraduate mathematics course. Often an original or imaginative step is required. Some variations of topics from year to year are expected. Topics may include: explicit summations of series, spherical trigonometry, advanced Euclidean geometry, elementary number theory, combinatorial problems, inequalities, continued fractions. There is a lot of classroom discussion. Homework is assigned regularly and makes a large contribution to the course grade. One threehour final is given.
Textbook(s)
ProblemSolving Through Problems by Loren C. Larson
Updated: 10/14 C. Manolescu
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Induction. Generalized induction. The pigeonhole principle.  
Week 2  Inequalities (AMGM, weighted AMGM, CauchySchwartz, Jensen, Holder, Minkowski).  
Week 3  Number theory. Modular arithmetic. Fermat’s little theorem, Euler’s theorem. The Chinese remainder theorem.  
Week 4  Algebra. Polynomials (factorization over different fields, Viete’s relations). Some abstract algebra (groups, rings).  
Week 5  Summation of series. Geometric progressions. Telescoping series and products. Taylor series.  
Week 6  Combinatorics. Binomial coefficients and combinatorial identities.  
Week 7  Recurent sequences (linear recurrences, generating functions). Discrete and continuous probability.  
Week 8  Geometry problems. Elementary methods. Analytic geometry, conics. Vectors and complex numbers.  
Week 9  Differential calculus. The extreme value theorem and the mean value theorem. Functional equations.  
Week 10  Integral calculus. Approximating integrals by Riemann sums. Integral functional equations. 
General Course Outline
Course Description
(4) Lecture, three hours. Prerequisite: Math 100 or significant experience with mathematical competitions. Advanced problem solving techniques and mathematical topics useful as preparation for Putnam Competition. Problems in abstract algebra, linear algebra, number theory, combinatorics, probability, real and complex analysis, differential equations, Fourier analysis. Regular practice tests given, similar in difficulty to the Putnam Competition. Enrollment is by permission of the instructor, based on a selection test or past Putnam results. May be repeated for maximum of 12 units. P/NP or letter grading.
Textbook(s)
R. Gelca & T. Andreescu. Putnam and Beynd, Springer Verlag
Updated 10/14: C. Manolescu
Schedule of Lectures
Lecture  Section  Topics 

1  Introduction to the Putnam Mathematical Competition. Selected test problems from previous years.  
2  Methods of proof: contradiction, induction, the pigeonhole principle, invariants.  
3  Algebra. Inequalities and identities. Real and complex polynomials.  
4  Linear Algebra. Eigenvalues, the CayleyHamilton Theorem. Abstract algebra (groups, rings).  
5  Geometry and trigonometry. Using vectors and complex numbers to solve gemetry problems.  
6  Number theory. Euler’s theorem. Diophantine equations.  
7  Combinatorics and combinatorial geometry. Generating functions. Probability.  
8  Real analysis problems. Sequences, series, continuity, derivatives and integrals. Convexity.  
9  Multivariable differential and integral calculus. Solving integrals using complex analysis.  
10  Differential equations and Fourier analysis. 
General Course Outline
Course Description
(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 33A, 33B. Course 103A is enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.
General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.
Math 103A: General Course Outline 
Assignments and Grading

Summary of Course Requirements 
Weekly Topics (Emphasis on the Teacher in the Classroom) Session 1: General Overview
Session 2: The Classroom Environment and Housekeeping
Session 3: Classroom Management
Session 4: Methods of Instruction
Session 5: Teacher Questioning; Wait Time
Session 6: (TeacherStudent Interaction)
Session 7: Teacher Content Knowledge and Final Reflection

Observation Protocol

Observation Reflection Guidelines

Reading Reflection and Critical Analysis Expectations

Problem of the Week WriteUp

MiniPortfolio Guidelines

CommentsOutline update: S. Hakansson 09/08 
For more information, please contact Student Services, ugrad@math.ucla.edu. 
General Course Outline
Course Description
(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 33A, 33B. Course 103A is enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.
General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.
Math 103B: General Course Outline 
Assignments and Grading

Summary of Course Requirements 
Weekly Topics (Emphasis on the Teacher in the Classroom) Session 1: General Overview
Session 2: Student Motivation
Session 3: Student Understanding
Session 4: Students? Mathematical Literacy
Session 5: Student Engagement/Student Expectations
Session 6: Student/Student and Student/Teacher Interaction and Final Reflection
Joint Math/Ed Breakfast and Mathematics for Teaching Conference: Winter Quarter 
Observation Protocol

Observation Reflection Guidelines

Reading Reflection and Critical Analysis Expectations

Problem of the Week WriteUp

MiniPortfolio Guidelines

CommentsOutline update: S. Hakansson 09/08 
For more information, please contact Student Services, ugrad@math.ucla.edu. 
General Course Outline
Course Description
(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 32B, 33A, 33B. Course 103A is an enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.
General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.
Math 103C: General Course Outline 
Assignments and Grading

Summary of Course Requirements 
Weekly Topics (Emphasis on the Teacher in the Classroom) Session 1: Overview of Assessment
Session 2: What Is Assessment?
Session 3: Formative Assessment
Session 4: High Stakes Tests?for the Student
Session 5: High Stakes Tests?for the School
Session 6: Summative Assessment
Session 7: Final Reflection

Observation Protocol

Observation Reflection Guidelines

Reading Reflection and Critical Analysis Expectations

Problem of the Week WriteUp

MiniPortfolio Guidelines

CommentsOutline update: S. Hakansson 09/08 
For more information, please contact Student Services, ugrad@math.ucla.edu. 
General Course Outline
Course Description
(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 110A (or 117), 120A (or 123), and 131A, with grades of C or better. Course 105A is requisite to 105B, which is requisite to 105C. Mathematical knowledge and researchbased pedagogy needed for teaching key geometry topics in secondary school, including axiomatic systems, measure, and geometric transformations. Introduction to professional standards and current research for teaching secondary school mathematics. Letter grading.
Description
Math 105A is a teamtaught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. This course also aims to teach you new mathematics content using various researchbased instructional strategies. It emphasizes problem solving and student presentation of solutions.
Math 105A also aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. This course includes readings of current math education research as well as state and national content standards for the teaching of secondary mathematics. It also requires observation in local secondary schools.
General Information
 senior mathematics majors with demonstrated success in the abovementioned upper division mathematics coursework and demonstrated interest in mathematics teaching
 graduate students in the GSE&IS Teacher Education Program
Required Texts/Supplies:
Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ
The Mathematics Framework for California Public Schools (available at http://www.cde.ca.gov/ci/ma/cf/documents/mathfrwkcomplete.pdf)
The National Council of Teachers of Mathematics Principles and Standards for School Mathematics (sign up for online access to this document at http://standardstrial.nctm.org/triallogin.asp)
J.D. Bransford, A.L. Brown, R.R. Cocking, Eds., How People Learn: Brain, Mind, Experience, and School, Expanded Edition. (2000) National Research Council, Washington, D.C.
J. Stigler, J. Hiebert, The Teaching Gap (1999) The Free Press, NY
TI 84 Plus graphing calculator (distributed by TI at a required training on October 28th)
Outline update: B. Rothschild, H. Dallas 09/08
Instructor Information:  
Bruce Rothschild Office: MS 6175 310) [82]53174  Heather Dallas Office: MS 2341 (310) [82]51702 
Meeting Information:
Mondays, 4 – 8 PM, MS 6221. Usually there will be a 20 minute break for nourishment.
Problems of the Week and Homework Exercises: 25%
Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week. When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.
Quizzes: 10%
A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.
Reading Summaries: 10%
Readings of math education research or professional standards will be assigned regularly, with brief summaries and reflections due via online submission.
Course and Lesson Design: 10%
Students will work in groups to write a course and unit plan for an Algebra or Prealgebra course which is in accordance with the California Framework and the NCTM Standards and Principles.
Secondary Classroom Observations: 10%
Students will observe for 5 hours in an assigned secondary classroom. Observation notes will be taken. Students will write a Standards in Practice paper identifying the California Standards and NCTM Principles and Standards covered in the observed classes.
Final: 25%
A final exam will be given in the first two quarters of the sequence and a final portfolio will be due in the third quarter of the sequence. Collection of the elements for the final portfolio will be incorporated throughout the three quarter 105 sequence, including work on a paper tracing the development of a mathematical idea through the secondary and undergraduate curriculum.
Participation: 10%
Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students, and engagement in and completion of the work of the model lessons plans will be assessed.
Please note the following policies:
No late assignments will be accepted.
For each of the above content pieces, the teaching, curriculum, and assessment of the content at the secondary level are introduced and analyzed in the context of current research and recommendations.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Intro to Problem Analysis; intro to definition  
Week 2  Number: integers ? history and algebraic structure; comparing methods for teaching (a)(b) = +ab  
Week 3  Number: rationals ? definition and algebraic structure; comparing models for rational division  
Week 4  Number: reals ? decimals, irrationals, countability; method for teaching rational operations  
Week 5  Attendance at all day Texas Instruments PTE  
Week 6  Number: complex ? polar, rectangular, and exponential representations and their advantages, De Moivre?s Theorem; model lesson to introduce i  
Week 7  Number: complex ? stereographic projection; model lesson on modeling probabilistic data with linear functions  
Week 8  Function: definitions; model lesson on modeling probabilistic data with exponential functions  
Week 9  Joint Meeting with the science team: modeling one dimensional motion with linear and quadratic functions  
Week 10  Function: model lesson on maximum box volume problem; review for final 
General Course Outline
Course Description
(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 105A, 110A (or 117), 120A (or 123), and 131A, with grades of C or better. Mathematical knowledge and researchbased pedagogy needed for teaching key polynomial, rational, and transcendental functions and related equations in secondary school; professional standards and current research for teaching secondary school mathematics. Letter grading.
Description
Math 105B is the second quarter in a teamtaught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. It also aims to teach you new mathematics content using various researchbased instructional strategies and to emphasize problem solving and student presentation of solutions.
Math 105B aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. The course includes readings and discussion of current math education research and requires observation in local secondary schools.
General Information
 senior mathematics majors with demonstrated success in the abovementioned upper division mathematics coursework and demonstrated interest in mathematics teaching
 graduate students in the GSE&IS Teacher Education Program
Required Texts/Supplies:
Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ
J.D. Bransford, A.L. Brown, R.R. Cocking, Eds., How People Learn: Brain, Mind, Experience, and School, Expanded Edition. (2000) National Research Council, Washington, D.C.
J. Stigler, J. Hiebert, The Teaching Gap (1999) The Free Press, NY
TI 84 Plus graphing calculator
Outline update: B. Rothschild, H. Dallas 09/08
Instructor Information:  
Bruce Rothschild Office: MS 6175 310) [82]53174  Heather Dallas Office: MS 2341 (310) [82]51702 
Meeting Information:
Mondays, 4 – 8 PM, MS 6221. Usually there will be a 20 minute break for nourishment.
Problems of the Week and Homework Exercises: 25%
Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week. When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.
Quizzes: 10%
A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.
Reading Summaries: 10%
Readings of math education research will be assigned regularly, with brief summaries and reflections due via online submission.
Course and Lesson Design: 10%
Students will work in groups to write two lesson plans employing methods taught in the course. After rounds of peer and instructor edits, groups will revise and submit final drafts.
Secondary Classroom Observations: 10%
Students will observe for 5 hours in an assigned secondary classroom. Observation notes will be taken. Students will choose one student to focus on, ask the students to complete a written response problem and subsequently interview them. Students will write a short paper analyzing the results of the interview.
Final: 25%
A final exam will be given in the first two quarters of the sequence and a final portfolio will be due in the third quarter of the sequence. Collection of the elements for the final portfolio will be incorporated throughout the three quarter 105 sequence, including work on a paper tracing the development of a mathematical idea through the secondary and undergraduate curriculum. A number of the portfolio components will be due at the end of the second quarter.
Participation: 10%
Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students (at least twice in the quarter), and engagement in and completion of the work of the model lessons will be assessed.
Please note the following policies:
No late assignments will be accepted.
A student who misses a final exam may receive an incomplete grade in the course providing the student (i) has completed all other grade components at a passing level, (ii) has an ironclad excuse (such as a medical emergency), and (iii), if possible, contacts one of the instructors on or before the day of the final exam to arrange a meeting.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Function: rational functions; def. of asymptotes; formative assessment in the classroom  
Week 2  Equation: preservation of solution sets; comparing strategies for teaching solving linear equations  
Week 3  Equation: preservation of solution sets; comparing strategies for teaching binomial multiplication  
Week 4  Equation: comparing methods for teaching factoring; the quadratic formula; solving the cubic  
Week 5  Axiomatic Systems: intro to Euclid; a model secondary lesson on developing the concept of axiom  
Week 6  Axiomatic Systems: a model secondary lesson on the triangle sum theorem in spherical geometry  
Week 7  Axiomatic Systems: the triangle sum theorem in the hyperbolic geometry  
Week 8  Measure: definition of area; evaluating student work on intro to integral project; model lesson to develop elementary polygon areas  
Week 9  Attendance at day long UCLA Mathematics and Teaching Conference  
Week 10  Attendance at annual UCLA California Math Teacher Program Reunion Dinner 
General Course Outline
Course Description
(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 105A, 105B, 110A (or 117), 120A (or 123), and 131A, with grades of C or better. Mathematical knowledge and researchbased pedagogy needed for teaching key analysis, probability, and statistics topics in secondary school; professional standards and current research for teaching secondary school mathematics. Letter grading.
Description
Math 105C is the third quarter in a teamtaught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. It also aims to teach you new mathematics content using various researchbased instructional strategies and to emphasize problem solving and student presentation of solutions.
Math 105C aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. The course includes readings and discussion of current math education research and requires observation in local secondary schools.
In Math 105C, students will complete the following performance components:
 presentation of model lessons both in class and in a secondary schoolroom
 presentation of a paper which traces a mathematical topic through the secondary and undergraduate curricula
General Information
 senior mathematics majors with demonstrated success in the abovementioned upper division mathematics coursework and demonstrated interest in mathematics teaching
 graduate students in the GSE&IS Teacher Education Program
Required Texts/Supplies:
Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ
J. Stigler, J. Hiebert, The Learning Gap (1999) The Free Press, NY
TI 84 Plus graphing calculator
Outline update: B. Rothschild, H. Dallas 09/08
Instructor Information:  
Bruce Rothschild Office: MS 6175 310) [82]53174  Heather Dallas Office: MS 2341 (310) [82]51702 
Meeting Information:
Mondays, 4:00 – 5:50 PM, MS 6221.
Tuesdays, 4:00 – 5:50, MS 6201
Problems of the Week and Homework Exercises: 20%
Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week. When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.
Quizzes: 10%
A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.
Reading Summaries: 10%
Readings of math education research will be assigned regularly, with brief summaries and reflections due via online submission.
Final portfolio: 50%
A portfolio consisting of:
 Two lesson plans (developed in Winter quarter and improved upon during spring quarter) along with analysis of video of one of these lessons in a secondary classroom (15%)
 Paper tracing a mathematical topic through the secondary and undergraduate curricula (15%)
 Exemplar Work including one POW, one Reading Summary and Reflection, the Winter quarter Student Interview Project, the Fall quarter Observation paper analyzing the CA and NCTM Standards addresses in secondary classrooms, and two class activities. A short reflection on each piece of exemplar work will be included in this portion of the portfolio. (10%)
 Presentation to the class of the analysis of the video of the lesson and subsequent improvements to the lesson (5%)
 Presentation to the class of the paper tracing a mathematical topic through the secondary and undergraduate curricula (5%)
Participation: 10%
Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students (at least twice in the quarter), and engagement in and completion of the work of the model lessons will be assessed.
——————————————————————————————————————
Please note the following policies:
No late assignments will be accepted.
A student who misses their final presentations may receive an incomplete grade in the course providing the student (i) has completed all other grade components at a passing level, (ii) has an ironclad excuse (such as a medical emergency), and (iii), if possible, contacts one of the instructors on or before the day of the presentation to arrange a meeting.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  More on Measure: Area: Pythagorean Theorem. Measure: Volume  
Week 2  Student Presentations of Lesson Plan  
Week 3  Measure: Volume  
Week 4  Student Presentations of Lesson Plan. Transformations: Symmetries.  
Week 5  Transformations: Congruence and Similarity. Transformations: in the Cartesian plane.  
Week 6  Trigonometry: Circular functions, similarity. Trigonometry and complex numbers.  
Week 7  Probability: finite. Probability: geometric.  
Week 8  Statistics  
Week 9  Student Presentations of videotaped teaching  
Week 10  Student Presentations of math paper 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, 32A. Roots of modern mathematics in ancient Babylonia and Greece, including place value number systems and proof. Development of algebra through Middle Ages to Fermat and Abel, invention of analytic geometry and calculus. Selected topics. P/NP or letter grading.
General Information. Math 106 focuses on the development of mathematics and its role in society through the ages. The presentation of topics in the course varies according to the instructor. However, there are four major topic areas that form the core of the course.
1. The history of numeral systems through various early civilizations, and the development of placevalue systems of numeration (the sexagesimal system of the Babylonians, and our own HinduArabic system).
2. The origins and evolution of the axiomatic method and proof in mathematics, beginning with the Greeks (Thales, Eudoxus, Euclid), with major advances in the nineteenth century when calculus was placed on a rigorous footing through the efforts of Cauchy, Weierstrass, and others.
3. The evolution of symbolic algebra, which includes solution of equations and the work of Diophantus, Cardano, Viete, and Descartes (who gave us the unknown quantity “x”).
4. The development of the calculus, which demonstrated its power by explaining the motion of the planets.
Math 106 is particularly recommended for students who are planning to teach in middle school and high school, since many of the topics treated in the course are directly related to the mathematics taught in the schools.
Textbook(s)
Stillwell, J., Mathematics and its History, 3rd Ed., Springer.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. Honors sequence parallel to courses 110A. P/NP or letter grading.
Course Information:
The following schedule anticipates 24 days of instruction, with 2 holidays and 4 days for exams and reviews. If there is extra time, one could do section 6.3 – the structure of R/I when I is prime or maximal and/or section 4.6 – irreducibility in R[x] or C[x].
Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.
An honors course sequence 110AH110BH runs parallel to 110A110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.
Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.
Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.
Students who take 110AH but not 110BH can take 110A or 117.
Math 110C, offered in the Spring, is designed for students completing either the 110A110B or the 110AH110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.
Textbook(s)
R. Elman, Lectures on Abstract Algebra
Book is Subject to Change Without Notice
Outline update: Gieseker, D. 12/15
Schedule of Lectures
Lecture  Section  Topics 

1  2,3, 4  The Integers: Well ordering and greatest common divisors. 
2  5,6  Equivalence relations, modular arithmetic 
3  8,9  Groups: Definitions and example 
4  10, 11  Cosets and homomorphisms 
5  12, 13  Isomorphism Theorems. 
6  14, 16  Finite abelian groups, Series 
7  18  Group actions: orbit decomposition theorem. 
8  20  Examples of Group actions 
9  21  Sylow theorems. Application of Sylow theorems 
10  22  Symmetric and Alternating groups 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. P/NP or letter grading.
Course Information:
The following schedule anticipates 24 days of instruction, with 2 holidays and 4 days for exams and reviews. If there is extra time, one could do section 6.3 – the structure of R/I when I is prime or maximal and/or section 4.6 – irreducibility in R[x] or C[x].
Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.
An honors course sequence 110AH110BH runs parallel to 110A110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.
Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.
Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.
Students who take 110AH but not 110BH can take 110A or 117.
Math 110C, offered in the Spring, is designed for students completing either the 110A110B or the 110AH110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.
Textbook(s)
Hungerford, T.,Abstract Algebra, 3rd Ed., Brooks Col.
Outline update: 4/98
Schedule of Lectures
Lecture  Section  Topics 

1  220  Division Algorithm, divisibility, primes, and unique factorization. 
2  2440  Congruence and congruence classes, modular arithmetic, Z/pZ when p is a prime. 
3  4262  Definition and examples of rings, basic properties. 
4  6679  Isomorphims and homomorphism of rings. Review and first midterm. [Note: The book does isomorphism first then homomorphism. The order should probably be inverted.] 
5  8092  Polynomials and the Division Algorithm, divisibility in F[x], irreducibles, and unique factorization. 
6  100115  Polynomial functions, roots, and reducibility, irreducibility in Q[x]. 
7  119123  Review, second midterm. Congruence in F[x] and congruence classes. 
8  123133  Congruence class arithmetic, the structure of F[x]/(p(x)) when p(x) is irreducible. 
9, 10  134153  Ideals and congruence, quotient rings and homomorphisms. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 110A or 117. Groups, structure of finite groups. P/NP or letter grading.
Course Information:
The course should cover essentially the material between pages 160 and 280 (excluding the section on the simplicity of the appropriate alternating groups; one can come back to this if there is enough time). If there is not enough time, the material at the beginning is more important than the material at the end.
Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.
An honors course sequence 110AH110BH runs parallel to 110A110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.
Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.
Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.
Students who take 110AH but not 110BH can take 110A or 117.
Math 110C, offered in the Spring, is designed for students completing either the 110A110B or the 110AH110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.
Textbook(s)
Hungerford, T., Abstract Algebra, 3rd Ed., Brooks Col.
Outline update: 4/98
Schedule of Lectures
Lecture  Section  Topics 

1  160180  Definition of groups, basic properties. 
2  181198  Subgroups, isomorphism, and homomorphism. 
3  199216  Congruence and Lagrange’s Theorem, normal subgroups. 
4  216222  Quotient groups, review, first midterm. 
5  222238  Quotient groups and homomorphism, symmetric and alternating groups. 
6  244261  Direct products, finite abelian groups. 
7  262265  The Sylow Theorems, review, second midterm. 
8  267273  Conjugacy and proof of the Sylow Theorems. 
9, 10  275283  The structure of finite groups, groups of small order. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 110A or 117. Groups, structure of finite groups. P/NP or letter grading.
Course Information:
The course should cover essentially the material between pages 160 and 280 (excluding the section on the simplicity of the appropriate alternating groups; one can come back to this if there is enough time). If there is not enough time, the material at the beginning is more important than the material at the end.
Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.
An honors course sequence 110AH110BH runs parallel to 110A110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.
Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.
Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.
Students who take 110AH but not 110BH can take 110A or 117.
Math 110C, offered in the Spring, is designed for students completing either the 110A110B or the 110AH110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.
Textbook(s)
Dummit and Foote, Abstract Algebra, 3rd Ed., Wiley & Sons.
Book is Subject to Change Without Notice
Outline update: R. Elman 9/14
Schedule of Lectures
Lecture  Section  Topics 

1  Definition of groups, basic properties.  
2  Subgroups, isomorphism, and homomorphism.  
3  Congruence and Lagrange’s Theorem, normal subgroups.  
4  Quotient groups, review, first midterm.  
5  Quotient groups and homomorphism, symmetric and alternating groups.  
6  Direct products, finite abelian groups.  
7  The Sylow Theorems, review, second midterm.  
8  Conjugacy and proof of the Sylow Theorems.  
9, 10  The structure of finite groups, groups of small order. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 110A, 110B. Field extensions, Galois theory, applications to geometric constructions, and solvability by radicals.
General Information. Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.
An honors course sequence 110AH110BH runs parallel to 110A110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.
Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.
Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.
Students who take 110AH but not 110BH can take 110A or 117.
Math 110C, offered in the Spring, is designed for students completing either the 110A110B or the 110AH110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: courses 110A. Algebraic number theory (including prime ideal theory), cyclotomic fields and reciprocity laws, Diophantine equations (especially quadratic forms, elliptic curves), equations over finite fields, topics in theory of primes, including prime number theorem and Dirichlet’s theorem. P/NP or letter grading.
General Information. Number theory is among the oldest and broadest branches of mathematics. It has roots going back to ancient babylonic cuneiform tablets, and it is the subject of several books in Euclid’s Elements. Number theory has played an important role in the development of mathematics. Today number theory cuts across virtually every field of contemporary mathematics.
The most important mathematical event of the past decade has been the resolution of a famous problem that had been around since Fermat stated that a certain Diophantine equation (Fermat’s equation, x^n+y^n=z^n for n larger that 2) does not have any positive integer solutions. The assertion defied numerous proof attempts over a period of 400 years, until recently it was proved as a result of work of Andrew Wiles and other mathematicians, using many of the modern techniques of number theory that have been developed over the past 30 years.
Perhaps the most famous remaining open problem in mathematics is the Riemann hypothesis on the location of the zeros of a specific meromorphic function, the Riemann zeta function. The location of the zeros has consequences for the asymptotic distribution of prime numbers.
Prime numbers are of great concern in connection with mathematical cryptography, entering into the construction of public key encryption codes. This illustrates how number theory ties in with various areas, ranging in this case from complex analysis to areas of current business and governmental security interest.
Because number theory is so vast, there is no one course that could serve as a good introduction to the entire field. Several possibilities for class syllabi are given, each of which focuses on a different emphasis. It may be that the course instructor will follow yet a different path.
General Course Outline
Course Description
(Formerly numbered 114A). Lecture, three hours; discussion, one hour. Requisite: course 110A or 131A or Philosophy 135. Effectively calculable, Turing computable, and recursive functions; Church/Turing thesis. Normal form theorem; universal functions; unsolvability and undecidability results. Recursive and recursively enumerable sets; relative recursiveness, polynomialtime computability. Arithmetical hierarchy. P/NP or letter grading.
General Information. If a function can be precisely defined, does that mean we can write a computer program for it? Math 114C looks at Turing machines and other models for making the concept of effective computability into genuine mathematics. The unsolvability of the halting problem demonstrates the existence of purely theoretical barriers to computability. There are decidable sets, effectively enumerable sets, and others.
Computability theory originated in groundbreaking work by Alonzo Church, Stephen Kleene, Emil Post, Alan Turing, and others, beginning in 1936. The topic is relevant to pure mathematics, theoretical computer science, and the philosophy of mathematics.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 110A or 131A or Philosophy 135. Introduction to mathematical logic, aiming primarily at completeness and incompleteness theorems of Godel. Propositional and predicate logic; syntax and semantics; formal deduction; completeness, compactness, and Lowenheim/Skolem theorems. Formal number theory: nonstandard models; Godel incompleteness theorem. P/NP or letter grading.
General Course Outline
Course Description
(Formerly numbered M112.) (Same as Philosophy M134.) Lecture, three hours; discussion, one hour. Prerequisite: course 110A or 131A or Philosophy 135. Axiomatic set theory as framework for mathematical concepts; relations and functions, numbers, cardinality, axiom of choice, transfinite numbers. P/NP or letter grading.
General Information Math 114S covers the basic facts about abstract sets, including the axiom of choice, transfinite recursion, and cardinal and ordinal arithmetic. It also makes a serious effort to explain how axiomatic set theory can be viewed as a “foundation of mathematics” — and, in particular, what this means.
Math 114S is especially useful for:
Undergraduate students who are preparing for graduate study in pure mathematics and graduate students in mathematics who have not had an opportunity to learn set theory in their undergraduate work. Real analysis, in particular, looks a lot more real if you know cardinal arithmetic and understand the meaning and uses of the axiom of choice.
Undergraduate students in mathematics or computer science who are preparing for graduate study in theoretical computer science, and CS graduate students who are veering towards theory and need to understand the mathematical justification of fixpoint theorems and the like.
Philosophy students with an interest in the philosophy of mathematics and a good mathematical background.
There is a strong tradition of research in logic — especially set theory — at UCLA, and both the Mathematics and Philosophy Departments offer a rich graduate program of study in the field.
Textbook(s)
Moschovakis, Y., Notes on Set Theory, 2nd Ed., Springer.
General Course Outline
Course Description
(5) Lecture, three hours; discussion, two hours. Requisite: course 33A. Techniques of proof, abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory. P/NP or letter grading.
Math 115A is a core mathematics course required of all the various mathematics majors. The course material can be regarded as an elaboration of the linear algebra already covered in Math 33A. However, the level of abstraction and the emphasis on proof technique make this a difficult course for many students. Successful students emerge from the experience not only with a better understanding of linear algebra, but also with a higher level of mathematical maturity, better equipped to deal with abstract concepts.
The material covered in Math 115A includes linear independence, bases, orthogonality, the GramSchmidt process, linear transformations, eigenvalues and eigenvectors, and diagonalization of matrices. These topics are all covered in Math 33A though only in the context of Euclidean space. Topics in Math 115A that go beyond Math 33A include inner product spaces, adjoint transformations, and the spectral decomposition theorem for selfadjoint operators.
Three or four sections of Math 115A are offered each term. Also, an honors version Math 115AH runs parallel to Math 115A in some quarters. The content of Math 115AH is as follows:
Vector spaces, subspaces, basis and dimension, linear transformations and matrices, rank and nullity, change of basis and similarity of matrices, inner product spaces, orthogonality and, orthonormality, GramSchmidt process, adjoints of linear transformations and dual spaces, quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of hermitian and symmetric matrices, eigenvectors and eigenvalues, and their computation, exponentiation of matrices and application to differential equations, least squares problems, trace, determinant, canonical forms. Systems of linear equations: solvability criteria, Gaussian elimination, rowreduced form, LU decomposition.
Textbook(s)
S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.
Outline Updated: June 2005
Schedule of Lectures
Lecture  Section  Topics 

1  1.2  Vector Spaces over a Field 
2  1.3  Subspaces 
3  1.4, 1.5  Linear Combinations and Systems of Linear Equations; Linear Dependence and Linear Independence 
4  1.5, 1.6  Linear Dependence and Linear Independence; Bases and Dimensions 
5  1.6  Bases and Dimensions 
6  1.6  Bases and Dimensions 
7  2.1  Linear Transformations, Null Spaces, and Ranges 
8  2.1  Linear Transformations, Null Spaces, and Ranges 
9  2.1, 2.2  Linear Transformations, Null Spaces, and Ranges; The Matrix Representation of a Linear Transformation 
10  .  Midterm #1 
11  2.2  The Matrix Representation of a Linear Transformation 
12  2.3  Composition of Linear Transformations and Matrix Multiplication 
13  2.4  Invertibility and Isomorphisms 
14  2.4, 2.5  Invertibility and Isomorphisms; The Change of Coordinate Matrix 
15  2.5  The Change of Coordinate Matrix 
16  4.4  Summary – Important Facts about Determinants 
17  5.1  Eigenvalues and Eigenvectors 
18  5.1  Eigenvalues and Eigenvectors 
19  5.2  Diagonalizability 
20  5.2  Diagonalizability 
21  5.2  Diagonalizability 
22  .  Midterm #2 
23  6.1  Inner Products and Norms 
24  6.1, 6.2  Inner Products and Norms; The GramSchmidt Orthogonalization Process and Orthogonal Complements 
25  6.2  The GramSchmidt Orthogonalization Process and Orthogonal Complements 
26  6.3  The Adjoint of a Linear Operator 
27  6.4  Normal and SelfAdjoint Operators 
28  6.4  Normal and SelfAdjoint Operators 
29  .  Catchup, Review 
General Course Outline
Course Description
(5) Lecture, three hours; discussion, two hours. Requisite: course 33A with grade of B or better. Techniques of proof, abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory. Honors course parallel to course 115A. P/NP or letter grading.
Math 115A is a core mathematics course required of all the various mathematics majors. The course material can be regarded as an elaboration of the linear algebra already covered in Math 33A. However, the level of abstraction and the emphasis on proof technique make this a difficult course for many students. Successful students emerge from the experience not only with a better understanding of linear algebra, but also with a higher level of mathematical maturity, better equipped to deal with abstract concepts.
The material covered in Math 115A includes linear independence, bases, orthogonality, the GramSchmidt process, linear transformations, eigenvalues and eigenvectors, and diagonalization of matrices. These topics are all covered in Math 33A though only in the context of Euclidean space. Topics in Math 115A that go beyond Math 33A include inner product spaces, adjoint transformations, and the spectral decomposition theorem for selfadjoint operators.
Three or four sections of Math 115A are offered each term. Also, an honors version Math 115AH runs parallel to Math 115A in some quarters. The content of Math 115AH is as follows:
Vector spaces, subspaces, basis and dimension, linear transformations and matrices, rank and nullity, change of basis and similarity of matrices, inner product spaces, orthogonality and, orthonormality, GramSchmidt process, adjoints of linear transformations and dual spaces, quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of hermitian and symmetric matrices, eigenvectors and eigenvalues, and their computation, exponentiation of matrices and application to differential equations, least squares problems, trace, determinant, canonical forms. Systems of linear equations: solvability criteria, Gaussian elimination, rowreduced form, LU decomposition.
Textbook(s)
S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.
Book is Subject to Change Without Notice
Outline Updated: June 2005
Schedule of Lectures
Lecture  Section  Topics 

1  1.2  Vector Spaces over a Field 
2  1.3  Subspaces 
3  1.4, 1.5  Linear Combinations and Systems of Linear Equations; Linear Dependence and Linear Independence 
4  1.5, 1.6  Linear Dependence and Linear Independence; Bases and Dimensions 
5  1.6  Bases and Dimensions 
6  1.6  Bases and Dimensions 
7  2.1  Linear Transformations, Null Spaces, and Ranges 
8  2.1  Linear Transformations, Null Spaces, and Ranges 
9  2.1, 2.2  Linear Transformations, Null Spaces, and Ranges; The Matrix Representation of a Linear Transformation 
10  .  Midterm #1 
11  2.2  The Matrix Representation of a Linear Transformation 
12  2.3  Composition of Linear Transformations and Matrix Multiplication 
13  2.4  Invertibility and Isomorphisms 
14  2.4, 2.5  Invertibility and Isomorphisms; The Change of Coordinate Matrix 
15  2.5  The Change of Coordinate Matrix 
16  4.4  Summary – Important Facts about Determinants 
17  5.1  Eigenvalues and Eigenvectors 
18  5.1  Eigenvalues and Eigenvectors 
19  5.2  Diagonalizability 
20  5.2  Diagonalizability 
21  5.2  Diagonalizability 
22  .  Midterm #2 
23  6.1  Inner Products and Norms 
24  6.1, 6.2  Inner Products and Norms; The GramSchmidt Orthogonalization Process and Orthogonal Complements 
25  6.2  The GramSchmidt Orthogonalization Process and Orthogonal Complements 
26  6.3  The Adjoint of a Linear Operator 
27  6.4  Normal and SelfAdjoint Operators 
28  6.4  Normal and SelfAdjoint Operators 
29  .  Catchup, Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Linear transformations, conjugate spaces, duality; theory of a single linear transformation, Jordan normal form; bilinear forms, quadratic forms; Euclidean and unitary spaces, symmetric skew and orthogonal linear transformations, polar decomposition. P/NP or letter grading.
Textbook(s)
S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.
Schedule of Lectures
Lecture  Section  Topics 

1  .  Review of Math 115A, Chapters I and II 
2  2.6  Dual Spaces (This section looks short but the concepts are new and thus will take two lectures to do well) 
3  2.6  Dual Spaces 
4  .  Review Sections 5.1 and 5.2 from 115A 
5  5.4  Invariant Subspaces and the Cayley Hamilton Theorem 
6  5.4  Invariant Subspaces and the Cayley Hamilton Theorem 
7  5.4  Invariant Subspaces and the Cayley Hamilton Theorem 
8  .  Review Sections 6.1 – 6.4 including more detail than was done in 115A 
9  .  Review Sections 6.1 – 6.4 including more detail than was done in 115A 
10  6.5  Unitary and Orthogonal Operators and their matrices 
11  6.5  Unitary and Orthogonal Operators and their matrices 
12  6.5  Unitary and Orthogonal Operators and their matrices 
13  6.6  Orthogonal Projections and the Spectral Theorem 
14  6.6  Orthogonal Projections and the Spectral Theorem 
15  6.6  Orthogonal Projections and the Spectral Theorem 
16  .  EXAM 
17  6.11  The Geometry of Orthogonal Operators 
18  6.11  The Geometry of Orthogonal Operators 
19  6.11  The Geometry of Orthogonal Operators 
20  7.1  Jordan Canonical Form I (This is a long and intricate presentation that takes time; do examples along the way!) 
21  7.1  Jordan canonical Form I 
22  7.1  Jordan canonical Form I 
23  7.3  The Minimal Polynomial (It might actually be better to do this section right after the Cayley Hamilton Theorem) 
24  7.3  The Minimal Polynomial 
2529  .  At the discretion of the teacher. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for Program in Computing 130. Introduction to mathematical cryptology using methods of number theory, algebra, probability. Topics include symmetric and publickey cryptosystems, oneway functions, signatures, key exchange, groups, primes, pseudoprimes, primality tests, quadratic reciprocity, factoring, rho method, RSA, discrete logs. P/NP or letter grading.
Course Information:
The course is planned for 28 lectures, 1 midterm exam, and 1 holiday.
Math 116 is the introduction to mathematical cryptology which uses methods of number theory, algebra, probability. Topics include: symmetric and publickey cryptosystems, oneway functions, signatures, key exchange, groups, primes, pseudoprimes, primality tests, quadratic reciprocity, factoring, rho method, RSA, and discrete logs.
Math 116 is not open for credit to students with credit for PIC 130.
Textbook(s)
Trappe, Intro to Cryptography with Coding Theory, Prentice Hall.
Outline update: D. Blasius, 2/02
NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

1  1.11.4, 2.12.2  Congruences, Classic Symmetric Ciphers, Intro to Probability. Read: Introduction, 1.11.4, 2.12.2. 
2  2.32.4, 4.4, 3.13.5  Probability (cont.), Applications to Attacks, Permutations. Read: 2.32.4, 4.4, 3.13.5. 
3  4.14.2, 6.16.3, 7.17.2  Symmetric Ciphers (Vigenere, DES, AES), Theory of Integers (Factorization, GCD, Euclidean Algorithm). Read: 4.14.2, 6.16.3, handout on AES (Rijndael), 7.17.2. 
4  7.37.8, 8.18.2  Theory of Integers (Euclidean Algorithm, Equivalence Relations, Integers mod n, Discrete logs, Primitive roots, Linear Algebra mod n), affine cipher. Read: 7.37.8, 8.18.2. 
5  10.110.5  Public Key Ciphers (RSA, DiffieHellman, ElGameal, Knapsack). Read 10.110.5. 
6  12.112.6  Midterm Monday. Roots mod p. Read: 12.112.5. 
7  13.113.3, 13.513.7, 15.115.5  Roots mod n, Quadratic Reciprocity. Read: 13.113.3, 13.513.7, 15.115.5. 
8  16.116.6  Pseudoprimes and Primality tests, Prime Generation. Read: 16.116.6. 
9  24.124.3  Factorization Attacks. Read: 24.124.3. 
10  27.127.3  Discrete logs, Review. Read: 27.127.3. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 110A. Integers, congruences; fields, applications of finite fields; polynomials; permutations, introduction to groups.
Course Information:
The following schedule is based on 26 lectures. The remaining three classroom meetings are for midterm exams and a review.
Math 117 is the “fast” course in abstract algebra, which focuses on topics that are of interest for applications. The topics covered include error correcting codes, fast polynomial multiplication, and the fast Fourier transform. The fast Fourier transform is absolutely critical for the efficient implementation of computer algorithms for signal processing and other engineering applications.
One section of Math 117 is offered each term. In the past several years the enrollments in the course have averaged about 35 students each term.
Textbook(s)
L. Childs, A Concrete Introduction to Higher Algebra, 3rd Ed., SpringerVerlag.
Note: The book contains a wealth of interesting topics (e.g. Sturm’s theorem, group theory), which can be substituted for material in the last five lectures at the instructor’s discretion.
Outline update: D. Gieseker, 1/97
Schedule of Lectures
Lecture  Section  Topics 

12  Ch 2 A–D  Induction and binomial theorem 
3  Ch 3 A  Division theorem, bases 
45  Ch 3 B–D; Ch 4 A, B  Euclidean algorithm, Bezout’s identity, unique factorization 
69  Ch 5; Ch 6  Congruences, congruence classes, and errorcorrecting codes 
1011  Ch 7  Rings and fields 
1213  Ch 9 A–D  Theorems of Euler and Fermat 
14  Ch 10 B  RSA codes 
1516  Ch 12 A, B  Chinese remainder theorem 
17  Ch 12 C  Application of Chinese remainder theorem to RSA cryptography 
1820  Ch 13, ch 14  Polynomials, unique factorization 
21  Ch 15 D, F, C  Complex numbers, fundamental theorem of algebra 
2223  Ch 17 A, B  Congruences modulo a polynomial and Chinese remainder theorem 
2426  Ch 18  Fast polynomial multiplication, fast Fourier transform 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 42 and 115A. Introduction to computational methods for data problems with a focus on linear algebra and optimization. Matrix and tensor factorization, PageRank, assorted other topics in matrices, linear programming, unconstrained optimization, constrained optimization, integer optimization, dynamic programming, and stochastic optimization. P/NP or letter grading.
Course Information:
Students will learn key processes of optimization and linear algebra which underlies data science. These include linear programming, unconstrained optimization, constrained optimization, integer optimization, dynamic programming, stochastic optimization, integer optimization, dynamic programming, and stochastic optimization.
Textbook(s)
Required:
1. Elden, Lrs. Matrix Methods in Data Mining and Pattern Recognition. The Society for Industrial and Applied Mathematics, 2007.
2. Chong, E and S. Zak. An Introduction to Optimization, 4th edition. Wiley, 2013.
Supplemental:
3. Hillier, Frederick S. and Lieberman, Gerald J. Introduction to Operations Research, 9th edition. McGrawHill Higher Education, 2009.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Elden, Chong & Zak  Review of linear algebra, least squares, orthogonality; QR decomposition; Singularvalue decomposition (SVD) Elden: Data Mining and Pattern Recognition, Vectors and Matrices (1.1) MatrixVector Multiplication, MatrixMatrix Multiplication, Scalar Product and Vector Norms, Matrix Norms, Linear Independence Bases, The Rank of a Matrix (1.2, 2.12.6) Linear Systems and Least Squares, LU Decomposition, Symmetric, Positive Definite Matrices, Perturbation Theory and Condition Number, Rounding Errors in Gaussian Elimination, Banded Matrices, The Least Squares Problem (3.13.6) Orthogonal Vectors and Matrices., Elementary Orthogonal Matrices, Number of Floating Point Operations, Orthogonal Transformations in Floating Point Arithmetic (4.14.4) Orthogonal Transformation to Triangular Form, Solving the Least Squares Problem, Computing or Not Computing Q, Flop Count for QR Factorization, Error in the Solution of the Least Squares Problem, Updating the Solution of a Least Squares Problem (5.15.2) Singular Value Decomposition, Fundamental Subspaces, Matrix Approximation, Principal Component Analysis, Solving Least Squares Problems, Condition Number and Perturbation Theory for the Least Squares Problem, RankDeficient and UnderDetermined Systems, Computing the SVD, Complete Orthogonal Decomposition (6.16.9) Chong & Zak: Real Vector Spaces, Rank of a Matrix, Linear Equations, Inner Products and Norms (2.12.4) Linear Transformation, Eigenvalues and Eigenvectors, Orthogonal Projections, Quadratic Forms, Matrix Norms (3.13.5) 
Week 2  Elden  Reducedrank least squares; Tensor decomposition; Nonnegative matrix factorization Elden: Truncated SVD: Principal Components Regression, Krylov Subspace Method (7.17.2) Introduction to Tensor Decomposition, Basic Tensor Concepts, A Tensor Singular Value Decomposition, Approximating a Tensor by HOSVD (8.18.4) 
Week 3  Elden  Data analysis applications; Pagerank Elden: The kMeans Algorithm, NonNegative Matrix Factorization (9.19.2) Handwritten Digits and a Simple Algorithm, Classification using SVD Bases, Tangent Distance (10.010.3) Preprocessing the Documents and Queries, The Vector Space Model, Latent Semantic Indexing, Clustering, NonNegative Matrix Factorization, LanczosGolubKahan Bidiagonalization, Average Performance (11.111.7) Pagerank, Random Walk and Markov Chains, The Power Method for Pagerank Computation, HITS (12.012.4) 
Week 4  Chong & Zak  Linear optimization: modeling; Standard form; Duality Chong & Zak: Introduction to Linear Programing, Simple Examples of Linear Programs, TwoDimensional Linear Programs, Convex Polyhedra and Linear Programming, Standard Form Linear Programs, Basic Solutions, Properties of Basic Solutions, Geometric View of Linear Programs (15.115.8) Solving Linear Equations Using Row Operations, The Canonical Augmented Matrix, Updating the Augmented Matrix, The Simplex Algorithm, Matrix Form of the Simplex Method, TwoPhase Simplex Method, Revised Simplex Method (16.116.7) Dual Linear Programs, Properties of Dual Problems (17.117.2) 
Week 5  Chong & Zak  Linear optimization solvers (Simplex Method, InteriorPoint Method) Chong & Zak: Introduction to Nonsimplex Methods, Khachiyan?s Method, Affine Scaling Method, Karmarkar?s Method (18.118.4) Introduction to Problems with Equality Constraints, Problem Formulation, Tangent and Normal Spaces, Lagrange Condition, SecondOrder Conditions, Minimizing Quadratics Subject to Linear Constraints (20.120.6) 
Week 6  Chong & Zak  Unconstrained optimization: optimality condition, localvs. global minimum, convex set and function; Solvers such as gradient descent and Newton Method Chong & Zak: Introduction to Convex Optimization Problems, Convex Functions, Convex Optimization Problems (22.122.3) 
Week 7  Chong & Zak  Constrained optimization: KKT condition; Solvers such as Gradient Projection Method, Penalty Method and Multipliers Method Chong & Zak: KarushKuhnTucker Condition, SecondOrder Conditions (21.121.2) Introduction to Algorithms for Constrained Optimization, Projections, Projected Gradient Methods, Penalty Methods (23.123.3, 23.5) 
Week 8  Hillier & Lieberman  Integer optimization: modeling, relaxations; Solvers such as cutting plane, BranchNBound/Cut Methods Hillier & Lieberman: Perspectives on Solving Integer Programming Problems (12.112.5) The BranchandBound Technique and Its Application to Binary Integer Programming (12.6) BranchandBound Algorithm for Mixed Integer Programming (12.7) 
Week 9  Hillier & Lieberman  Dynamic programming Hillier & Lieberman: A Prototype Example for Dynamic Programming (11. 1) Characteristics of Dynamic Programming Problems (11.2) Deterministic Dynamic Programming (11.3) 
Week 10  Chong & Zak  Neural networks Chong & Zak: Introduction (13.1) Single Neuron Training (13.2) (needs 12.3 – aolution to Ax=b minimizing x and 12.4 Kaczmarz?s Algorithm) Backpropagation Algorithm (13.3) 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 131A. Course 120A is requisite to 120B. Curves in 3space, Frenet formulas, surfaces in 3space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.
Differential geometry can be viewed as the study of space and curvature. The course depends heavily upon calculus, it uses the tools of linear algebra, and it develops geometric insight. As such it is a good course for students who want to strengthen their understanding of the core mathematics curriculum.
Differential geometry is a crucial tool in modern physics. The idea of curved space is at the foundation of Einstein’s theory of gravitation (general relativity). Several more recent developments in physics, as YangMills theory and string theory, involve differential geometry.
The courses 120A and 120B deal with differential geometry in a special context, curves and surfaces in 3space, which has a firm intuitive basis, and for which some remarkable and striking theorems are available.
The course begins with curves in the plane and in 3space, which already have some interesting geometric features. Curvature and torsion measure how curves bend and twist. There are some beautiful theorems that if a curve in 3space forms a closed loop, it has to bend at least a certain amount, and if it forms a knot, it has to bend at least a larger certain amount. Another beautiful theorem is the celebrated isoperimetric theorem, that among all closed curves of a fixed length, the circle encloses the largest area.
There are several notions of curvature for surfaces in 3space. Mean curvature shows up in the problem of determining the surface of the smallest area with a fixed prescribed boundary. (The solution can be illustrated with soap bubbles.) Gaussian curvature shows up in the problem of determining which surfaces can be represented by a flat map.
Another problem treated in the course is how to determine the shortest route on a surface between two points. In the plane the shortest path is a straight line, and on a sphere the shortest path is an arc of a great circle.
The theorem of highschool geometry that the sum of the angles of a triangle is 180 degrees turns out to have a very beautiful generalization to a triangle on any surface (as a spherical triangle). The generalization is the GaussBonnet theorem, which is one of the highpoints of undergraduate mathematics. The theorem provides an identity with a sum of angles and a correction term that takes into account how curved the sides of the triangle are and how much the surface is curved inside the triangle. One of the remarkable features of the GaussBonnet theorem is that it asserts the equality of two quantities, one of which comes from differential geometry and the other of which comes from topology.
Math 120AB is highly recommended for mathematics students who want to go on to graduate school.
Textbook(s)
Millman & Parker, Elements of Differential Geometry, Prentice Hall
Book is Subject to Change Without Notice
Outline update: P. Petersen, 9/14
(Requisites updated 5/98)
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Curves  
Week 2  Curvation of Curves  
Week 3  Global Theory of Curves  
Week 4 + 5  Surfaces  
Week 6  Examples of Surfaces  
Week 7  First Fundamental Form  
Week 8  Curvature of Surfaces  
Week 9 + 10  Second Fundamental Form 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 120A, 131A. Curves in 3space, Frenet formulas, surfaces in 3space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.
Differential geometry can be viewed as the study of space and curvature. The course depends heavily upon calculus, it uses the tools of linear algebra, and it develops geometric insight. As such it is a good course for students who want to strengthen their understanding of the core mathematics curriculum.
Differential geometry is a crucial tool in modern physics. The idea of curved space is at the foundation of Einstein’s theory of gravitation (general relativity). Several more recent developments in physics, as YangMills theory and string theory, involve differential geometry.
The courses 120A and 120B deal with differential geometry in a special context, curves and surfaces in 3space, which has a firm intuitive basis, and for which some remarkable and striking theorems are available.
The course begins with curves in the plane and in 3space, which already have some interesting geometric features. Curvature and torsion measure how curves bend and twist. There are some beautiful theorems that if a curve in 3space forms a closed loop, it has to bend at least a certain amount, and if it forms a knot, it has to bend at least a larger certain amount. Another beautiful theorem is the celebrated isoperimetric theorem, that among all closed curves of a fixed length, the circle encloses the largest area.
There are several notions of curvature for surfaces in 3space. Mean curvature shows up in the problem of determining the surface of the smallest area with a fixed prescribed boundary. (The solution can be illustrated with soap bubbles.) Gaussian curvature shows up in the problem of determining which surfaces can be represented by a flat map.
Another problem treated in the course is how to determine the shortest route on a surface between two points. In the plane the shortest path is a straight line, and on a sphere the shortest path is an arc of a great circle.
The theorem of highschool geometry that the sum of the angles of a triangle is 180 degrees turns out to have a very beautiful generalization to a triangle on any surface (as a spherical triangle). The generalization is the GaussBonnet theorem, which is one of the highpoints of undergraduate mathematics. The theorem provides an identity with a sum of angles and a correction term that takes into account how curved the sides of the triangle are and how much the surface is curved inside the triangle. One of the remarkable features of the GaussBonnet theorem is that it asserts the equality of two quantities, one of which comes from differential geometry and the other of which comes from topology.
Math 120AB is highly recommended for mathematics students who want to go on to graduate school.
Textbook(s)
Millman & Parker, Elements of Differential Geometry, Prentice Hall
Book is Subject to Change Without Notice
Outline update: P. Petersen, 9/14
(Requisites updated 5/98)
Schedule of Lectures
Lecture  Section  Topics 

Week 1+2  Geodesics  
Week 3+4  Theorema Egregium  
Week 5+6  Hyperbolic Geometry  
Week 7+8  Minimal Surfaces  
Week 9+10  The GaussBonnet Theorem 
General Course Outline
Course Description
(4) Requisite: course 131A. Metric and topological spaces, completeness, compactness, connectedness, functions, continuity, homeomorphisms, topological properties.
Course Information:
The following sample schedule, with textbook sections and topics, is based on 25 lectures. Assigned homework problems play an important role in the course, and there is usually a midterm exam.
Topology is the study of the properties of spaces (such as surfaces, or solids) that are invariant under homeomorphisms (such as stretchings). One striking theorem in topology is that any compact orientable twodimensional surface is topologically a sphere with a certain number of handles attached. The number of handles completely characterizes the topological type of the surface. This leads to the adage that a topologist is a person who cannot tell the difference between a teacup and a doughnut. Topologically speaking, each is a sphere with one handle, and each can be continuously deformed to the other.
While topology is classified under geometry, the language of topology is fundamental to analysis. Many of the issues addressed by topology, such as compactness of spaces and continuity of functions, are treated in a simpler setting in the analysis courses 131AB.
One method for studying topological spaces is to assign algebraic objects, such as groups or vector spaces, to a topological space. One such object is the “fundamental group” of a topological space, which measures in some sense the number of holes in the space. Thus topology interacts also with algebra, leading to a branch of mathematics called “algebraic topology.”
Math 121 is a flexible course, and the selection of topics might be organized quite differently by different instructors. The subject matter for a standard syllabus breaks into three parts.
The first part treats metric spaces, which are closest to the intuition and to the development presented in 131AB. The fundamental concepts are completeness, compactness, continuity, and uniform continuity. The principal theorems are the Baire category theorem, the characterization of compact metric spaces, the theorem that continuous functions on a compact space are uniformly continuous, and the contraction mapping principle, which is perhaps the most important and useful tool in analysis.
The second part of the standard course covers pointset topology. Topological spaces are introduced, along with the separation axioms and various notions as compactness, local compactness, connectedness, and path connectedness. Product and quotient spaces are defined. The most important theorem in pointset topology is Tychonoff’s theorem that the product of a family of compact topological spaces is compact.
The third part of the standard course consists of an elementary introduction to algebraic topology. The fundamental group is introduced, and covering spaces are used to compute it for some special spaces. Some simple applications of the algebraic invariants are given.
Math 121 is offered once each year, usually in the Spring Quarter. Course enrollments run between 10 and 35.
Textbook(s)
T. Gamelin and R. Greene, Introduction to Topology, 2nd Ed., Dover.
Outline update: T. Gamelin, 5/96
NOTE: While this outline only suggests one midterm exam, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

13  1.14  Metric spaces, open and closed sets; completeness, Baire category theorem; euclidean space 
45  1.5  Compactness, characterization of compact metric spaces 
6  1.6  Continuous functions 
79  1.78  Normed linear spaces; linear operators, principle of uniform boundedness; contraction mapping principle 
10  2.12  Topological spaces, subspaces 
11  2.3  Continuous functions 
12  2.4  Base for a topology 
13  2.5  Separation axioms 
14  2.6  Compactness 
15  2.7  Locally compact spaces 
16  2.8  Connectedness 
17  2.9  Path connectedness 
18  2.1  Finite product spaces 
1920  2.1112  Transfinite induction; infinite product spaces, Tychonoff’s theorem 
21  2.13  Quotient spaces 
2223  3.14  Homotopic paths, fundamental group 
2425  3.56  Covering spaces; index of circle maps; applications of the index 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Prerequisite: course 115A. Axioms and models, Euclidean geometry, Hilbert axioms, neutral (absolute) geometry, hyperbolic geometry, Poincare model, independence of parallel postulate.
Course Information:
The purpose of Math 123 is to study the classical geometries from an axiomatic perspective, with particular attention paid to Euclid’s parallel postulate and to geometric systems that violate it. These systems are called NonEuclidean Geometries. Among them, the Hyperbolic Geometry is the most important today. Here is some background.
In his Elements, Euclid (~365BC~300BC) built his geometry using five axioms. The first 4 are:
(1) Any two points can be joined by a (straight) line.
(2) Any segment can be extended continuously in a (straight) line.
(3) Given any point and distance, there is a circle centered at the point with radius equal to the distance.
(4) All right angles are equal to each other. These are easily understood as Euclid gave them.
The fifth was less obvious, but was found to be equivalent to (5) Given a line L and a point P not on the line, there exists one and only one line which passes through P and is parallel to (i.e. does not intersect) L. Axiom (5), in this version, is called the Parallel Postulate (and also Playfair’s Axiom).
From near the beginning, it seemed as if Euclid’s 5th axiom might be a consequence of the first 4, but no proof was ever found. Finally, in the nineteenth century Bolyai, Gauss and Lobachevsky independently put the question to rest by showing that a new geometry, Hyperbolic Geometry, satisfies the first 4 axioms but not the 5th. Thus, one of the goals of Math 123 is to study the concept of a “geometry” and to illustrate the implementation of this concept in examples.
The course can be useful for prospective secondary school teachers, in that it illustrates how a mathematical structure can be built upon an axiom system, and how the Euclidean geometry that is traditionally studied in the schools is only one of many possible “geometries”.
Math 123 is a flexible course, and it is taught quite differently by different instructors. For example, some instructors may approach the course primarily through the classical axiom systems, while others may take the Kleinian approach according to which geometries are classified by their symmetry groups.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Recommended: course 115A. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining three classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.
Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.
There is an honors sequence Math 131AH131BH running parallel to 131A131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.
Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.
Textbook(s)
K.A. Ross, Elementary Analysis: The Theory of Calculus, 2nd Ed.
Outline update: J. Ralston, 8/08(*1) Include Section 23, if time permits. The instructor can pick which convergence tests to cover in Sections 14 and 15.
Schedule of Lectures
Lecture  Section  Topics 

1  1,2  Induction and Rational Numbers. 
2  3,4,5  Real Numbers, Least Upper Bound Axiom 
3  7,8,9  Limits of Sequences, Limit Theorems. 
4  10  Monotone Sequences, Cauchy Sequences, Midterm I. 
5  11,12  Subsequences, BolzanoWeierstrass, Limsup and Liminf. 
6  14(*1),15,17  Convergence Tests, Continuous Functions. 
7  18,19,20  Limit Theorems, Uniform Continuity. 
8  28,29  Derivative, Mean Value Theorem, Midterm II. 
9  31,32,33  Taylor’s Theorem, Riemann Integral, Properties of Riemann Integral. 
10  34  Fundamental Theorem of Calculus, Review of Course. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites for course 131AH: courses 32B and 33B, with grades of B or better. Recommended: course 115A. Honors sequence parallel to courses 131A. P/NP or letter grading. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining three classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.
Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.
There is an honors sequence Math 131AH131BH running parallel to 131A131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.
Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.
Textbook(s)
Rudin, W., Principles of Mathematical Analysis, 3rd Ed, McGrawHill Higher Education
Copson, E. Metric Spaces, Cambridge University Press
Outline update:D. Gieseker, 9/14(*1) Include Section 23, if time permits. The instructor can pick which convergence tests to cover in Sections 14 and 15.
Schedule of Lectures
Lecture  Section  Topics 

1  Induction and Rational Numbers.  
2  Real Numbers, Least Upper Bound Axiom  
3  Limits of Sequences, Limit Theorems.  
4  Monotone Sequences, Cauchy Sequences, Midterm I.  
5  Subsequences, BolzanoWeierstrass, Limsup and Liminf.  
6  Convergence Tests, Continuous Functions.  
7  Limit Theorems, Uniform Continuity.  
8  Derivative, Mean Value Theorem, Midterm II.  
9  Taylor’s Theorem, Riemann Integral, Properties of Riemann Integral.  
10  Fundamental Theorem of Calculus, Review of Course. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. P/NP or letter grading. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of fourth and eighth weeks of instruction, plus reviews for the final exam.
Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.
There is an honors sequence Math 131AH131BH running parallel to 131A131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.
Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.
Textbook(s)
Rudin, W., Principles of Mathematical Analysis, 3rd Ed
Copson, E. Metric Spaces, Cambridge University Press
Section 14.8 is the proof of the Weierstrass Approximation Theorem. This should probably be left for the Honors Section.
This is rather difficult, but it introduces summation by parts. Using summation by parts to prove Dirichlet’s Test (and hence the Alternating Series Test) is an alternative to Abel’s Theorem.
This is a lot, but Sections 17.1 is just a review of linear transformations and 17.2 and 17.3 contain only one theorem.
Outline update: D. Gieseker, 9/14
Schedule of Lectures
Lecture  Section  Topics 

1  Metric Spaces, Some PointSet Topology and Relative Topology  
2  Cauchy Sequences and Completeness, Compact Metric Spaces, Continuous Functions on Metric Spaces  
3  Continuity on Product, Connected and Compact Metric Spaces  
4  Uniform Convergence, Midterm I  
5  Uniform Convergence and Continuity, the “Sup” Norm, Series of Functions, Uniform Convergence in Integration and Differentiation3  
6  Formal Power Series, Real Analytic Functions, Abel’s Theorem (Optional)4, Multiplication of Power Series  
7  Exponential and Logarithmic Functions, Trigonometric Functions, Periodic Functions  
8  Inner Products on Periodic Functions, Trigonometric Polynomials, Hour Exam II  
9  Periodic Convolutions, L2 convergence of Fourier Series and Plancherel’s Theorem, Differentiability of Functions of Several Variables  
10  The Several Variable Chain Rule, Clairaut’s Theorem, Review of Course 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. P/NP or letter grading. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of fourth and eighth weeks of instruction, plus reviews for the final exam.
Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.
There is an honors sequence Math 131AH131BH running parallel to 131A131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.
Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.
Textbook(s)
Tao, T., Analysis II, 3rd Ed., Hindustan
Section 3.8 is the proof of the Weierstrass Approximation Theorem. This should probably be left for the Honors Section.
This is rather difficult, but it introduces summation by parts. Using summation by parts to prove Dirichlet’s Test (and hence the Alternating Series Test) is an alternative to Abel’s Theorem.
This is a lot, but Sections 6.1 is just a review of linear transformations and 6.2 and 6.3 contain only one theorem.
Outline update: J. Ralston, 9/19
Schedule of Lectures
Lecture  Section  Topics 

1  1.1, 1.2, 1.3  Metric Spaces, Some PointSet Topology and Relative Topology 
2  1.4, 1.5, 12.1  Cauchy Sequences and Completeness, Compact Metric Spaces, Continuous Functions on Metric Spaces 
3  2.2, 2.3, 2.4  Continuity on Product, Connected and Compact Metric Spaces 
4  3.1, 3.2  Uniform Convergence, Midterm I 
5  3.3, 3.3, 3.6, 3.7  Uniform Convergence and Continuity, the “Sup” Norm, Series of Functions, Uniform Convergence in Integration and Differentiation3 
6  4.1, 4.2, 4.3, 4.4  Formal Power Series, Real Analytic Functions, Abel’s Theorem (Optional)4, Multiplication of Power Series 
7  4.5, 4.6, 4.7, 5.1  Exponential and Logarithmic Functions, Trigonometric Functions, Periodic Functions 
8  5.2, 5.3  Inner Products on Periodic Functions, Trigonometric Polynomials, Hour Exam II 
9  5.4, 5.5, 6.1, 6.2, 6.3  Periodic Convolutions, L2 convergence of Fourier Series and Plancherel’s Theorem, Differentiability of Functions of Several Variables 
10  6.4, 6.5  The Several Variable Chain Rule, Clairaut’s Theorem, Review of Course 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 131A, 131B or 131AH, 131BH. Covers multivariable calculus and applications to ordinary differential equations.
Math 131C studies primarily multivariable analysis: definition of differentiability in several variables, partial derivatives, chain rule, Taylor expansion in several variables, inverse and implicit function theorems, equality of mixed partials, multivariable integration, change of variables formula, differentiation under the integral sign, analysis on curves and surfaces. Further topics to be chosen, usually including basic applications to ordinary differential equations (existence and uniqueness theorems for solutions) and the Green, Gauss and Stoke theorems.
Textbook(s)
Conway, J., A First Course In Analysis, Cambridge University Press
Coddington, E., An Introduction to Ordinary Differential Equations, Dover Publications
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Introduction to basic formulas and calculation procedures of complex analysis of one variable relevant to applications. Topics include Cauchy/Riemann equations, Cauchy integral formula, power series expansion, contour integrals, residue calculus.
Course Information:
The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and a midterm exam. These are scheduled by the individual instructor. Often there are a review and a midterm exam about the end of the fifth week of instruction, plus a review for the final exam.
General Information. Complex analysis is one of the most beautiful areas of pure mathematics, at the same time it is an important and powerful tool in the physical sciences and engineering. The course Math 132 is aimed primarily at students in applied mathematics, engineering, and physics, and it is satisfies a major requirement for students in Electrical Engineering.
The topics covered in Math 132 include: analytic functions, CauchyRiemann equations, harmonic functions, branch points, branches of multiplevalued functions, Cauchy’s theorem, integral representation formulae, power series of analytic functions, zeros, isolated singularities, Laurent series, poles, residues, use of residue calculus to evaluate real integrals, use of argument principle to locate zeros, fractional linear transformations, and conformal mapping.
Students entering Math 132 are assumed to have some familiarity with complex numbers from high school, including the polar form of complex numbers. Students in Math 132 are also assumed to have a strong background in single and multivariable calculus, including infinite series, power series, radius of convergence (ratio and root tests), integration term by term of power series, parametrized curves, line integrals, and Green’s theorem. Some of this material is reviewed in Math 132, though at a fast pace.
Several sections of Math 132 are offered each term.
Textbook(s)
T. Gamelin, Complex Analysis, Springer/Verlag.
*The book is subject to change. Check with the UCLA Bookstore.
The students should be familiar with the elementary properties of complex numbers from high school. They have been introduced to the complex exponential function in Math 33B. They should be familiar with power series, including radius of convergence, the ratio and root tests, and integration term by term.
The idea of gluing sheets together at branch cuts to form a surface is important, but it can be omitted at this stage. At most it should be treated only at an intuitive level, to introduce the idea to the students and to arouse their interest.
The idea of conformality can be treated lightly if short on time. The results of the section on conformality are used primarily to see that fractional linear transformations map orthogonal circles to orthogonal circles.
With respect to uniform convergence, the only thing that is really needed is the Weierstrass Mtest, together with the integration term by term of a uniformly convergent series of functions.
The material in Section VIII.1 on the argument principle is important to electrical engineers and should not be omitted. Rather omit Section VII.3 if short of time at the end of the course.
Outline update: T. Gamelin, 3/04
NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

1  I.12  Complex numbers, polar form, complex multiplication, roots of complex numbers (much of this is review) 
2  I.3  Stereographic projection 
35  I.48  Elementary functions, including power, root, exponential, logarithm, and trigonometric functions 
6  II.12  Complex derivatives, basic rules of differentiation 
710  II.34  CauchyRiemann equations; inverse functions; harmonic functions; conformality; fractional linear transformations 
11  III.13  Review line integrals and Green’s theorem; harmonic conjugates 
1213  IV.12  Complex line integrals, MLestimate, fundamental theorem of complex calculus 
1415  IV.36  Cauchy’s theorem, Cauchy integral formulae, Liouville’s theorem, Morera’s theorem (statement only) 
1617  .  Catch up, review, midterm exam 
1821  V.17  Weierstrass Mtest, power series, radius of convergence, operations on power series, order of zeros 
2224  VI.14  Laurent decomposition, isolated singularities, orders of poles and zeros, partial fractions decomposition 
2527  VII.14  Residue theory, applications of residue calculus to evaluate integrals 
28  VIII.1  Argument principle, location of roots 
29  Catch up, review for final exam. 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, and 131A with grades of B or better. This course is specifically designed for students who have strong commitment to pursue graduate studies in mathematics. Introduction to complex analysis with more emphasis on proofs. Honors course parallel to course 132. P/NP or letter grading.
Textbook(s)
Complex Analysis by Stein and Shakarchi.
Schedule of Lectures
Lecture  Section  Topics 

13  1.11.2  Complex numbers and the complex plane (Basic properties, convergence, sets in the complex plane); Functionas on the complex plane (continuous functions, holomorphic functions, power series) 
46  1.3  Integration along curves 
78  2.12.2  Goursat’s theorem; Local existence of primitives and Cauchy’s theorem in a disc 
911  2.32.4  Evaluation of some integrals; Cauchy’s integral formulas 
1214  3.13.2  Zeros and poles; The residue formula 
1516  Midterm/Continuation  
1718  3.3  Singularities and meromorphic functions 
1921  3.43.6  The argument principle and applications; Homotopies and simply connected domains; The complex algorithm 
2224  8.18.4  Conformal equivalence and examples (the disc and upper halfplace, further examples, the Dirichlet problem in a strip); The Schwarz lemma and automorphisms of the disc and upper halfplace (Automorphisms of the disc, automorphisms of the upper halfplace); The Riemann mapping thoerem (Necessary conditions and statement of theorem, Montel’s theorem, proof of Riemann mapping theorem; Conformal mappings onto polygons (Some examples, the SchwarzChristoffel integral, boundary behavior, the mapping formula, return to elliptic integrals) 
2527  TBA  Catchup, Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B, 131A. Fourier series, Fourier transform in one and several variables, finite Fourier transform. Applications, in particular, to solving differential equations. Fourier inversion formula, Plancherel theorem, convergence of Fourier series, convolution. P/NP or letter grading.
Course Information:
This syllabus is based on a single midterm; instructors who wish to give a second midterm may adjust the syllabus appropriately, or give the second midterm in section. The lecturer may also wish to expand the applications components (lectures 1112, 2224, 2628) or move them earlier in the course.
Math 133 is the introduction to Fourier series, the Fourier transform in one and several variables, finite Fourier transform, applications, in particular to solving differential equations. Fourier inversion formula, Plancherel’s theorem, convergence of Fourier series, convolution.
Textbook(s)
E. Stein and R. Shakarchi, Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume 1), Princeton University Press.
Schedule of Lectures
Lecture  Section  Topics 

1  Review: Complex numbers (esp. Euler’s formula); periodic functions; functions on an interval; functions on a circle; continuous functions; continuously differentiable functions; Riemann integrable functions (or at least piecewise continuous functions). No  
2  Does every function have a Fourier series? Formal computation of Fourier coefficients. Inversion formula for trigonometric polynomials. Examples of Fourier series (esp. Dirichlet kernel).  
3  Review of convergence, uniform convergence. Do Fourier series converge back to the original function? Injectivity of the Fourier transform for continuous functions.  
4  Uniform convergence for absolutely summable Fourier coefficients. Relationship between differentiation and the Fourier transform. Uniform convergence for C^2 functions. (Optional) Some foreshadowing of future convergence results.  
5  Convolutions of continuous periodic functions: examples and basic properties. Connections with Fourier coefficients. Connection between partial sums and the Dirichlet kernel.  
6  Convolutions of integrable periodic functions: approximation of integrable functions by continuous ones. Approximation via convolution by good kernels.  
7  Badness of the Dirichlet kernel; Gibbs’ phenomenon. Cesaro means; Fejer kernel. Fejer’s theorem. Uniform approximation of continuous functions by trigonometric polynomials.  
8  Leeway  
9  Review of vector spaces, inner product spaces, orthonormal sets, CauchySchwarz inequality, Pythagoras’s theorem. Orthonormality of the Fourier basis. Bessel’s inequality. Best meansquare approximation by trigonometric polynomials.  
10  Meansquare convergence of Fourier series for continuous functions. Meansquare convergence of Fourier series for Riemannintegrable functions. Plancherel’s theorem, Parseval’s theorem. RiemannLebesque lemma.  
1112  Applications and further properties of Fourier series, at instructor’s discretion. Some suggestions: Summation of 1/n^2; local convergence of Fourier series at smooth points; smoothness of a function versus decay of Fourier coefficients; a continuous func  
13  Leeway/review  
14  Midterm.  
15  From Fourier series to Fourier integrals – an informal discussion. Review of improper integrals. Functions of moderate decrease. Functions of rapid decrease. Schwartz functions. Definition of the Fourier transform.  
16  Basic algebraic properties of the Fourier transform. Preservation of the Schwartz space.  
17  Fourier transform of Gaussians. Gaussians as good kernels.  
18  Multiplication formula. Fourier inversion formula. Bijectivity on Schwartz space.  
19  Fourier transform and convolutions. Plancherel’s theorem. Extension to functions of moderate decrease.  
2021  Integration on R^d; Fourier transform on R^d; key properties.  
2224  Applications to PDE: heat equation; Laplace’s equation. (Optional) The wave equation (in 1D or higher dimensions).  
25  Z_N. The finite Fourier transform; key properties.  
2628  Applications and further properties of Fourier transforms, at instructor’s discretion. Some suggestions: The fast Fourier transform; fast multiplication; Heisenberg uncertainty principle; Comparison of Fourier and Laplace transforms; The FourierBessel tr  
29  Leeway/review. 
General Course Outline
Course Description
(4) (Formerly numbered 135A.) Lecture, three hours; discussion, one hour. Requisites: course 33B. Dynamical systems analysis of nonlinear systems of differential equations. One and two dimensional flows. Fixed points, limit cycles, and stability analysis. Bifurcations and normal forms. Elementary geometrical and topological results. Applications to problems in biology, chemistry, physics, and other fields. P/NP or letter grading.
Textbook(s)
S. Strogatz, Nonlinear Dynamics and Chaos (2nd Ed.), Perseus Books Group.
J. Crawford, Introduction to Bifurcation Theory, Reviews of Modern Physics, vol. 63. (Recommended supplement).
For those instructors wishing to incorporate a final project, lectures 9 and 10 can be skipped and the last four lectures can be used for final project poster presentations.
If time is available for more lectures than those outlined, additional lectures could cover section 7.6 (on weakly nonlinear oscillations and perturbation theory) or selected sections from chapter 9 (on chaos and the Lorenz equations).
Outline update: C. Topaz, 4/04, updated, 3/05
NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

1  General course overview.  
2  1.0 – 1.3  Definition of dynamical systems. Discussion of importance and difficulty of nonlinear systems. Examples of applications giving rise to nonlinear models. 
3  2.0 – 2.3  Elementary onedimensional flows. Flows on the line, fixed points, and stability. Application to population dynamics. Discussion of how geometric “dynamical systems” approach is different from approach in Math 33. 
4  2.4 – 2.6  “Advanced” onedimensional flows. Linear stability analysis (with numerous examples), existence and uniqueness, impossibility of oscillations. 
5  2.6 – 2.7  Potentials. Introduction to the idea of numerical solutions of nonlinear equations, including discussion of basic methods, software tools (Matlab, Maple, Mathematica, DSTool, xppaut, etc.). Advertisement for Math 151A/B. 
6  3.0 – 3.1  Introduction to bifurcations, saddlenode bifurcation. Physical relevance of bifurcations, introduction to bifurcation diagrams, notion of normal forms. For saddlenode bifurcation, incorporate treatment in Crawford. 
7  3.2 – 3.3  Transcritical bifurcation. Incorporate treatment in Crawford. Extended example on laser threshold. 
8  3.4 – 3.5  Pitchfork bifurcation. Incorporate treatment in Crawford. Extended example on overdamped bead on rotating hoop. 
9  3.5  Dimensional analysis. Basic technique. Relate to overdamped bead example. 
10  3.6 – 3.7  Imperfect bifurcations. Basic theory and bifurcation diagrams. Insect outbreak model, time permitting. 
11  4.0 – 4.3  Flows on the circle. Definition, beating, nonuniform oscillators, ghosts and bottlenecks. 
12  4.4 – 4.6  Oscillator examples. Instructor should choose one or two of the examples (overdamped pendulum, fireflies, superconducting Josephson junctions) to cover in depth. 
13  5.0 – 5.1  Introduction to twodimensional linear systems. Motivating examples, mathematical setup, definitions, different types of stability. Phase portraits, stable and unstable eigenspaces. 
14  5.2  Classification of linear systems. Eigenvalues, eigenvectors. Characteristic equation, trace and determinant. Different types of fixed points. (Suggestion: cover example material in Section 5.3 and related problems on homework.) 
15  Midterm  
16  6.0 – 6.2  Introduction to twodimensional nonlinear systems. Phase portraits and nullclines. Existence, uniqueness, and strong topological consequences for twodimensions. 
17  6.3  Equiliria and stability. Fixed points and linearization. Effect of nonlinear terms. Hyperbolicity and the HartmanGrobman theorem. 
18  6.5 – 6.6  Special nonlinear systems. Conservative and reversible systems. Heteroclinic and homoclinic orbits. 
19  6.7  Extended application of nonlinear phase plane analysis to classic pendulum problem without restricting to smallangle regime. (Alternatively: another application of the instructor’s choice.) 
20  6.8  Index theory. Discussion of local versus global methods. Definition and useful properties of the index, with examples. 
21  7.0 – 7.1  Introduction to limit cycles. Definition. Polar coordinates. Van der Pol oscillator and other examples. 
22  7.2  Ruling out limit cycles. Gradient systems, Liapunov functions, and Dulac’s criterion, with examples. 
23  7.3  Proving existence of closed orbits. PoincareBendixson theorem, trapping regions. Examples. Impossibility of chaos in the phase plane. 
24  8.0 – 8.1  Bifurcations in two (and more) dimensions. Revisitation of saddlenode, transcritical, and pitchfork bifurcations, with examples. 
25  8.2 – 8.3  Hopf bifurcation. Definition. Supercritical, subcritical, and degenerate types. Application to oscillating chemical reactions if time permits. 
26  8.4  Global bifurcations of cycles. Saddlenode, infiniteperiod, and homoclinic bifurcations. Scaling laws for amplitude and period of limit cycle. 
27  leeway  
28  Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B. Selected topics in differential equations. Laplace transforms, existence and uniqueness theorems, Fourier series, separation of variable solutions to partial differential equations, SturmLiouville theory, calculus of variations, two point boundary value problems, Green’s functions. P/NP or letter grading.
General Information. Differential equations are of paramount importance in mathematics because they are equations whose solutions are functions – not numbers. Differential equations are thus widely used in mathematical models of systems where one wants to determine functional relationships. For example, the concentration of chemical reactants as a function of the time, the temperature on the surface of a heat shield as a function of position, or the size of a loan payment as a function of the interest rate. In fact, in nearly all of the courses in the physical sciences and engineering, and in many courses in the social sciences, differential equations play a fundamental role.
One of the goals of this course is to present solution techniques for differential equations that go beyond what is taught in 33B. In particular, the Laplace transform technique for solving linear differential equations is covered. This technique transforms the task of solving linear differential equations to one of solving algebraic problems. It is also a technique that can be used to solve differential equations containing generalized functions (e.g. discontinuous or Dirac delta functions). Other solution techniques include the method of Fourier series, the method of eigenfunction expansions and perturbation methods.
Another goal of this course is to introduce students to the theory of ordinary differential equations. A key part of this theory is the determination of the existence and uniqueness of solutions to differential equations. Just as it’s a fact that not all algebraic equations have solutions, it’s also a fact that not all differential equations have solutions. The theorems covered are especially useful, as they allow one to determine the existence and uniqueness of solutions without having to solve the differential equation.
Textbook(s)
G. Simmons, Differential Equations with Applications and Historical Notes, 3rd Ed., McGrawHill.
Footnotes
1. The book does not include a review of partial fractions. Most calculus textbooks provide a suitable discussion of the technique.
2. The book only states a limited form of the Heaviside expansion theorem in problem 5 of section 53. The more general statement can be found in standard texts devoted to Laplace transforms.
3. The book provides a limited description of the use of the unitstep function and unit impulse functions. A better treatment can be found in Redheffer’s book Differential Equations.
4. The proof of Theorem B is easier than Theorem A (the local existence theorem) since one doesn’t have to worry about the Picard iterates leaving the domain where f(x,y) is Lipschitz. Thus, discussing and proving Theorem B before Theorem A is recommended.
5. The book glosses over some of the mathematical details required by the convergence proofs so one must supplement the material in the text as needed.
Additional Notes
An energetic instructor may want to cover two point boundary value problems and Green’s functions in more depth instead of spending the last three lectures on the calculus of variations. Alternately, one could replace the lectures on the calculus of variations with lectures on regular perturbation theory. A reference for this latter topic is Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers, Chapter 7.
Outline update: C. Anderson, 5/05
NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

1  General course overview.  
2  17, 18  Review of solution methods and properties of solutions for linear constant coefficient equations. 
3  48, 50, 51  Laplace transform. Forward transform, inverse transform. Examples of transform pairs. 
4  48, 50, 51  The Laplace transform of a differential equation. The use of Laplace transforms for the solution of initial value problems. 
5  48, 50, 51  Computation of the inverse Laplace transform. Partial fraction expansions revisited1. 
6  49  Existence and uniqueness of Laplace transforms. Sectionally continuous functions. Exponentially bounded functions. 
7  52, 53  Proof of the convolution theorem. The Heaviside expansion theorem2. 
8  52, 53  The Heaviside function and Dirac distribution. Unit impulse response functions. Use of the unit impulse response function3. 
9  68, 69  Existence and uniqueness theory. Examples of differential equations without unique solutions or global solutions. Lipschitz condition; determination of Lipschitz constants. 
10  68, 69  Statement of a global existence and uniqueness theorem — when f(x,y) is Lipschitz in [a,b] x [8, 8]4. Examples of the application of the existence and uniqueness theorem. 
11  68, 69  Outline of the proof of existence and uniqueness theorem. Proof preliminaries; max norm, uniform convergence, Weierstrauss Mtest. Equivalence of the differential equation to an integral equation5. 
12  68, 69  Picard iteration. Proof of existence and uniqueness. 
13  68, 69  Local existence and uniqueness theorems. Applications of local existence and uniqueness theorems. 
14  Midterm  
15  33  Periodic functions and Fourier series. The inadequacy of power series approximations for periodic functions. Fourier series coefficient formulas. Examples of Fourier series. 
16  35, 36  Derivation of Fourier series coefficient formulas. Fourier series for periodic functions over arbitrary intervals. 
17  37  Function inner products. Orthogonal functions. Derivation of Fourier series coefficient formulas using inner products. 
18  34, 38  Convergence theorems for Fourier series: Pointwise convergence. 
19  34, 38  Convergence theorems for Fourier series: L2 convergence (Mean convergence). 
20  40  Eigenvalues and Eigenfunctions of two point boundary value problems. 
21  41  Separation of variables solution to one dimensional heat equation. 
22  42  Separation of variables solution to Laplace’s equation in a disk. 
23  43  SturmLiouville problems. 
24  43  Leeway 
25  65, 66, 67  Calculus of Variations: Introduction. 
26  65, 66, 67  Euler’s differential equation for an extremal. 
27  65, 66, 67  Isoperimetric problems. 
28  Review 
General Course Outline
Course Description
Lecture, three hours; discussion,one hour. Prerequisites: courses 33A, 33B. Linear partial differential equations, boundary and initial value problems; wave equation, heat equation, and Laplace equation; separation of variables, eigenfunction expansions; selected topics, as method of characteristics for nonlinear equations.
General Information. Math 136 is offered once each year, in the Spring. Together with 135A in the Fall and 135B in the Winter, it is the third of a natural sequence of courses in differential equations. Note however that the courses 135AB are not required for 136.
Enrollments in Math 136 have oscillated between 30 and 100 over the past several years.
Textbook(s)
W.A. Strauss, Partial Differential Equations, 2nd Edition, John Wiley and Sons.
The course covers Chapters 1, 2, parts of 3, and most of 46.
Schedule of Lectures
Lecture  Section  Topics 

1  1.11.2  The notion of a partial differential equation (PDE), the order of a PDE, linear PDE, examples. First order linear PDE. 
23  1.2  Homogeneous first order linear PDE with constant coefficients. The method of characteristics (geometric method) and the coordinate method. First order linear PDE with variable cofficients. Characteristic curves and the geometric method in the case of variable cofficients. The solvability of the Cauchy problem for a first order linear PDE (the statement only). 
4  1.3  PDE from Physics. Examples: the heat equation (derivation using Fourier’s law), vibrating strings and drumheads, the wave equation and the Laplace equation. Schrodinger’s equation. 
5  1.4, 1.6  Initial and boundary conditions for PDE. Classification of second order linear PDE with constant coefficients. Elliptic and hyperbolic PDE. 
67  2.1  The wave equation on the real line.Traveling waves. The Cauchy problem for the wave equation and the d’Alembert formula. Examples. 
8  2.2  The causality principle for the wave equation. The domain of dependence and the domain of influence. Conservation of energy. 
910  2.32.5  The diffusion/heat equation on the real line. The maximum principle and the uniqueness of the Dirichlet problem for the heat equation. The heat kernel and the solution of the initial value problem for the heat equation on the real line. The smoothing property of the heat flow and the comparison of the main properties of the wave and heat equations. 
11  3.1  The heat equation on the halfline. The Dirichlet and Neumann boundary conditions. The method of reflections. 
12  3.2  The wave equation on the halfline. Reflected waves. (The first part of Section 3.2). 
1314  3.3, 3.4  The inhomogeneous heat equation on the real line. The inhomogeneous wave equation on the real line and the operator method. Duhamel’s principle. (Section 3.4: the proof of Theorem 1 using the operator method) 
15  Review before the midterm.  
16  Midterm.  
17  4.1  Spectral methods for boundary problems on finite intervals. Separation of variables and the wave equation with Dirichlet boundary conditions. The eigenvalues and eigenfunctions on a bounded interval with Dirichlet boundary conditions. The heat equation with Dirichlet boundary conditions. Formal eigenfunction expansions. 
18  4.2  The Neumann boundary conditions for the wave and the heat equations. The eigenvalues and eigenfunctions of on a bounded interval with Neumann boundary conditions. 
19  4.3  The eigenvalues and eigenfuctions on a bounded interval with Robin boundary conditions: a cursory discussion. 
2021  5.15.2  Fourier series and Fourier coefiicients of periodic functions in real and complex form. Fourier series expansions for functions defined on an interval of the form via even and odd extensions. Since and cosine expansions. Examples. 
2224  5.35.4  Symmetric boundary conditions and the orthogonality of eigenfunctions. Convergence theorems for Fourier series, the notions of uniform and L^2convergence. The least square approximation, Bessel’s inequality, and Parseval’s identity. One word about the pointwise convergence of Fourier series. 
2526  6.1  The Laplace equation and harmonic functions. The maximum principle and the uniqueness of the Dirichlet problem. The Laplace operator in polar coordinates and the Newtonian potential in 2D and 3D. 
27  6.2  The Laplace equation and separation of variables in a rectangle. (Section 6.2, may be omitted due to time constraints). 
2829  6.3  The Dirichlet problem in the disc and Poisson’s formula. The mean value property for harmonic functions and their differentiability properties. 
30  Review. 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeavor, such as the physical sciences, biology, economics, and traffic dynamics.
General Information. One section of Math 142 is offered each term. For the past several years the enrollments in the course have run between 35 and 100 students each term.
Textbook(s)
Haberman, R., Mathematical Models, Society for Industrial and Applied Mathematics.
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Prerequisite: courses 32B, 33B. Integral equations, Green’s function, and calculus of variations. Selected applications from control theory, optics, dynamical systems, and other engineering problems.
General Information. The content of Math 146 varies depending on the instructor. The course is usually offered once each year, in Spring Quarter.
Textbook(s)
Troutman, J., Variational Calculus and Optimal Control: Optimization with Elementary Convexity, 2nd Ed., Springer.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct methods for solving linear systems. Matlab programming. Letter grading.
Assignments Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using Matlab.
General Information. Math 151AB is the standard course sequence in numerical analysis, suitable for all Applied Mathematics majors. It trains students in the design and use of algorithms for obtaining approximate solutions to problems in all areas. As such graduates in Applied Mathematics who embark on quantitative careers often find Math 151AB to be very useful.
Math 151A and Math 151B are usually offered every quarter.
Textbook(s)
R. Burden and J. Faires, Numerical Analysis, 10th Ed., Brooks/Cole.
Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using Matlab.
* This topic is not in Burden and Faires. It can be found in CheneyKincaid, Numerical Mathematics and Computing, Brooks/Cole, section 4.2.
Topics in parenthesis are optional and can be included under the discretion of the instructor.
Outline update: J. Qin, 06/2015
NOTE: This outline includes only one midterm. The instructor may prefer to offer 2 midterms. In this case, the syllabus might be modified to cover the content of the first three lectures in two lectures, or by covering the content of lectures 24 through 27 in three lectures, or another change may be made.
Schedule of Lectures
Lecture  Section  Topics 

1  1.2  General course overview and machine numbers 
2  1.2  Errors 
3  1.3  Algorithms and convergence 
4  2.1  The bisection method 
5  2.2  Fixedpoint iteration 
6  2.3  Newton’s method 
7  2.3  Secant method, and method of False Position 
8  2.4  Convergence order. Multiple roots 
9  2.5  Accelerating convergence 
10  2.6  Zeros of polynomials. Horner’s method 
11  2.6, 3.1  Deflation and Lagrange polynomials 
12  3.1, 3.2  Lagrange polynomials and Neville’s method 
13  3.3  Divided differences 
14  3.3  Interpolation nodes and finite difference 
15  Midterm  
16  3.4  Hermite Interpolation 
17  3.5  Cubic spline interpolation 
18  4.1  Forward/backward difference 
19  4.1  Finitedifference formulas 
20  4.2, 4.3.  Richardson’s extrapolation. Interpolation based numerical integration 
21  4.3, 4.4  NewtonCotes formulas. Composite integration formulas 
22  4.5  Romberg integration 
23  4.7  Gaussian quadrature 
24  6.1  Solving linear systems 
25  6.2  Pivoting 
26  6.6  Special types of matrices 
27  7.1, 7.3  Review of matrix algebra. Jacobi’s method 
28  7.3  GaussSeidel method 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 151A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation. Numerical solution of ordinary differential equations. Iterative solution of linear systems. Computation of least squares approximations. Discrete Fourier approximation and the fast Fourier transform. Matlab programming. Letter grading.
Assignments Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.
General Information. Math 151AB is the standard course sequence in numerical analysis, suitable for all Applied Mathematics majors. It trains students in the design and use of algorithms for obtaining approximate solutions to problems in all areas. As such graduates in Applied Mathematics who embark on quantitative careers often find Math 151AB to be very useful.
Math 151A and Math 151B are usually offered every quarter.
Textbook(s)
R. Burden and J. Faires, Numerical Analysis, 10th Ed., Brooks/Cole.
Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.
AS: The topics of stiffness and of absolute stability are not well presented in Burden and Faires. Other textbooks should be consulted.
DLS: The matrix form of the discrete least squares problem is not presented in Burden and Faires. Other textbooks should be consulted.
Outline update: J. Qin, 06/2015
NOTE: This outline includes only one midterm. The instructor may prefer to offer 2 midterms. In this case, the syllabus will be modified slightly.
Schedule of Lectures
Lecture  Section  Topics 

1  5.1  Initial value problem 
2  5.2  Euler’s method 
3  5.3, 5.10  Higherorder Taylor methods. Error analysis of onestep methods 
4  5.10, 5.4  Stability of onestep methods. Taylor Theorem in two variables 
5  5.4  RungeKutta methods 
6  5.4  Butcher tableau. Design a RungeKutta method 
7  5.5  RungeKuttaFehlberg method. 
8  5.6  AdamsBashforth/AdamsMoulton multistep methods 
9  5.6, 5.10  Predictorcorrector methods. Analysis of general multistep methods 
10  5.10, 5.11  Stability of multistep methods. Stiff differential equations 
11  5.11  Region of absolute stability 
12  5.9  Highorder differential equations. Systems of differential equations 
13  11.1  Boundary value problems. Linear shooting method 
14  11.2, 11.3  Nonlinear shooting method. Finite difference methods for linear BVP 
15  Midterm  
16  11.4  Finitedifference methods for nonlinear BVP 
17  10.1, 10.2  Solving nonlinear systems of equations. Newton’s method 
18  10.3  QuasiNewton method – Broyden’s method 
19  10.4  Steepest descent method 
20  10.5  Homotopy and continuation methods 
21  9.1, 9.2  Linear algebra, Eigenvalues, orthogonal matrices and similarity transformations 
22  9.3  Power method. Inverse Power method 
23  9.4  Householder’s transformation. Householder’s method 
24  9.5  QR factorization. QR algorithm 
25  8.1  Discrete least squares approximation. Linearly independent functions 
26  8.2  Orthogonal polynomials and least squares approximation 
27  8.5  Continuous and discrete trigonometric polynomial approximation. 
28  8.6  Fast Fourier transform I 
29  8.6  Fast Fourier transform II 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 131A, Programming in Computing 10A or equivalent. Rigorous introduction to numerical algorithms including necessary skills to apply algorithms in statistics, imaging, data science, engineering and related fields. Root Finding, solving linear systems, interpolation, quadrature and finding eigenvalues. MatLab programming. P/NP or letter grading.
Textbook:
L. Ridgway Scott, Numerical Analysis, Princeton University Press.
General Course Outline/Schedule of Lectures:
Week  Chapter  Topics 
1  1, 18  Introduction to finite precision arithmetic and algorithms. Convergence and Stability. Floating point numbers, their arithmetic and errors. Big “O” notation. 
2  2  Fixedpoint algorithms. Applications to rootfinding. Newton’s method and the secant method. Connections with optimization. Error analysis. 
3  5, 6  Review of linear algebra. Vector spaces and norms. Infinite dimensional vector spaces. Operators and operator norms. Inner products. Powers and convergence of matrices. 
4  3  Basic numerical methods for linear systems. Guassian elimination. Triangular matrices and the LU decomposition. Pivoting rules. Cholesky decomposition. Application to banded matrices. 
5  8  Iterative methods for linear systems. Jacobi and GaussSeidel methods. Convergence analysis for these algorithms. Application to sparse linear systems. Matrix splittings in general. 
6  7  System of nonlinear equations. Functional iteration. Newton’s method and quasiNewton’s method. Bilevel procedures for fixed point problems. 
7  10, 11  Polynomial interpolation. Connection to linear systems. Relationship between Taylor polynomials and Lagrange polynomials. Higher order interpolation schemes such as Hermite polynomials. Approximation with trigonometric series. 
8  12  Introduction to approximation theory. Lebesgue and Sobolev spaces of functions. Weierstrass Thoerem. Bernstein polynomials. Splines. Connection between polynomials approximation and least squares. 
9  13  Numerical quadrature. “Basic” schemes such as trapezoidal and Simpson’s. Gaussian quadrature. Composite schemes. 
10  14  Introduction to eigenvalue problems. Some sample applications. Gershgorin’s disks. Finding all vs. finding highest eigenvalue. Power method. Hessenberg factorizations and finding all eigenvalues. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 115A or 115AH, 131A or 131AH, 151A or 151AH, Computer Science 31 or Programming in Computing 10A, with grades of B or better. Rigorous introduction to numerical algorithms including necessary skills to apply algorithms in statistics, imaging, data science, engineering and related fields. Finding eigenvalues, finding numerical solutions to ordinary differential equations, the least squares problem and the fast Fourier transform. MatLab programming. Honors course parallel to course 151B. P/NP or letter grading.
Course Objectives
1. Students will acquire an understanding of the background theory, the derivation, and the implementation, of foundational methods of numerical approximation.
2. Students will learn to analyze concrete problems that arise in practice, and choose and implement appropriate numerical methods for their solution.
3. Students will learn how to assess the accuracy of approximations as function of the algorithms employed and the data used.
4. Areas covered in 151BH include numerical methods for finding eigenvalues and eigenvector/eigenvalue pairs, methods for numerical solution of ordinary differential equations, including systems and boundary value problems, solution of least squares problems, and elementary Fourier theory, including the Fast Fourier Transform and some of its applications.
Textbook:
L. Ridgway Scott, Numerical Analysis, Princeton University Press. (LSR)
R. Burden and J. Faires, Numerical Analysis, 10th Ed., Cengage. (BF)
Grade policy:
Homework 40%
Midterm 25%
Final exam 35%
General Course Outline/Schedule of Lectures:
Week  Chapter  Topics 
1  BF: 8.1 – 8.5  Brief review of linear algebra. The least squares problem. QR decompositions, Householder trans formations.

2  BF: 8.1 – 8.5 LSR: 9  The conjugate gradient method. The Kacsmarz method. Ridge regression and LASSO.

3  LSR: 14  Introduction to eigenvalue problems. Some sample applications. Gershgorin’s disks. Finding all vs. finding highest eigenvalue. Power method. Hessenberg factorizations and

4  LSR: 15  Eigenvalue algorithms. Power method, inverse iteration and deﬂation. Singular Value Decomposition. Finding all eigenvalues using QR decomposition and using Jacobi iteration.

5  BF: 5.9 LSR: 16  Ordinary diﬀerential equations. Existence and uniqueness of solutions. Euler and implicit Euler methods. Error estimates.

6  BF: 5.4 LRS: 17  Systems of diﬀerential equations and higher order diﬀerential equations. Higher order solvers for initial value problems. RungeKutta.

7  BF: 5.6, 5.10, 5.11  Stability for numerical ODE solvers. Implicit schemes such as AdamsMoulton. Multistep and predictor corrector schemes. Stability.

8  BF: 11.1 – 11.4  Boundary value problems. Linear and nonlinear shooting methods. Finite diﬀerence methods.

9  BF 8.5, 8.6  Trigonometric polynomial approximation. Elementary Fourier theory. The fast Fourier transform.

10 
 Review and catchup.

Core Competencies: This course addresses Critical Thinking, Informational Literacy and Quantitative Reasoning:
• Critical Thinking: students learn to apply methods of the field of numerical algorithms in a principled manner to problems arising in statistics, imaging, data science, engineering and related fields.
• Informational Literacy: students will learn key concepts, methods, and results of the theory and practice of numerical algorithms.
• Quantitative Reasoning: all content and problems in this course are quantitative in nature.
Learning Outcomes: By the end of the course, students will:
• Recognize eigenvalue problems in realworld applications. Understand various schemes for finding eigenvalue/eigenvector pairs, such as the power method and Lanczos method. Know when various methods are appropriate.
• Understand the existence and uniqueness for ordinary differential equations (ODE’s). Be able to apply simple ODE solvers (such as Euler’s method) as well as more sophisticated methods (such as RungeKutta or PredictorCorrector) to initial value problems. Understand the difference between in initial value problem and a boundary problem. Be able to apply shooting methods and finite difference methods to boundary value problems.
• Appreciate the importance of the least squares problem. Be able to solve least squares problems using QR factorization, conjugate gradient method, gradient descent and Kaszmarz method.
• Realize the ubiquity of the Fourier transform in modern communication and computation systems. Understand and be able to implement the Fast Fourier Transform.
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A. Imaging geometry. Image transforms. Enhancement, restoration, and segmentation. Descriptors. Morphology. P/NP or letter grading.
Math 155 is an introductory course on mathematical models for image processing and analysis. The students will become familiar with basic concepts (such as image formation, image representation, image quantization, change of contrast, image enhancement, noise, blur, image degradation), as well as with mathematical models for edge and contour detection (such as the Canny edge detector), filtering, denoising, morphology, image transforms, image restoration, image segmentation, and applications. All theoretical concepts will be accompanied by computer exercises.
Textbook(s)
R. Gonzalez and R. Woods, Digital Image Processing, New edition, PrenticeHall. Book is Subject to Change Without Notice.
Outline update: L. Vese, 2/03
Schedule of Lectures
Lecture  Section  Topics 

1  2.22.3  Introduction: A Simple image model (2.2); Sampling and Quantization (2.3) 
2  2.5  Imaging Geometry 
3  3.1  Introduction to the Fourier Transform 
4  3.2  The Discrete Fourier Transform 
5  3.3  Some Properties of the TwoDimensional Fourier Transform 
6  3.3  Some Properties of the TwoDimensional Fourier Transform 
7  3.4  The Fast Fourier Transform 
8  3.5  Other Separable Image Transforms 
9  3.5  Other Separable Image Transforms 
10  3.6  The Hotelling Transform 
11  4.1  Image Enhancement 
12  4.2  Enhancement by Point Processing 
13  4.2  Enhancement by Point Processing 
14  4.3  Spatial Filtering 
15  4.4  Enhancement in the Frequency Domain 
16  5.1  Image Restoration: Degradation Model 
17  5.4  Inverse Filtering 
18  7.1  Detection of Discontinuities 
19  7.2  Edge Linking and Boundary Detection 
20  7.3  Thresholding 
21  7.4  RegionOriented Segmentation 
22  7.5  The Use of Motion in Segmentation 
23  8.1  Representation Schemes 
24  8.2  Boundary Descriptors 
25  8.3  Regional Descriptors 
26  8.4  Morphology 
27  8.4  Morphology 
28  8.4  Morphology 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Requisite: course 115A, 164, 170E (or 170A or Statistics 100A) and Programming in Computing 10A of Computer Science 31. Strongly recommended requisite: Program in Computing 16A or Statistics 21. Introductory course on mathematical models for pattern recognition and machine learning. Topics include parametric and nonparametric probability distributions, curse of dimensionality, correlation analysis and dimensionality reduction, and concepts of decision theory. Advanced machine learning and pattern recognition problems, including data classification and clustering, regression, kernel methods, artificial neural networks, hidden Markov models, and Markov random fields. Projects in MATLAB to be part of final project presented in class. P/NP or letter grading.
Textbook(s)
?Pattern Recognition and Machine Learning?, by Christopher M. Bishop, Springer, 2006 (ISBN13: 9780387310732), plus complementary sources where necessary (?n/a?).
Schedule of Lectures
Lecture  Section  Topics 

1  1.2, 1.51.6, 2.32.5  Introduction, Definitions, Prerequisites. Course Introduction, recap on Linear Algebra, probabilities. Gaussian, exponential pdf; Learning parametric pdf. Learning nonmetric pdf. 
2  12.112.4  Correlation Analysis, dimensionality reduction, PCA. PCA: maximum variance, minimum error, highdimensional PCA. Probablilistic PCA (MLPCA, EM, Bayesian PCA). Nonlinear latent variable models: ICA, kPCA 
3  3.1, 3.3, 3.5  Regression. Linear Basis Function Models, least squares and maximum likelihood. Bayesian linear regression. Evidence Approximation. 
4  4.1, 4.3, 14.3  Classification. Disriminant functions; least squares. Logistic regression. Mixture of linear classifiers: Boosting and Bagging. 
5  9.19.2  Clustering. KMeans, Gaussian mixture model, ExpectationMaximization, Spectral clustering. 
6  6.16.2, 6.4, 7.1  Kernel methods. Dual representation, kernel trick; Constructing kernels. Gaussian processes, GP regression, GP classification. Support vector machines, kSVM. 
7  4.1.7, 5.15.3  Artificial neural networks. Biological motivation; the perceptron; Feedforward Network. Single Layer network training. Multilayer perceptron training: Backpropogation. 
8  8.1, 8.3, 13.113.2  Markov models. Bayesian Networks. Markov Random Fields; Iterated conditional modes (SA, graphcuts). Hidden Markov Models; forwardbackward, Viterbi algorithm. 
9  N/A  Advanced Topics (optional). Reinforcement learning, Bellman optimality. VapnikChervonenkis (VC) dimension; overfit and underfit. Probably approximately correct (PAC) learning. 
10  N/A  Leeway (to accommodate midterm and holidays in the preceding weeks). Review. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A, 131A. Not open for credit to students with credit for Electrical Engineering 136. Fundamentals of optimization. Linear programming: basic solutions, simplex method, duality theory. Unconstrained optimization, Newton’s method for minimization. Nonlinear programming, optimality conditions for constrained problems. Additional topics from linear and nonlinear programming. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 27 lectures. The remaining classroom meetings are for leeway, reviews, and a midterm exam. These are scheduled by the individual instructor.
General Information. Math 164 provides an introduction to the theory and algorithms concerned with finding extrema (maxima and minima) of functions subject to constraints.
After a review of topics from multivariable calculus such as the gradient, Hessian, Jacobian, Taylor series, and linear algebra, the course offers the students a working knowledge of optimization theory and methods for linear and nonlinear programming, that is, how to find extrema of linear and nonlinear functions subject to various kinds of constraints.
There are ample opportunities for the students to improve their ability to read and write mathematical proofs as well as to solve applied and theoretical problems.
Textbook(s)
E. K.P. Chong and S. Zak, An Introduction to Optimization, 4th Edition, Wiley.
Outline update: W. Yin, 6/15
Schedule of Lectures
Lecture  Section  Topics 

1  15 6  Review vector space, transforms geometry, calculus Optimization models, constraints 
2  7  1D search methods 
3  8  Gradient methods, steepest descent method 
4  9  Newton’s method 
5  10  The Conjugate Direction methods 
6  12  Midterm 
7  15  Intro. to linear programming, polyhedron 
8  –  Continuation 
9  20  Nonlinear optimization with equality constraints 
10  23 TBA  Algorithms for constrained optimization Catchup, Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Quantitative modeling of strategic interaction. Topics include extensive and normal form games, background probability, lotteries, mixed strategies, pure and mixed Nash equilibria and refinements, bargaining; emphasis on economic examples. Optional topics include repeated games and evolutionary game theory. P/NP or letter grading.
Outline update: D. Blasius, 5/02
NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.
Schedule of Lectures
Lecture  Section  Topics 

1  2.1  Example and graphical solution 
2  2.22.3  Make eplicit definitios in 2×2 case and minmax. Minmax Statement 
3  2.4  Solving 0sum games. 
4  2.4  2.4 Continued 
5  2.5  Nash equilibria (mutual best responses) 
6  2.6  Proof of minmax: assume separating hyperplane theorem and derive proof. Do planar n by 2 case first with pictures. 
7  Prove separating hyperplanes  
8  Work on good problems in class  
9  3.1  Work on 3.1 material 
10  3.2  Work on 3.2 material 
11  3.2  Introduce noncooperative (aka general sum) game 
12  4.1  Basic 2 x 2 examples (PD, DoveHawk, etc.) 
13  4.2  Solve two player NE’s (2×2, 3×3 case) 
14  Review for Midterm  
15  Midterm  
16  4.3  Many player NE’s 
17  4.3  Many player NE’s (cont.) 
18  4.4  Potential games 
19  4.5  Tragedy of the Commons. 
20  Beginning of proof of NE’s: definition of a convex correspondence.  
21  
22  7.1  Proof NE’s exist based on Kakutani (convex correspondences have fixed points) 
23  7.2  Review for Midterm 
24  Midterm 2 (L13)  
25  Price of Anarchy, Chapter 8  
26  
27  Stable matching, Chapter 10  
28  Review 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 115A, 170A or Electrical and Computer Engineering 131A or Statistics 100A. Introduction to network science (including theory, computation, and applications), which can be used to study complex systems of interacting agents. Study of networks in technology, social, information, biological, and mathematics involving basic structural features of networks, generative models of networks, network summary statistics, centrality, random graphs, clustering, and dynamical processes on networks. Introduction to advance topics as time permits. P/NP or letter grading.
Course Information:
Students will develop a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of networks. The study of networks is predominantly a modern subject, so the students will also be expected to develop the ability to read and understand current research papers in the field. They will also have a chance to explore a topic in depth in a final project. Topics include basic structural features of networks, generative models of networks, centrality, random graphs, clustering, and dynamical processes on networks.
Textbook(s)
Mark E. J. Newman, Networks 2nd Edition, 2018 [primary text]
Mason A. Porter and James Gleeson, Dynamical Systems on Networks: A Tutorial, 2016
Supplementary material from survey, review, and tutorial articles.
Schedule of Lectures
Lecture  Section  Topics 

13 
Newman 16, 8 
Introduction and Basic Concepts 
46 
Newman 7.9, 15.1 
SmallWorld Networks 
710 
Newman 14 
Models of Network Formation 
1113 
Newman 7 + supplementary material 
Network Summary Statistics 
1417 
Newman 1213 
Random Graphs 
1821 
Newman 11 + supplementary articles 
Clustering in Networks 
2225 
Newman 1618; Porter & Gleeson 
Dynamical Processes on Networks 
2627 

Introduction to Advanced Topics 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33A, 131A. Not open to students with credit in course 170E, Electrical Engineering 131A or Statistics 100A. Rigorous presentation of probability theory based on real analysis. Probability space, probability and conditional probability, independence, Bayes? rule, discrete and continuous random variables and their distributions, expectation, moments and variance, conditional distribution and expectation, weak law of large numbers. P/NP or letter grading.
Course Information:
The course discusses the foundations of probability as a mathematical discipline rooted in undergraduate real analysis. At the end of the course, the students will have the tools and ability to formulate, analyze an answer questions in probability and prove the validity of their reasoning in full mathematical rigor.
Textbook(s)
Probability: An Introduction (2nd ed.). by Grimmett, G. R., & Welsh, D. J. (2014).Oxford: Oxford University Press.
Outline update: T. Austin, 01/20
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Sections 1.11.5, 1.9  Sample space, events, probability 
Week 2  Sections 1.61.7  Conditional probability and independence 
Week 3  Sections 1.8, 1.10  Partition theorem and Bayes rule, examples 
Week 4  Sections 2.12.4  Discrete random variables, their functions, expectation and variance 
Week 5  Sections 2.5, 3.13.2  Conditional expectation, multivariate discrete distributions 
Week 6  Sections 3.33.5  Independent discrete random variables, indicators 
Week 7  Sections 5.15.6  Cumulative distribution function, continuous random variables 
Week 8  Sections 6.16.4  Multivariate distributions and their marginals 
Week 9  Sections 6.56.7  Change of variables, conditional expectation 
Week 10  Sections 6.8, 7.3, 8.18.2  Mutlivariate normal distribution, weak law of large numbers 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Enforced requisite: courses 170A, 131A. Continuation of rigorous presentation of probability theory based on real analysis. Moments and generating functions; laws of large numbers, the central limit theorem, and convergence in distribution; branching processes; random walks; Poisson and other random processes in continuous time. Advance topics in probability theory. P/NP or letter grading.
Additional Information Advanced Topics:
Instructor selection of three more advanced topics. Possibilities include, but are not limited to:
(1) order statistics, extreme values, and Poisson processes;
(2) basics of entropy and information theory;
(3) theory of sampling and confidence in statistics;
(4) the probabilistic method in combinatorics, including lower bounds for Ramsey numbers;
(5) BorelCantelli lemmas, the strong law of large numbers, and Borel’s normal number theorem.
Instructor will provide notes or reference materials.
Textbook(s)
Probability: An Introduction (2nd ed.). Oxford: Oxford University Press. by Grimmett, G. R., & Welsh, D. J. (2014).
Schedule of Lectures
Lecture  Section  Topics 

13  Review from 170A: probability spaces, random variables, and distributions; multivariate distributions and independence; discrete and continuous conditional probability  
46  4.2–4.4, 7.1–7.4, 7.6  Moments, probability and moment generating functions, characteristic functions 
79  8.1–8.3, 8.5  Inequalities. Laws of large numbers. The central limit theorem. Convergence in distribution. 
1012  9.1–9.5  Branching processes and the method of generating functions, probability of extinction 
1315  10.1–10.4  Random walks on the integers: recurrence vs transience, gambler’s ruin 
1618  11.1–11.4  Poisson processes and their interarrival times. Population growth, birth processes 
1921  11.5, 11.6  Birthanddeath processes, queueing models 
2224  *Advanced Topic 1  
2527  *Advanced Topic 2  
2830  *Advanced Topic 3 
General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Not open to students with credit for course 170A, Electrical and Computer Engineering 131A, or Statistics 100A. Introduction to probability theory with emphasis on topics relevant to applications. Topics include discrete (binomial, Poisson, etc.) and continuous (exponential, gamma, chisquare, normal) distributions, bivariate distributions, distributions of functions of random variables (including moment generating functions and central limit theorem). P/NP or letter grading.
Textbook(s)
Hogg, Tanis, Zimmerman Probability and Statistical Inference (10th Edition)
Outline Updated 10/17
Schedule of Lectures
Lecture  Section  Topics 

1  1.1  Basic Properties of Probability 
2  1.2  Methods of Counting 
3  1.3  Conditional Probability 
4  1.4  Independence 
5  1.5  Bayes’ Theorem 
6  2.1  Discrete Random Variables 
7  2.2  Expectation 
8  2.3  Examples of Expectation 
9  2.4  Binomial Distribution 
10  2.5  Negative Binomial Distribution 
11  2.6  Poisson Distribution 
12  3.1  Continuous Random Variables 
13  3.2  Examples: exponential, Gamma, Chisquare 
14  Midterm on Chapters 1 and 2  
15  3.3  Normal Distribution 
16  3.4  Add’l models: failure rate, mortality, insurance 
17  4.1  Discrete bivariate distributions 
18  4.2  Correlation 
19  4.3  Conditional Distributions 
20  4.4  Continuous Bivariate Distributions 
21  4.5  Bivariate Normal Distribution 
22  5.1  Functions of a random variable 
23  5.2  Transformations of 2 random variables 
24  5.3  Several Random variables 
25  5.4  Moment generating functions 
26  5.5  Random functions associated to normal distributions 
27  5.6  Central Limit Theorem 
28  5.7  Approximations for Discrete distributions 
29  5.8  Chebyshev’s inequality and convergence in probability 
General Course Outline
Course Description
Lecture, four hours. Requisites: courses 31A, 31B, and 170E. The Math 170E and 170S two quarter probability and statistics sequence is aimed to equip MathEcon and Financial Actuarial majors with essential skills in these areas. Math 170S is an introduction to statistics. Topics include sampling; estimation and the properties of estimators; construction of confidence intervals and hypotheses testing. It is designed to meet the Society of Actuaries’ VEE Requirements for Mathematical Statistics. Letter grading.
Textbook(s)
Hogg, Tanis, Zimmerman Probability and Statistical Inference (10th Edition)
Outline Updated 10/17
Schedule of Lectures
Lecture  Section  Topics 

1  6.1  Descriptive Statistics 
2  6.2  Exploratory Data Analysis 
3  6.3  Order Statistics 
45  6.4  Maximum Likelihood Estimation 
6  6.5  A Simple Regression Problem 
7  6.7  Sufficient Statistics 
89  6.8  Bayesian Estimation 
10  7.1  Confidence Interval for Means 
11  7.2  Confidence Intervals for the Difference of Two Means 
12  7.3  Confidence Interval for Proportions 
13  7.4  Sample Size 
14  7.5  DistributionFree Confidence Intervals for Percentiles 
15  Midterm 1 on Chapters 6 and 7  
1617  8.1  Tests About One Mean 
18  8.2  Tests of the Equality of Two means 
19  8.3  Tests About Proportions 
20  8.4  The Wilcoxon Tests 
21  8.5  Power of a Statistical Test 
22  8.6  Best Critical Regions 
23  8.7  Likelihood Ratio Test 
24  9.1  ChiSquare GoodnessofFit Tests 
25  9.2  Contingency Tables 
2627  9.3  OneFactor Analysis of Variance 
28  9.4  TwoWay Analysis of Variance 
General Course Outline
Course Description
(Formerly numbered 151.)Lecture, three hours; discussion, one hour. Requisites: courses 33A, 170A (or Statistics 100A). Discrete Markov chains, continuoustime Markov chains, renewal theory. P/NP or letter grading.
Additional Information Probability and stochastic processes are used to create and analyze models in a broad range of fields, including statistics, economics, finance, engineering, biology and physics. Mathematics 170AB and 171 are designed to give a firm foundation in this area for students who will work and/or do graduate work in one of these fields. They also provide an excellent background for graduate work in probability and related areas of mathematics.
These courses are particularly well suited to students who plan to take the exams in actuarial science. The second exam in this series (number 110) is on probability and statistics. Mathematics 170AB covers roughly 2/3 of the material on that exam.
Course 170A is multiply listed with Statistics. Usually, two sections are offered each Fall Quarter, one by Mathematics and one by Statistics. Total enrollment in the two sections tends to be about 50.
The three courses are intended as a yearlong sequence. However, it is possible, and not unusual, to take 171 without 170B. In fact, the enrollments in 171 are sometimes larger than in 170B (both are in the 1020 range). Mathematics 170B is offered each Winter Quarter, and Mathematics 171 is offered each Spring.
Course Description
(Formerly numbered 174.) Lecture, three hours; discussion, one hour. Enforced requisites: courses 33A, and 170E (or Math 170A or Statistics 100A). Not open for credit to students with credit for course 174A, Economics 141, or Statistics C183/C283. Mathematical modeling of financial securities in discrete and continuous time. Forwards, futures, hedging, swaps, uses and pricing (tree models and BlackScholes) of European and American options, Greeks and numerical methods. P/NP or letter grading.
Textbook(s)
Hull, John C., Options, Futures and Other Derivatives, 10th Edition. Pearson 2018.
It is recommended to run course with one midterm in Week 6 and quizzes in discussion section in Weeks 2, 4, 8, and 10 whose total value is one midterm.
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Ch. 13  Forwards, Futures, Options; Types of Traders; Examples of positions. 
Week 2  Ch. 34  Hedging Using Futures, Interest Rates (zero, forward, term structure) Bonds (duration, convexity) 
Week 3  Ch. 7 & 10  Swaps, Mechanics of Option Markets, Basic Properties of Stock Options (PutCall Parity, Upper and Lower Bounds for Prices, Effect of Dividends) 
Week 4  Ch. 12  Trading Strategies 
Week 5  Ch. 13  Binomial Tree Model of Option Pricing (include Proof in Appendix of Black Scholes model) 
Week 6  Ch. 14  Wiener Process (Brownian Motion) and Ito?s Lemma (include proof as per Appendix) 
Week 7  Ch. 15  BlackScholes model (include risk neutral derivation in appendix) 
Week 8  Ch. 19  The Greeks 
Week 9  Ch. 17 or Ch. 20  Instructor Choice: Do topics from Chapter 17 (Options on Stock Indices and Currencies) and Chapter 20 Volatility Smiles (Concerns deviation of realworld pricing from BlackScholes model). 
Week 10  Ch. 21  Basic Numerical Procedures 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 32B. Types of interest, time value of money, annuities and similar contracts, loans, bonds, portfolios and general cash flows, rate of return, term structure of interest rates, duration, convexity and immunization, interest rate swaps, financial derivatives, forwards, futures, and options. Letter grading.
Course Information:
An introductory course on financial mathematics, Math 177 lays the foundation and prepares students for the series of courses required for the Financial Actuarial Mathematics major. By the end of this course, students should be familiar with numerous foundational concepts of financial mathematics, especially those from the theory of interest rates. Since one goal of the course is to help students prepare for the challenging Financial Mathematics (FM) exam) for the Society of Actuaries (SOA), two lectures before the midterms will be devoted to analysis of complex FM exam problems. While the basic ideas are mathematically elementary, their applications can be complex. The class is suitable for students who seek a career in financial engineering, the actuarial field, banking, etc., or are seeking to improve their financial literacy in a highly quantitative way.
Textbook(s)
Broverman, Samuel A. Mathematics of Investment and Credit. 7th ed., Actex Publications, 2017.
Bean, Michael A. (FSA, CERA FCIA, FCAS, PHD). Determinants of Interest Rates. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
https://www.soa.org/Files/Edu/2017/fmdeterminantsinterestrates.pdf
Alps, Robert (ASA, MAAA). Using Duration and Convexity to Approximate Change in Present Value. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
https://www.soa.org/Files/Edu/2016/edu2016fm2417usingdurationconv…
Beckley, Jeffrey (FSA, MAAA). Interest Rate Swaps. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
Https://www.soa.org/Files/Edu/2017/fminterestrateswaps.pdf
Schedule of Lectures
Lecture  Section  Topics 

1  Intro, 1.01.4  Simple, compound, nominal and effective interest rates. Accumulation. Equation of value, actuarial notation. 
2  1.5, 1.6  Effective and nominal discount rates, force of interest. 
3  Determinants of interest rates.  
4  2.1  Level payment annuities. 
5  2.2, 2.3  Nonconstant payments and other generalizations. 
6  2.4  Yield and reinvestment rates, depreciation. 
7  3.1, 3.2  Amortization of loans. 
8  4.1, 4.2  Determination of bond prices and amortization of a bond. 
9  4.3, 4.4  Examples of bonds and applications. 
10  5.1  Internal rate of return defined and net present value. 
11  5.2, 5.3  Other methods (dollarweighted and time weighted) and examples of rate of return. 
12  Review/Leeway.  
13  Advanced problem analysis from Weeks 14.  
14  Advanced problem analysis from Weeks 14.  
15  Midterm  
16  6.1, 6.2  Basic definitions, spot rates. 
17  6.3  Forward rates. 
18  6.4  Applications and examples of arbitrage, forward rate agreements and atpar yield. 
19  7.1.17.1.2  Macaulay duration and modified duration. 
20  7.1.37.1.5, 7.2  Application to valuation of cash flows, dependence on term structure. 
21  7.2  Convexity and immunization. 
22  Definitions, determining swap rate.  
23  Case of constant notional amount, net payments.  
24  Market value of a swap.  
25  Advanced problem analysis from Weeks 68.  
26  Advanced problem analysis from Weeks 68.  
27  Midterm  
28  9.19.4  Derivatives, dividend discount model, short sale of stock, equity investments, financial derivatives. 
29  9.5  Forward contracts. 
30  9.6, 10.110.2  Futures, options. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 32B, 175 or 177, 170A or 170E or Statistics 100A. An introductory course on to the mathematics associated with long term insurance coverages. Single and multiple life survival models, annuities, premium calculations and policy values, reserves, pension plans and retirement benefits. Letter grading.
Course Information:
A core sequence course for the Financial Actuarial Mathematics major, Mathematics 178A and the first half of Mathematics 178B cover the syllabus of the Society of Actuaries (SOA) LongTerm Actuarial Mathematics (LTAM) exam. By the end of this course, students will be able to value and set premiums for insurance instruments of numerous types using traditional actuarial models. They will also understand the typical models of life contingencies which are used in the calculations.
Textbook(s)
Dickson, David C.M., Hardy, Mary R. and Waters, Howard R, Actuarial Mathematics for Life Contingent Risks. 2nd ed., Cambridge University Press, 2013.
Schedule of Lectures
Lecture  Section  Topics 

1  1  Life insurance and annuity contracts, pension benefits, mutual and proprietary insurers. 
2  2.2  Future lifetime random variable. 
3  2.3  Force of mortality. 
4  2.42.5  First actuarial notation and basic properties of TX. 
5  2.62.7  Curtate future lifetime, further discussion and exercises. 
6  3.13.3.1  Life tables, fractional age assumptions. 
7  3.3.23.6  National life tables, survival models for life insurance holders, survival models for life insurance, life insurance underwriting. 
8  3.73.9  Select and ultimate survival models, select life tables. 
9  3.103.13  Heterogeneity in mortality, mortality trends and sample problems. 
10  4.14.4.3  Whole life insurance (continuous, annual, 1/mthly case). 
11  4.4.44.4.7  Recursions, term insurance, pure endowment and endowment insurance. 
12  4.4.84.5.2  Deferred insurance benefits, uniform distribution of deaths assumption, claims acceleration approach. 
13  4.64.8  Pure endowment, endowment insurance, deferred insurance benefits. 
14  Advanced problem analysis.  
15  Midterm  
16  5.15.4.2  Whole life annuity due and term life annuity. 
17  5.4.35.7  Whole life immediate, term life immediate, whole life continuous, term continuous, payable 1/ mthly cases, comparison by payment frequency. 
18  5.55.10  Deferred, guaranteed, increasing cases. 
19  5.115.14  Evaluating annuity functions, recursions, applying UDD assumption, Woolhouse?s formula. 
20  6.16.4  Present value of future loss random variable. 
21  6.56.6  Net and gross premiums. 
22  6.7  Profit 
23  6.86.10  Portfolio percentile maximum principle, extra risks. 
24  7.17.3.1  Policies with annual cash flows, future loss random variable. 
25  7.3.2  Case of policies with annual cash flows. 
26  7.3.3  Recursive formulas for policy values. 
27  7.3.4, 7.4  Annual profit by source, case of policies with cash flows at 1/mthly. 
28  7.5  Case of continuous cash flows. 
29  7.87.10  Negative policy values, deferred acquisition expenses and modified premium reserves, net premium approach. 
30  Leeway/Review. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: 170S (or 170B or Statistics 100B), 178A. The second of the three quarter sequence 178ABC. Multiple state models, pensions, health insurances, profit testing. Topics in statistics used in actuarial work: methods of estimation and probability distributions. Letter grading.
Course Information:
Mathematics 178A and the first half of Mathematics 178B will almost completely cover the syllabus of the LongTerm Actuarial Mathematics exam by the Society of Actuaries. At the end of Mathematics 178A, students learned to value and set premiums for different types of insurances using traditional actuarial models. They were also exposed to typical models and calculations used in life contingencies. Mathematics 178B first extends this work to multistate models and then covers pensions, health insurances, and profittesting. The last three weeks of the course will cover the probability distributions employed in most common actuarial theory and begins the study of the Short Term Actuarial Mathematics syllabus by the Society of Actuaries.
Textbook(s)
(DHW)
Dickson, David C.M., Hardy, Mary R. and Waters, Howard R., Actuarial Mathematics for Life Contingent Risks. 2nd ed., Cambridge University Press, 2013.
(Hardy)
Hardy, Mary R., LongTerm Actuarial Mathematics Study Note. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Long Term Actuarial Mathematics Supplementary Note.
https://www.soa.org/Files/Edu/2018/2018ltamsupplementarynote.pdf
(KPW)
Klugman, Stuart A., Panjer, Harry H. and Willmot, Gordon E., Loss Models: From Data to Decisions. 3rd Edition, Wiley, 2012.
Schedule of Lectures
Lecture  Section  Topics 

1  DHW 8.1 – 8.3  Examples, assumptions and notations of multiple state models. 
2  DHW 8.4 – 8.6  Probability formulae and computations, Kolmogorov equations, premiums. 
3  DHW 8.7  Policy values, Thiele?s Differential Equation 
4  DHW 8.8 – 8.9  Multiple decrement models 
5  DHW 8.10 – 8.12  Multiple decrement tables 
6  DHW 8.13  Discrete time models 
7  Hardy 2  Disability income, long term care, critical illness insurance, continuing care communities 
8  Hardy 3  Policy value recursions 
9  Hardy 4.1 – 4.3  Mortality improvement modelling 
10  DHW 9.1 – 9.4  Joint life and last survivors benefits, independent future lifetimes 
11  DHW 9.4 – 9.5  Independent future lifetimes (cont.), multiple state model for independent future lifetimes 
12  DHW 9.6 – 9.8  Model with dependent future lifetimes, common shock model 
13  DHW 10.1 – 10.4  Salary scale function, DC contribution 
14  DHW 10.5  Service table 
15  DHW 10.6  Benefit valuation 
16  DHW 10.7  Funding benefits 
17  Review/Leeway  
18  Midterm  
19  DHW 12.1 – 12.4  Introduction to profit testing and principles 
20  DHW 12.5 – 12.8  Profit measures. Using profit test to calculate premium and reserves, case of multiple state models 
21  KPW 3.1 – 3 3  Introduction to tails 
22  KPW 3.1 – 3.3  Basic distributions (moments, percentiles, generating functions and sums of random variables) 
23  KPW 3.4  Tails and their classifications 
24  KPW 3.5  Measures of risk (value at risk, tail value at risk) 
25  KPW 4.1 – 4.2  Continuous actuarial models, background probability 
26  KPW 5.1 – 5.2  Examples of continuous models, creating new distributions 
27  KPW 5.3 – 5.4  Relations between distributions. Linear exponential family 
28  KPW 6.1 – 6.3  Poisson and negative binomial distributions 
29  KPW 6.4 – 6.5  Binomial distributions and (a, b,0) class 
30  KPW 6.6  Truncation and modification at 0 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: 178B. This course is the third of the three quarter sequence 178ABC. 178C studies loss models associated with actuarial problems. It covers severity, frequency, and aggregate loss models, parameter estimation (frequentist, Bayesian), model selection and credibility. Letter grading.
Course Information:
The three quarter sequence 178ABC is the actuarial core of the FAM major. 178C covers topics associated with short term actuarial risk. With 178B, most of the topics 17 on the SOA STAM exam are covered.
Textbook(s)
S. Klugman, H. Panjer, G. Willmot, Loss Models: From Data to Decisions. 3rd Edition, Wiley, 2012.
Hardy, Mary R., LongTerm Actuarial Mathematics Study Note. Society of Actuaries, 2017.
Education and Examination Committee of the Society of Actuaries – Long Term Actuarial Mathematics Supplementary Note.
https://www.soa.org/Files/Edu/2018/2018ltamlossmodels.pdf
Schedule of Lectures
Lecture  Section  Topics 

1  KPW 8.18.2  Deductibles 
2  KPW 8.38.4  Loss elimination ratio, policy limits 
3  KPW 8.58.6  Coinsurance, deductibles, limits, impact of deductibles on claim frequency 
4  KPW 9.19.2  Introduction to aggregate loss models and model choice 
5  KPW 9.3  Compound model 
6  KPW 9.3  Continued and examples. 
7  KPW 9.4  Other closed form results 
8  KPW 9.5, 9.69.6.5 (exclude 9.6.1)  Recursive method, arithmetic discretization 
9  KPW 9.79.8.2  Effect of modifications and individual risk model 
10  Empirical distributions, grouped data  
11  Right censored data  
12  Left truncated data  
13  Approximations for large data sets  
14  Maximum likelihood estimation of decrement probabilities  
15  Estimation of transition intensities  
16  Review/Leeway  
17  Midterm  
18  KPW 13.2  Maximum likelihood estimation 
19  KPW 13.4  Nonnormal confidence intervals and exercises 
20  KPW 14.114.2  Frequentist estimation: Poisson and negative binomial cases 
21  KPW 14.3, 14.4, 14.6  Binomial and (a, b,1) cases and effect of exposure 
22  KPW 15.1  Bayes? Theorem 
23  KPW 15.2  Bayesian inference and prediction 
24  KPW 15.3  Conjugate priors 
25  KPW 16.116.3  Model selection: introductory concepts 
26  KPW 16.4 (except 16.4.2)  Hypothesis testing 
27  KPW 16.5  Selecting a model 
28  KPW 17.117.5  Classical Credibility 
29  KPW 18.2  Conditional Distributions 
30 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 174E. Continuation of Mathematics of Finance. In depth study of risk measures and the instruments of risk management in investment portfolios and corporate financial structure. Exotic and real options, value at risk, meanvariance analysis, portfolio optimization, risk analysis, capital asset pricing model, market efficiency and the ModiglianiMiller theory. P/NP or letter grading.
Textbook(s)
Hull, J. Optios., Futures and Other Derivatives, 10th edition. Pearson, 2018.
Berk, J. and P. DeMarz., Corporate Finance, 4th edition. Pearson, 2017.
White, Toby AMeasures of Investment Risk, Monte Carlo Simulation, and Empirical Evidence on the Efficient Markets Hypothesi Society of Actuaries, 2018. Education and Examination Committee of the Society of Actuaries ? Investment and Financial Markets Study Note.
https://www.soa.org/Files/Edu/2018/ifm2118studynote.pdf
Schedule of Lectures
Lecture  Section  Topics 

1  Hull p.2212, p.2378, p.249, p.3435, p.4603  Effect of Dividends on stock prices and option valuation 
2  Hull 26.126.3, p.598600  Exotic Options 1 
3  Hull 26.426.7, p.601603  Exotic Options 2 
4  Hull 26.826.11, p.603609  Exotic Options 3 
5  Hull 26.1226.14, p.609612  Exotic Options 4 
6  Hull 22.122.3, p. 494504  Value at Risk 
7  Hull 22.422.6, p. 504512  Value at Risk 
8  Hull 22.722.9, p. 512517  Value at Risk 
9  Hull 28.128.3, p. 655660  Market Price of Risk 
10  Hull 35.135.3, p. 792796  Real Options 
11  Hull 35.435.5, p. 796803  Real Options 
12  Berk & DeMarzo 10.110.4, p. 318335  Risk, Return, Diversification 
13  Berk & DeMarzo 10.510.8, p. 335350  Risk, Return, Diversification 
14  Berk & DeMarzo 11.111.3, p. 357369  Portfolio Optimization: Variance and Covariance 
15  Berk & DeMarzo 11.411.5, p. 369381  Portfolio Optimization: Risk versus Return 
16  Midterm  
17  Berk & DeMarzo 11.611.8, p. 381395  Efficient Portfolio, Capital Asset Pricing Model and Risk Premium 
18  Berk & DeMarzo 12.112.2, p. 404413  Cost of Capital: Equity Cost and Market Portfolio 
19  Berk & DeMarzo 12.312.4, p. 407420  Beta Estimation and Debt Cost of Capital 
20  Berk & DeMarzo 12.512.7, p. 420433  Project Cost and Project Risk 
21  Berk & DeMarzo 13.113.4, p. 445455  Role of Investor Behavior 
22  Berk & DeMarzo 13.513.6, p. 456469  Market Portfolio and Efficiency 
23  Berk & DeMarzo 13.713.8, p. 469479  Multifactor Models of Risk 
24  Berk & DeMarzo 14.114.2, p. 487498  ModiglianiMiller: Equity vs. Debt Financing 
25  Berk & DeMarzo 14.314.5, p. 498511  Leverage, Risk, Cost of Capital 
26  Berk & DeMarzo 8.5, p. 258265  Project Analysis: Sensitivity, BreakEven, Scenario 
27  Berk & DeMarzo 16.116.3, p. 551561  Default, Bankruptcy, and Distress 
28  Berk & DeMarzo 16.416.6, p. 562575  Optimal Capital Structure and Leverage 
29  Berk & DeMarzo 16.716.9, p. 575588  Agency Costs and Asymmetric Information 
30  Review/Leeway 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, and 61. Strongly recommended preparation: 115A. Graphs and trees. Planarity, graph colorings. Set systems. Ramsey theory. Random graphs. Linear Algebra methods. Ideal for students in computer science and engineering. P/NP or letter grading.
Course Information:
The following schedule, with textbook sections and topics, is based on 25 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.
Textbook(s)
J. Matousek and J. Nesetril, Invitation to Discrete Mathematics, 2nd Ed., Oxford
Outline update: I. Pak, 12/15
Schedule of Lectures
Lecture  Section  Topics 

1  13  Basic counting methods (induction, pigeonhole principle). 
2  4.1 – 4.3  Graphs, subgraphs, graph isomorphism. Connectivity. Score. 
3  4.4 – 4.7  Eulerian graphs, diagraphs. Hamiltonian cycles. 2connected graphs. 
4  5  Trees, their characterizations, isomorphism. Minimal spanning tree problem. 
5  6  Planar graphs. Euler’s formula. Examples of nonplanar graphs. Five color theorem. 
6  7  Sperner’s Lemma. Set systems. Sperner’s theorem via LYM inequality. 
7  10, 9.4  Probabilistic method (expectation, independence). 2Colorings. Random sorting. Turan’s theorem. 
8  11  Ramsey’s theorem (upper bound, lower bound). 
9  13  Linear algebra methods. Cycle space of a graph. GrahamPollak theorem. Matrix tree theorem. 
10  13.6, 9  Probabilistic checking. Finite projective planes. Applications to graphs with no 4cycles. 
General Course Outline
Course Description
(4) Lecture, three hours; discussion, one hour. Requisite: course 3C or 32A and 61. Lecture, three hours; discussion, one hour. Requisite: course 3C or 32A, and 61. Not open for credit to students with credit for Computer Science 180. Graphs, greedy algorithms, divide and conquer algorithms, dynamic programming, network flow. Emphasis on designing efficient algorithms useful in diverse areas such as bioinformatics and allocation of resources. P/NP or letter grading.
Textbook(s)
Kleinberg, Tardos: Algorithm Design, Addison Wesley
Schedule of Lectures
Lecture  Section  Topics 

Week 1  Introduction, Stable Marriage Problem, GaleShapley algorithm.  
Week 2  Orders of magnitude (Big O notation). Estimating the running time for simple algorithms looking up an entry in a sorted list, mergesort.  
Week 3  Basic graph definitions. Directed graphs, trees, paths. Data structures as graphs: stacks, heaps. Breadth first search, Depth First search, test of bipartitness, DAG’s.  
Week 4  Introduction to the four main classes of algorithms: Greedy, Divide and Conquer, Dynamic programming, Network flow. Application of greedy algorithms to interval scheduling and shortest path problems, minimum spanning trees.  
Week 5  Divide and conquer algorithms. Mergesort, counting inversions, closest pairs of points. Recurrences.  
Week 6  Dynamic programming, weighted interval scheduling, Knapsack problems.  
Week 7  Dynamic programming continued, RNA secondary structures, sequence alignment.  
Week 8  Network flow: Maximum flow problem. Min cuts. Circulations.  
Week 9  Network flow: Airline scheduling, Image segmentation, Project selection.  
Week 10  Introduction to P and NP. 
General Course Outline
Course Description
(Formerly numbered 180.) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, 61 and 115A. Permutations and combinations, counting principles, recurrence relations and generating functions. Application to asymptotic and probabilistic enumeration. Ideal for students in mathematics and physics. P/NP or letter grading.
Textbook(s)
M. Bona, Introduction to Enumerative Combinatorics , 2nd Ed., Chapman and Hall/CRC
Outline update: Pak, I., 12/15
Schedule of Lectures
Lecture  Section  Topics 

1  1.1 – 1.3  Basic counting methods (induction, pigeonhole principle). 
2  1.4 – 2.2  Binomial coefficients, multinomial coefficients, set partitions, Stirling numbers. 
3  2.3 – 2.4  Integer partitions, partitions into odd and distinct numbers. Euler’s Pentagonal theorem. 
4  3.1 – 3.4  Ordinary and exponential generating functions. 
5  4.1 – 4.3  Permutations, Number of cycles and descents. Derangements via InclusionExclusion Principle. 
6  4.4 – 4.5  Inversions. Counting permutation by a cycle type. 
7  5.1 – 5.2  Counting labeled trees. Different proofs of Cayley’s formula. 
8  5.3  Catalan numbers. Plane and binary trees. 
9  5.4 – 5.5  Chromatic polynomial. Enumerations of connected graphs and Eulerian graphs. 
10  9.1 – 9.2  Sequences. Unimodality. Logconcavity. 
General Course Outline
Course Description
(1) Tutorial, three hours. Limited to students in College Honors Program. Designed as adjunct to upperdivision lecture course. Individual study with lecture course instructor to explore topics in greater depth through supplemental readings, papers, or other activities. May be repeated for maximum of 4 units. Individual honors contract required. Honors content noted on transcript. Letter grading.
General Course Outline
Course Description
(Formerly Math 197). Seminar, three hours. Math 191 is a variable topics research course in mathematics. Courses will cover material not covered in the regular mathematics upper division curriculum. Reading, discussion, and development of culminating project. May be repeated for credit with topic and/or instructor change. P/NP or letter grading.
General Course Outline
Course Description
(Formerly Math 190). Math Seminar, three hours. Participating seminar on advanced topics in mathematics. Content varies from year to year. May be repeated for credit by petition. P/NP or letter grading.
General Information Math 191H, the Honors Seminar: Mathematics, is offered once a year, quarter to be determined. The course is open to all Upper Division students who have done reasonably well in their other mathematics courses. Enrollment may be restricted by the instructor. Student participation is required, and the students are charged with presenting most of the material.
The instructor and the topic vary from year to year. The instructor for Spring 2004 is M. Takesaki. Topics treated in years past, and the instructors, are:
Spring 2005: The BanachTarski Paradox, G. Hjorth
Spring 2004: Introduction to Harmonic Analysis, M. Takesaki
Spring 2003: Introduction of Functional Analysis, M. Takesaki
Spring 2002: Matrix Groups, G. Hjorth
Spring 2001: Participating Seminar in Numerical Analysis, C. Anderson
Spring 2000: Introduction to Coding Theory and Information Theory, D. Blasius
Fall 1998: The Theory of Groups and Quantum Mechanics, M. Takesaki
Fall 1997: Basic Examples in Dynamical Systems, R. PerezMarco
Fall 1996: Knots and their Invariants, S. Popa
Winter 1996: Control Theory: Pure and Applied, P. Petersen
Spring 1995: Rational Points on Elliptic Curves, R. Elman
Winter 1994: Fractal Geometry — Mathematical Foundations and Applications, L. Young
Winter 1993: Topics in Elementary Number Theory, M. Green
Winter 1992: Complex Dynamical Systems, T. Gamelin
Winter 1991: Continued Fractions, L. Carleson
Spring 1990: Mathematical Principles of Scientific Computing, B. Engquist
Winter 1990: An Introduction to Chaotic Dynamical Systems, D. Babbitt
General Course Outline
Course Description
Tutorial, to be arranged. Limited to juniors/seniors. Internship to be supervised by Center for Community Learning and Mathematics Department. Students meet on a regular basis with instructor, provide periodic reports of their experience, have assigned readings on mathematics education, and complete final paper. The final paper is a substantial part of course, and will require a significant investment of time during the quarter. May not be repeated and may not be applied toward major requirements. Individual contract with supervising faculty member required. P/NP grading.
General Course Outline
Course Description
(2 to 4 units). Tutorial, three hours per week per unit. Limited to juniors/seniors. At discretion of chair and subject to availability of staff, individual intensive study of topics suitable for undergraduate course credit but not specifically offered as separate courses. Scheduled meetings to be arranged between faculty member and student. Assigned reading and tangible evidence of mastery of subject matter required. May be repeated for maximum of 12 units, but no more than one 197 or 199 course may be applied toward upper division courses required for majors offered by Mathematics Department. Individual contract required. P/NP or letter grading.
General Information The Math 197 title has been used to cover coursework for a course that is listed in the catalog but not given in a particular year. However, Math 197 cannot be used to duplicate the coverage of a regularly offered course. University regulations specify that courses labeled 197 are open only to juniors and seniors with a 3.0 GPA in their major field. Math 197 is intended for students who have already taken a number of Math and PIC courses.
In order to enroll in a 197 course, the student’s petition must receive the approval of the sponsoring faculty member and of the Undergraduate Vice Chair. The petition should spell out student’s obligations are for successful completion of the course, including what will be covered in the course, how often the student will meet with the faculty sponsor, and what written material will be required.
While the 197 course is meant to be flexible, to cover students or groups of students with special interests and in special situations, there is a list of criteria that the Undergraduate Vice Chair considers before giving approval to a 197 petition. Some of these conditions have been mentioned above. Exceptions to these conditions are rare. The conditions are:
1. Math 197 is intended for students who have already taken a number of Math and PIC courses.
2. Math 197 cannot be used to duplicate the coverage of a regularly offered course.
3. The 197 course should be sponsored by a regular Mathematics Department faculty member.
4. Before agreeing to sponsor a 197 course, the faculty member should have some good grounds upon which to assess the student’s potential and level of ability, such as having had the student in another course.
5. There should be roughly 30 hours work for each unit credit.
6. The faculty sponsor and the student should meet on a regular basis, which should be specified in the petition. For four units credit, weekly meetings are appropriate, while for two units credit, biweekly meetings suffice.
7. There must be some written work, specified in the petition, that is submitted to the sponsoring faculty member and available to the Undergraduate Vice Chair upon the conclusion of the course.
8. Math 197 credit will not be given for work also turned in for another course.
9. Math 197 is not appropriate for field study credit, except in conjunction with a project for a Mathematics Department faculty member that also has a major component of reading on an advanced topic.
General Course Outline
Course Description
(2 or 4 units). Tutorial, three hours per week per unit. Limited to juniors/seniors. Supervised individual research under guidance of faculty mentor. Scheduled meetings to be arranged between faculty member and student. Culminating report required. May be repeated for maximum of 12 units, but no more than one 197 or 199 course may be applied toward upper division courses required for majors offered by Mathematics Department. Individual contract required. P/NP or letter grading.
General Information The research course, Math 199, provides an excellent opportunity for an advanced student to do research on a mathematical topic under the guidance of a faculty member. In a typical situation, the student finds an interesting topic in a mathematics course and wants to pursue the topic in the subsequent term. If the student has shown initiative and done well in the course, the faculty member may agree to direct the student’s further research through a 199 course. Occasionally a group of students will approach a professor to take a 199 course together, doing research on some aspect of a course they are currently taking from the professor.
University regulations specify that courses labeled 199 are open only to juniors and seniors with a 3.0 GPA in their major field. Math 199 is intended for students who have already taken a number of Math and PIC courses.
In order to enroll in a 199 course, the student’s petition must receive the approval of the sponsoring faculty member and of the Undergraduate Vice Chair. The petition should spell out student’s obligations are for successful completion of the course, including what will be covered in the course, how often the student will meet with the faculty sponsor, and what written material will be required.
While the 199 course is meant to be flexible, to cover students or groups of students with special interests and in special situations, there is a list of criteria that the Undergraduate Vice Chair considers before giving approval to a 199 petition. Some of these conditions have been mentioned above. Exceptions to these conditions are rare. The conditions are:
1.Math 199 is intended for students who have already taken a number of Math and PIC courses.
2.The 199 course should be sponsored by a regular Mathematics Department faculty member.
3.Before agreeing to sponsor a 199 course, the faculty member should have some good grounds upon which to assess the student’s potential and level of ability, such as having had the student in another course.
4.There should be roughly 30 hours work for each unit credit.
5.The faculty sponsor and the student should meet on a regular basis, which should be specified in the petition. For four units credit, weekly meetings are appropriate, while for two units credit, biweekly meetings suffice.
6.There must be some written work, specified in the petition, that is submitted to the sponsoring faculty member and available to the Undergraduate Vice Chair upon the conclusion of the course.
7.Math 199 credit will not be given for work also turned in for another course.
8.Math 199 credit will not be given for standard programming work alone. While a computer project can form part of the work, there should also be a major component of research on an advanced topic.
9.Math 199 is not appropriate for field study credit, except in conjunction with a project for a Mathematics Department faculty member that also has a major component of research on an advanced topic.
pic courses
General Course Outline
Catalog Description
(5) Lecture, three hours; discussion, two hours; laboratory, eight hours. Recommended requisite for students with no prior computing experience: course 1. Students with credit for course 3 will receive only two units of credit for this course. No prior programming experience assumed. Basic principles of programming, using C++; algorithmic, procedural problem solving; program design and development; basic data types, control structures and functions; functional arrays and pointers; introduction to classes for programmerdefined data types. P/NP or letter grading. Usually offered every quarter.
This class is intended for those of you who need to know how to write your own computer programs. This class will teach you how to design and develop computer programs using sound programming techniques. This class does not assume prior programming knowledge, but if you don’t have at least some familiarity with computers, consider taking PIC 1 first.
General Course Outline
Catalog Description
(5) Lecture, three hours; discussion, two hours; laboratory, eight hours. Requisite: course 10A or Computer Science 31. Abstract data types and their implementation using C++ class mechanism; dynamic data structures, including linked lists, stacks, queues, trees, and hash tables; applications; objectoriented programming and software reuse; recursion; algorithms for sorting and searching. P/NP or letter grading.
General Course Outline
Catalog Description
(5) Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10B. More advanced algorithms and data structuring techniques; additional emphasis on algorithmic efficiency; advanced features of C++, such as inheritance and virtual functions; graph algorithms. P/NP or letter grading
(5) (Formerly numbered Programming in Computing 16.) Lecture, three hours; discussion, two hour. Requisites: course 10A, Computer Science 31 or equivalent. In depth introduction to the Python programming language for students who have already taken a beginning programming course in a strongly typed, compiled language (C++, C or Fortran). Core Python language constructs, applications, text processing, data visualization, interaction with spreadsheets and SQL data bases, and creation of graphical user interfaces. P/NP or letter grading.
Course Objectives
The student will be familiar with the core Python language components, including program syntax, fundamental data types, flow control, file and console I/O, the creation of functions, the creation and use of classes. The student will be familiar with the Python interpreter, Python notebooks, and Python integrated development environments. In addition, the student will be capable of both creating and using Python modules.
Textbook
Online Resources:
 The Python Tutorial (https://docs.python.org/2/tutorial/index.html)
 RegexOne – Learning Regular Expressions (http://regexone.com/)
 TkInter Tutorial (http://effbot.org/tkinterbook/tkinterindex.htm)
General Course Outline/Schedule of Lectures
Week  Topics 
Lecture 1  Introduction to PIC 16 
Lecture 2  Python Basics – Getting Started. Basic Data Types 
Lecture 3  Python Basics – Control Flow, Functions 
Lecture 4  Python Basics – Data Structures 
Lecture 5  Python Basics – Functional Programming 
Lecture 6  Python Basics – Exception 
Lecture 7  Python Basics – Classes and Objects 
Lecture 8  Python Basics – Magic Methods 
Lecture 9  Python Basics – Iterators and Generators 
Lecture 10  Input/Output (Console, text files, CSV) 
Lecture 11  Regular Expressions – Basics 
Lecture 12  Regular Expressions – Groups and Quantifiers 
Lecture 13  Regular Expressions – Captured Groups, Data Cleaning 
Lecture 14  Matplotlib – Data Visualization 
Lecture 15  NumPy I – Generating and Manipulating Arrays 
Lecture 16  NumPy II – NumPy functions and methods 
Lecture 17  Pandas I (Data Processing) – Series and DataFrames, Manipulating Data 
Lecture 18  Pandas II – Pandas functions and methods 
Lecture 19  Multidimensional Data Visualization – Contour Plots, Masks 
Lecture 20  Network Data Visualization – NetworkX 
Lecture 21  Plotly (Plotting Data) – Generating plots, Modifying Appearance 
Lecture 22  Inheritance – Subclassing and Super 
Lecture 23  GUI TkInter I – Drawing Lines and Shapes 
Lecture 24  GUI TkInter II – Widgets 
Lecture 25  GUI TkInter III – Events and Bindings 
Lecture 26  GUI TkInter IV – Application Windows 
Lecture 27  Database Programming – SQLite 
Lecture 28  NLTK (Natural Language Toolkit) – Concordance, Contexts, Dispersion 
Lecture 29  Review 
Lecture 30  Final Exam 
Grades:
Quizzes: 10%
Homework: 40%
Midterm: 20%
Final: 30%
Core Competencies
Critical Thinking: The ability to identify Python constructs that are necessary to implement Python programs to solve a specific problem. The ability to formulate a logical sequence of computational activities that result in the computational solution of a specific problem or computational implementation of a specific task. The capability for abductive reasoning (i.e., the ability to formulate and/or modify logical inferences about Python program properties and behavior based on computational evidence).
Information Literacy: The ability to extract and correctly interpret program error messages and warnings generated by an interpreter or integrated development environment. The ability to interpret correctly runtime error messages. The ability to locate, evaluate, and incorporate information from online and text resources as it pertains to the effective use and development of Python programs. The ability to identify, evaluate and incorporate Python modules from external sources.
Quantitative Reasoning: The ability to apply mathematical concepts to the interpretation and analysis of quantitative information obtained during the implementation and use of Python programs. The ability to formulate estimates of expected outcomes of mathematical procedures implemented in Python.
Written Communication: Communication by means of written language the concept and/or details underlying specific Python constructs. The ability to compose in written form descriptions of Python program behavior for the purposes of user documentation or for the identification of Python program problems.
Oral Communication: Communication by means of spoken language the concept and/or details underlying specific Python constructs. The ability to describe program behavior to inform others about proper Python program usage or to support the identification of problems in Python programs.
Learning Outcomes:
After successfully completing this course, students will be able to write programs in the Python programming language to accomplish a variety of tasks, including some of the following:
 text processing using regular expressions;
 implementation of numerical algorithms, numerical manipulation of multidimensional data;
 displaying plots of multidimensional data;
 loading/saving data from/to various formats, including text files, spreadsheets, and SQL data bases;
 creating graphical user interfaces (GUI);
 natural language processing.
In addition, students will be able to seek out and install supporting Python modules that facilitate the construction of Python programs for a wider variety of applications than those covered in the course.
General Course Outline
Catalog Description
(5) Lecture, three hours; discussion, two hours. Requisites: course PIC 16A or equivalent. In depth application of Python programming language to problems arising in a variety of areas of current interest such as machine learning, computer vision, statistical analysis, numerical analysis, and data acquisition. Advanced Python programming techniques to improve computational efficiency. P/NP or letter grading.
Course Objectives
Students will become proficient in using the Python programming language to solve nontrivial problems that arise in areas of great current interest such as statistical analysis, machine learning, computer vision, and the solution of differential equations. Students will develop an intuitive understanding of the algorithms that are the basis for the Python modules that they incorporate into their programs. Students will also develop a deeper understanding of those aspects of the computational environment, both hardware and software, that are essential for the successful creation of Python programs.
Online Resources:
 The Python Tutorial (https://docs.python.org/3/tutorial/)
 SymPy Tutorial (https://docs.sympy.org/latest/tutorial/index.html)
 Scikitlearn Tutorial (https://scikitlearn.org/stable/tutorial/ )
 OpenCV Tutorial (https://opencvpythontutroals.readthedocs.io/)
 Scrapy Tutorial (https://docs.scrapy.org/en/latest/intro/tutorial.html)
Week  Topics 
Lecture 1  Introduction to PIC 16 B 
Lecture 2  Sympy – Symbolic Math I 
Lecture 3  Sympy – Symbolic Math II 
Lecture 4  SciPy – Input/Output and FFT 
Lecture 5  SciPy – Linear Algebra and Integration 
Lecture 6  SciPy – Interpolation and Optimization 
Lecture 7  SQLite – Database programming 
Lecture 8  SQLIte – Database programming II 
Lecture 9  Pandasql – Data analysis 
Lecture 10  Plotly – Interactive plotting 
Lecture 11  NetworkX – Network data 
Lecture 12  Social Network Analysis 
Lecture 13  Midterm Exam 
Lecture 14  Introduction to statistical analysis 
Lecture 15  Scikitlearn – Machine learning I 
Lecture 16  Scikitlearn – Machine learning II 
Lecture 17  Scikitlearn – Machine learning III 
Lecture 18  Machine learning interpretability I 
Lecture 19  Machine learning interpretability II 
Lecture 20  OpenCV – Computer Vision I 
Lecture 21  OpenCV – Computer Vision II 
Lecture 22  OpenCV – Computer Vision III 
Lecture 23  Scrapy – Web Scraping I 
Lecture 24  Scrapy – Web Scraping II 
Lecture 25  threading – Multithreading I 
Lecture 26  threading – Multithreading II 
Lecture 27  Socket – Networking 
Lecture 28  Review 
Lecture 29  Final Exam 
Grades:
Quizzes: 30%
Homework: 20%
Midterm: 20%
Final: 30%
Core Competencies
Critical Thinking: The ability to identify those Python constructs that are necessary to implement Python programs to solve a specified problem. The ability to formulate a logical sequence of computational activities that result in the computational solution of a specified problem or computational implementation of a specified task. A capability for abductive reasoning, i.e. the ability to formulate and/or modify logical inferences about Python program properties and behavior based upon computational evidence.
Information Literacy: The ability to locate, evaluate, and incorporate information from online and text resources as it pertains to the effective use and development of Python programs. The ability to identify, evaluate and incorporate Python modules from external sources. The ability to extract and interpret correctly program error messages and warnings generated by an interpreter or integrated development environment. The ability to interpret correctly runtime error messages.
Quantitative Reasoning: The ability to apply mathematical concepts to the interpretation and analysis of quantitative information obtained during the implementation and use of Python programs. The ability to formulate estimates of expected outcomes of mathematical procedures implemented in Python.
Written Communication: Communication by means of written language the concept and/or details underlying specific Python constructs. The ability to compose in written form descriptions of Python program behavior for the purposes of user documentation or for the identification of Python program problems.
Oral Communication: Communication by means of spoken language the concept and/or details underlying specific Python constructs. The ability to describe program behavior to inform others about proper Python program usage or to support the identification of problems in Python programs.
Learning Outcomes:
After successfully completing this course, students will be able to write programs in the Python programming language to accomplish a variety of tasks, including some of the following:
• solving systems of linear and nonlinear equations, ordinary differential equations, and linear programming problems from realworld examples;
• performing statistical data analysis using database programming and interactive plotting
• machine learning, for making decisions based on a model generated from data;
• computer vision, including object tracking;
• threading for achieving multitasking during a program’s execution.
• scraping and processing data from websites;
• communicating between networked computers;
In addition, students will be able to seek out and install supporting python modules that facilitate the construction of Python programs for a wider variety of applications than those covered in the course.
General Course Outline
Catalog Description
(5) (Formerly numbered 20.) Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Not open for credit to students with credit for course 3. Introduction to Java computer language. Class and interface hierarchies; graphics components and graphical user interfaces; streams; multithreading; event and exception handling. Issues in class design and design of interactive Web pages. P/NP or letter grading.
General Course Outline
Catalog Description
(5) (Formerly numbered 40.) Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Recommended: course 10B. Introduction to core technologies of Internet, with focus on clientside Web programming. Fundamental protocols, static Web pages, Perl language, Common Gateway Interface, XML. P/NP or letter grading.
General Course Outline
Catalog Description
(4) Lecture, one to three hours; discussion, zero to one hour. Enforced requisite: course 10A. Variable topics in programming not covered in regular program in computing courses. May be repeated for credit with topic change. P/NP or letter grading. (See schedule for any current listings.)
A section of PIC 97 is offered occasionally, for example, as an experimental version of a course before it becomes a regular course.
General Course Outline
Catalog Description
(Formerly numbered 197). Lecture, three hours; discussion, one hour. Variable topics in programming and mathematics of programming not covered in regular program in computing courses. May be repeated for credit with topic change. P/NP or letter grading. (See schedule for any current listings.)
Watch the Schedule of Classes each quarter for interesting offerings with this course number. Recent courses have included Modern Heuristics, Introduction to Scientific Research, and Introduction to UNIX System Administration.