## Summer Courses

#### message from the Department

Hello Bruins and future Bruins:

A good education in mathematics is more important than ever: Even during these difficult times, we are dedicated to offering you high quality instruction through remote and online learning. Our summer classes will help you to stay on track with your career goals and aspirations.

- For seat availability and times, please visit UCLA’s Schedule of Classes for Mathematics or Program in Computing.
- For students who have completed their Lower Division Math courses and are ready for the Upper Division courses, we recommend taking Math 115A first. Math 115A is offered during both session A & C.

I hope you will join us this summer!

Best regards,

UCLA Mathematics Department

#### Course Descriptions

**Session A (6/20 – 7/29) Courses**:

**MATH 31A – Differential and Integral Calculus**

Lecture, three hours; discussion, one hour. Preparation: at least three- and one-half 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. Differential calculus and applications; introduction to integration. P/NP or letter grading.

**MATH 32A – Calculus of Several Variables**

Lecture, three hours; discussion, one hour. Enforced requisite: course 31A with grade of C- or better. Introduction to differential calculus of several variables, vector field theory. P/NP or letter grading.

**MATH 33A – Linear Algebra and Applications**

Lecture, three hours; discussion, one hour. Enforced requisite: course 3B or 31B or 32A with grade of C- or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. P/NP or letter grading.

**MATH 33B – Differential Equations**

Lecture, three hours; discussion, one hour. Enforced requisite: course 31B with grade of C- or better. Highly recommended: course 33A. First-order, linear differential equations; second-order, linear differential equations with constant coefficients; power series solutions; linear systems. P/NP or letter grading.

**MATH 61 – Introduction to Discrete Structures**

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.

**MATH 106 – History of Mathematics**

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.

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.

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.

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. P/NP or letter grading.

**MATH 132 – Complex Analysis for Applications**

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.

**MATH 134 – Linear and Nonlinear Systems of Differential Equations**

Lecture, three hours; discussion, one hour. Requisite: 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.

**MATH 135 – Ordinary Differential Equations**

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, Sturm/Liouville theory, calculus of variations, two-point boundary value problems, Green’s functions. P/NP or letter grading.

**MATH 142 – Mathematical Modeling**

Lecture, three hours; discussion, one hour. Requisites: 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 physical sciences, biology, economics, and traffic dynamics.

**MATH 151A – Applied Numerical Methods**

Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A or Computer Science 31. 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. Letter grading.

Lecture, three hours; discussion, one hour. Requisites: courses 115A, 164, 170A or 170E or Statistics 100A, and Computer Science 31 or Program in Computing 10A. 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.

Lecture, three hours; discussion, one hour. Enforced requisites: courses 115A, 131A. Not** **open for** **credit to students with credit for former Electrical Engineering 136. Fundamentals of optimization. Linear programming: basic solutions, simplex method, duality theory. Unconstrained optimization, Newton method for minimization. Nonlinear programming, optimality conditions for constrained problems. Additional topics from linear and nonlinear programming. P/NP or letter grading.

**MATH 167 – Mathematical Game Theory**

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.

**MATH 170E – Introduction to Probability and Statistics 1: Probability**

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, chi-square, normal) distributions, bivariate distributions, distributions of functions of random variables (including moment generating functions and central limit theorem). P/NP or letter grading.

**MATH 179 – Advanced Topic in Financial Mathematics**

Lecture, three hours; discussion, one hour. Requisite: course 174E. Continuation of course 174E. In-depth study of risk measures and instruments of risk management in investment portfolios and corporate financial structure. Exotic and real options, value at risk, mean-variance analysis, portfolio optimization, risk analysis, capital asset pricing model, market efficiency, and Modigliani-Miller theory. P/NP or letter grading.

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.

**Program in Computing – Session A (6/20 – 7/29) Courses:**

**COMPTNG 10A – Introduction to Programming**

Lecture, three hours; discussion, two hours; laboratory, eight hours. 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 programmer-defined data types. P/NP or letter grading.

**COMPTNG 16A – Python with Applications I**

(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.

#### Session C (8/1 – 9/9) Courses:

Lecture, three hours; discussion, one hour. Preparation: three years of high school mathematics. Requisite: successful completion of Mathematics Diagnostic Test. Function concept. Linear and polynomial functions and their graphs, applications to optimization. Inverse, exponential, and logarithmic functions. Trigonometric functions. P/NP or letter grading.

**MATH 31B – Integration and Infinite Series**

Lecture, three hours; discussion, one hour. Requisite: course 31A with 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.

**MATH 32B – Calculus of Several Variables**

Lecture, three hours; discussion, one hour. Enforced requisites: courses 31B and 32A, with grades of C- or better. Introduction to integral calculus of several variables, line and surface integrals. P/NP or letter grading.

**MATH 33B – Differential Equations**

Lecture, three hours; discussion, one hour. Enforced requisite: course 31B with grade of C- or better. Highly recommended: course 33A. First-order, linear differential equations; second-order, linear differential equations with constant coefficients; power series solutions; linear systems. P/NP or letter grading.

**MATH 95 – Transition to Upper-Division Mathematics**

Lecture, three hours; discussion, one hour. Enforced requisites: courses 32A, 32B. Not open for credit to students with credit for course 131A or 132. Introduction to rigorous methods of proof-based upper-division mathematics courses. Basic logic; structure of mathematical proofs; sets, functions, and cardinality; natural numbers and induction; construction of real numbers; topology of real numbers; sequences and convergence; continuity. May not be applied toward major requirements. P/NP or letter grading.

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.

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. P/NP or letter grading.

**MATH 151B – Applied Numerical Methods**

Lecture, three hours; discussion, one hour. Requisite: course 151A. 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. Letter grading.

**MATH 170S – Introduction to Probability and Statistics 2: Statistics**

Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, and 170A or 170E or Statistics 100A. Not open to students with credit for Statistics 100B. Introduction to statistics. Topics include sampling, estimation (maximum likelihood and Bayesian), properties of estimators, regression, confidence intervals, hypotheses testing, analysis of variance. P/NP or letter grading.

**MATH 174E – Mathematics of Finance for Mathematics/Economics Students**

Lecture, three hours; discussion, one hour. Enforced requisites: courses 33A, and 170A or 170E 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 Black-Scholes) of European and American options, Greeks and numerical methods. P/NP or letter grading.

**MATH 177 – Theory of Interest and Applications**

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.

**MATH 296G – Research Seminar: Analysis**

Seminar, two hours. Seminars and discussion by staff and students. May be repeated for credit. S/U grading.