## Qualifying Exam Dates & system

#### qualifying exam dates: Spring 2023 (Subject to Change)

**Last day to sign up for qualifying exams is: March 10, 2023. **

Date | Exam | Time | |
---|---|---|---|

Monday, March 27, 2023 | Basic (MS 6221) | 9 AM – 1 PM | |

Logic (MS 6943) | 10 AM – 2 PM | ||

Tuesday, March 28, 2023 | Numerical Analysis (MS 6221) | 9 AM – 1 PM | |

Algebra (MS 6221) | 2 PM – 6 PM | ||

Wednesday, March 29, 2023 | Geometry/Topology (MS 6221) | 9 AM – 1 PM | |

Applied Differential Equations (MS 6221) | 2 PM – 6 PM | ||

Thursday, March 30, 2023 | Analysis (MS 6221) | 9 AM – 1 PM | |

Optimization/Numerical Linear Algebra (MS 6221) | 2 PM – 6 PM | ||

Qualifying exams are four hour written exams, and are given twice a year, in September right before the start of the Fall quarter, and in March right before the start of the Spring quarter. The Logic qualifying exam is generally offered only in the Fall. Students may petition to have a Spring exam in special circumstances, for example when this is necessary for meeting Satisfactory Progress deadlines. Petitions should be made to the GVC by the end of January.

#### Topics:

The Algebra Qualifying Exam will be based on the following syllabus (updated in Fall 2019). The sequence Math 210ABC will prepare students according to this syllabus.

**Exam Instructions**

Out of the 10 questions, the students will have to complete 8 (as opposed to 10 as it is currently).

**Group Theory: **basic notions and results (e.g., homomorphisms, subgroups, quotient groups, cosets, conjugation, generators, Lagrange’s theorem), isomorphism theorems,automorphism groups, symmetric groups, linear groups, group actions and permutation representations, the Sylow theorems and applications to simple groups, free groups and presentations, composition series and the Jordan-Holder theorem, nilpotent and solvable groups, semidirect products and group extensions.

**Ring and Module Theory: **basic notions and results (e.g., homomorphisms, ideals and generators, subrings, quotient rings), maximal and prime ideals, units, division rings, fields of fractions, PIDs, UFDs, polynomial rings and factorization (e.g., Gauss’s lemma, Eisenstein criterion), structure of finitely generated modules over a PID, canonical forms, duality and bilinear pairings, tensor products, localization, Nakayama’s lemma, torsion and rank, chain conditions and noetherian and artinian rings and modules, exterior powers and determinants.

**Field and Galois Theory: **basic notions and results, algebraic and transcendental extensions, separable, inseparable and normal extensions, field embeddings, algebraic closure, Galois theory, cyclotomic extensions, finite fields, solvability by radicals, norm and trace, discriminants of polynomials, Hilbert’s Theorem 90, Kummer theory.

**Category Theory: **basic definitions and examples, full and faithful functors, monomorphisms and epimorphisms, natural transformations, Yoneda’s lemma, representable functors, adjoint functors, equivalences of categories, limits and colimits (e.g., for limits: products, equalizers, pullbacks, initial objects).

**Homological Algebra: **exact sequences, splittings, snake and five lemmas, projective, injective, and flat modules, complexes, (co)homology.

**Commutative Ring Theory: **localizations, Hilbert’s basis theorem, integral extensions, radicals of ideals, Zariski topology and Hilbert’s Nullstellensatz, Dedekind rings, DVRs.

**Noncommutative Ring Theory: **Associative and graded algebras, endomorphism rings, group rings, semisimple rings, irreducible and indecomposable modules, central simple algebras, Artin-Wedderburn theorem, Jacobson radicals.

**Representation Theory: **basic definitions and examples (e.g,. representations, regular and trivial representations), Schur’s lemma, Maschke’s theorem, characters of finite groups, class functions, orthogonality relations, character tables, induced characters.

**References**

• Dummit and Foote, *Abstract Algebra.*

• Grillet, *Abstract Algebra.*

• Hungerford, *Algebra.*

• Jacobson, *Basic Algebra I and II.*

• Lang, *Algebra*.

**Click here for pdf version:** Geometry Topology

*General Information for Student *

The structure of the exam consists of four 5 point questions on topics in undergraduate numerical analysis and four 10 point questions based upon graduate course topics

**Exam instructions**

There are 8 problems. Problems 1-4 are worth 5 points and problems 5-8 are worth 10 points. All problems will be graded and counted towards the final score.

As stated in the instructions, it is expected that students will attempt all problems on the exam. The exam questions are designed with a level of difficulty so that a well prepared student who has taken the core graduate courses and done a bit of studying on their own will be able to answer all questions correctly. Historically a passing score has always been in the neighborhood of 45/60, with generally a fail being being between 40 and 45 depending on the difficulty of the exam.

The NA qualifying exam is not a cumulative final exam for just the graduate numerical analysis courses 269ABC. Answering some questions may require students to learn on their own some topics not explicitly covered in specific offerings of 269ABC (topics not covered due to time constraints and instructor preferences). In addition, formulating successful responses to some questions will require familiarity with material taught in other courses, e.g. material from 266ABC and undergraduate numerical analysis courses.

**Basic Topics**

Course material-Mathematics 151AB and 269A.

**Topics**

Interpolation and approximation: divided differences, Chebycheff systems, Lagrangian interpolation, splines; numerical differentiation and integration: elementary quadrature, Simpson’s, Gauss’s and Romberg’s rules; solutions of nonlinear equations: Newton’s method and its variations, estimate of rate of convergence; error analysis: methods of approximation of round-off errors and fixed and floating point arithmetic; numerical methods in Linear Algebra: Gaussian elimination, diagonalization of symmetric matrices, conditioning; numerical methods for ordinary differential equations; initial value problems, 2 point boundary value problems and eigenvalue problems; introduction to numerical methods for partial differential equations.

**More Advanced Topics**

Course material- Mathematics 151 AB and 269ABC.

Difference methods for time dependent problems: stability, consistency, convergence, initial boundary value theory, and nonlinear problems; finite element methods; initial and boundary value problem, approximation theory, linear algebra considerations.

**References – Basic Level**

- Conte and de Boor (1980). Elementary Numerical Analysis (3rd edition,) McGraw Hill.
- Dahlquist and Bjorck (1974). Numerical Methods, Prentice Hall.
- Henrici (1964). Elements of Numerical Analysis, Addison Wesley.
- Ralston, J. (1965). A First Course in Numerical Analysis, McGraw Hill.

**References – More Advanced Topics**

- Johnson (1987), Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge U. Press.
- Kreiss and Oliger (1973), Methods for the Approximate Solution of Time Dependent Problems, Garp.
- Richtmyer and Morton (1967), Difference Methods for Initial- Value Problems, Wiley.
- Sod (1985), Numerical Methods in Fluid Dynamics, Cambridge U. Press

**Click here for pdf verison: ONLA **

#### about the exams

There are two types of qualifying exams: the Basic exam and the Area exams. The Basic exam is designed to be passed by well-trained students before they commence study at UCLA. It examines fundamental topics of the undergraduate mathematics curriculum. The Area exams are graduate level exams. For each Area exam there is a preparatory course sequence. There are Area exams in Algebra, Analysis, Applied Differential Equations, Numerical Analysis, Geometry/Topology, Logic, and Optimization/Numerical Linear Algebra. Students may attempt any number of examinations in each examination period.

MA students must pass the Basic Exam only. PhD students must pass the Basic exam and two Area exams. MA students must pass the Basic by the beginning of the sixth quarter of study. PhD students must pass the Basic by the fourth quarter of graduate study. A PhD student must pass the one Area examination by the sixth quarter. A PhD student must pass the second Area examination by the seventh quarter of graduate study.

The exams are offered in the Fall and in the Spring, usually just before the beginning of those quarters. Precise dates and times are posted well in advance of the exams. Students must sign up for the exams in the Graduate Office. Each exam lasts 4 hours. Copies of past exams may be downloaded from our website by clicking here: https://ww3.math.ucla.edu/past-qualifying-exams. It is very useful to prepare for exams by doing questions from previous exams. Recent exams are likely to be more relevant since the syllabi have changed over time. However, it is essential to combine problem-solving with studying the exam syllabus theme by theme. In order to learn the material well, you will need to do exercises well beyond those on previous exams.

Each exam is written and graded by a committee created for that purpose. The Graduate Studies Committee approves exam results (passing or failing), taking into account recommendations of the examination committee. Shortly after the Graduate Studies Committee’s decision, students are notified of their exam results. Students are reminded that the grading of exams is a complex matter, and that final result (Pass or Fail) is not usually determined by the total score of all work on all problems. Students should read and follow carefully the instructions of an exam.

Graded exams are kept in the Graduate Office for six months and then destroyed. They may be examined in the Graduate Office during this time. After the results of the exams are announced, there is a one week appeal period during which students may petition, in writing, to a Qual Committee for regrading of problems. Appeals must be submitted via the Graduate Office. The Qual Committee will respond, usually in writing, to any appeal within one week.

Currently, most UCLA PhD students pass all their exams on schedule. However, the few students who fail to pass exams by the required deadlines are deemed not to be making Satisfactory Progress. Each such student is discussed individually by the Graduate Studies Committee at a meeting shortly after the above period of appeals is over. Students who have missed a deadline, or otherwise failed to make Satisfactory Progress, will receive a letter from the Graduate Vice Chair indicating any action that was taken, and detailing any schedule for performance that must be satisfied in order to continue in the program. Only in unusual circumstances will a PhD student who is more than six months behind the schedule of Satisfactory Progress be permitted to remain in the PhD program. Students who are facing negative actions are encouraged to write to GSC, and to speak to the Graduate Vice Chair, before GSC meets, to explain any extenuating circumstances that could positively influence it.