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. Language exams are given twice a year, usually in the Fall and either the Winter or Spring quarters. They last two hours and consist of translating one of the given papers. A dictionary is allowed.

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Basic Examination Algebra Analysis Applied Differential Equations

Geometry/Topology Logic Numerical Analysis

There are two types of qualifying exam: 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, and Logic. 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: Download Exams. However, it is strongly recommended that students prepare for exams by studying their syllabi theme by theme, and by doing numerous exercises other than those on old exams. Experience shows that study organized around working old exams is not as efficacious as thematically organized study.

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.

Here follow the syllabi for the Examinations. Each examination may test on any of the topics of its syllabus.

The following is the syllabus for the Basic Examination

Fundamentals of analysis:

1. One-variable calculus foundations: completeness of the real numbers, sequences, series, limits, continuity, including epsilon-delta arguments, maxima and minima, uniform continuity, definition of the derivative, the mean value theorem, Taylor expansion with remainder, Riemann integral, mean value theorem for integrals, fundamental theorem of calculus, sequences and series of functions, uniform convergence and integration, differentiation under the integral sign, contraction maps, fixed point theory, with applications to Newton's method and solutions of non-linear equations, numerical integration with error estimation.
2. Metric space topology and analysis, primarily in R^n: open and closed sets, completeness, convergence of sequences of numbers and functions, closure, compactness, connectedness, uniform continuity, equicontinuity, countability and uncountability (e.g. of the reals), spaces of functions. Basic arguments and theorems of undergraduate analysis using these concepts, including the Bolzano-Weierstrass, the Stone-Weierstrass, and the Arzela-Ascoli theorems.
3. Multivariable calculus: definition of differentiability in several variables (approximating linear transformation), partial derivatives, chain rule, Taylor expansion in several variables, inverse and implicit function theorems, equality of mixed partials, multivariable integration, change of variables formula.

Starting Fall 2009, the Algebra Area Exam will be based on the following syllabus. The 2008-2009 sequence Math 210A, Math 210B, and Math 210C will prepare students entering in Fall 2008 according to this syllabus. Algebra Area Exams given prior to September 2009 will cover the older syllabus which can be found here.

Categories and Functors: basic definitions and examples, universal properties, natural transformations, representable functors, Yoneda lemma, adjoint functors, products

Group Theory: basic notions and results (esp. those pertaining to automorphisms,  homomorphisms, normal subgroups, factor groups, and conjugacy); the isomorphism theorems; group actions, and the Sylow and Jordan-Holder theorems; symmetric groups and permutation representations; free, nilpotent, solvable, simple groups, finitely generated groups and their presentations, esp. abelian (with classification); semi-direct product groups and group extensions.

Ring Theory: commutative case.  Ideals and homomorphisms, localization and completion, free and projective modules, basic theorems about factorization and UFD's , structure theory of modules over a PID, including applications to canonical forms of a matrix, chain conditions, Hilbert basis theorem, integral ring extensions, Hilbert Nullstellensatz, Dedekind rings, tensor products, duality and bilinear pairings, esp. symmetric and alternating forms.

Field Theory. General field theory including separable and inseparable extensions, normal extensions, transcendental extensions, cyclotomic extensions, finite fields, and algebraic closure; Galois theory, solvability by radicals, cyclic extensions and Kummer theory.

Ring Theory: non-commutative case. Semisimple rings, irreducible modules, and the Artin-Wedderburn theorem; non-semisimple rings, indecomposable modules and the Krull-Schmidt theorem; group rings.

Representations of Groups, esp. finite groups. Basic definitions, matrix coefficients, Schur orthogonality, invariant inner products and complete reducibility of representations, characters of finite groups and parametrization of complex representations by characters, character tables, Peter-Weyl theorem.

References

Course material: Mathematics 245AB, the first half of Mathematics 245C, and Mathematics 246AB.

Real Analysis Topics: Lebesgue integration; convergence theorems (uniform convergence, Ego- roff’s theorem, Lusin’s theorem, Lebesgue dominated convergence theorem, monotone conver- gence theorem, Fatou’s lemma); Fubini’s theorem; covering lemmas and differentiation of measures (Lebesgue decomposition theorem, Radon-Nikodym theorem, Jordan decomposition theorem, rela- tions to functions of bounded variation, signed measures and Hahn decompositions); approximate identities; basic functional analysis (linear functionals, Hahn-Banach theorem, open mapping the- orem, closed graph theorem, uniform boundedness principle, strong, weak, and weak∗ topologies); elementary point set topology including Urysohn’s lemma, the Tychonoff theorem, the Baire Cat- egory theorem and the Stone-Weierstrass theorem. The spaces C(X), the Riesz representation theorem, and the compact subsets of C(X), (Arzela-Ascoli theorem); Hilbert space, self-adjoint linear operators and their spectra; Lp spaces (duality, distribution functions, weak Lp spaces, Hölder’s inequality, Jensen’s inequality, linear operators); basic Fourier analysis (orthonormal sys- tems, trigonometric series, convolutions on Rn, Plancherel’s theorem, Riemann-Lebesgue lemma Poisson summation formula); abstract measure theory; Hausdorff measures.

Complex Analysis Topics: Analytic functions: Examples, sums of power series, exponential and logarithm functions, M¨obius transformations, and spherical representation. Cauchy’s theorem: Goursat’s proof, consequences of Cauchy integral formula, such as Liouville’s theorem, isolated singularities, Casorati-Weierstrass theorem, open mapping theorem, maximum principle, Morera’s theorem, and Schwarz reflection principle. Cauchy’s theorem on multiply connected domains, residue theorem, the argument principle, Rouch´e’s theorem, and the evaluation of definite inte- grals. Harmonic functions: conjugate functions, maximum principle, mean value property, Poisson integrals, Dirichlet problem for a disk, Harnack’s principle, Schwarz lemma and the hyperbolic metric. Compact families of analytic and harmonic functions: series and product developments, Hurwitz theorem, Mittag-Leffler theorem, infinite products, Weierstrass product theorem, Poisson- Jensen formula. Conformal mappings: Elementary mappings, Riemann mapping theorem, mapping of polygons, reflections across analytic boundaries, and mappings of finitely connected domains. Subharmonic functions and the Dirichlet problem. The monodromy theorem and Picard’s theorem. Elementary facts about elliptic functions.

Note: Also all material for the Basic Examination. To prepare, students are advised to work problems from as many old examinations as possible.

References

Basic Topics
Course material- Mathematics 266AB; additional sources are Mathematics 135AB, 136, and 146. Topics- Spectrum theory of regular boundary value problems and examples of singular Sturm-Liouville problems, related integral equations, special functions; Fourier series, Fourier and Laplace transforms; phase plane analysis of nonlinear equations; asymptotic methods for obtaining approximate solutions of ordinary differential equations; solution of simple initial and boundary value problems for potential, heat and wave equations, Green's functions, separation of variables.

More Advanced Topics
Course material- Mathematics 266ABC.
Topics- All M.A. level topics as well as: first order partial differential equations; classification and theory of linear and nonlinear higher order partial differential equations; well-posed problems; classical potential theory, Dirichlet and Neumann problems; fundamental solutions; wave equations, Cauchy problem, initial-boundary value problems, energy estimates, method of characteristics, principle of linearization; variational problems; maximum principles; equations of fluid mechanics.

References

• Bender C. M. & Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers, New York: McGraw-Hill.
• Boyce, W. E. & Diprima, R. C. (1986). Elementary Differential Equations and Boundary Value Problem (4th edition), New York: Wiley.
• Courant and Hilbert, Methods of Mathematical Physics, Vols. I, II.
• Garabedian, Partial Differential Equations.
• Haberman, Elementary Applied Partial Differential Equations.
• John, Partial Differential Equations.
• Stakgold, Boundary Value Problems of Mathematical Physics, Vols. I, II.
• Zauderer, Partial Differential Equations of Applied Mathematics.
• Haberman, Elementary Applied Partial Differential Equations

Course material: Mathematics 225ABC. The basic concepts of metric space point-set topology will be presumed known but will not be covered explicitly in the examination. For students unfamiliar with point-set topology, Mathematics 121 is suggested, although the topics covered in the analysis part of the Basic Examination are nearly sufficient. Geometry/Topology Area Exams given prior to September 2009 will cover the older syllabus which can be found here.

Topics: The topics covered fall naturally into three categories , corresponding to the three terms of Math. 225. However, the examination itself will be unified, and questions can involve combinations of topics from different areas.

1) Differential topology: manifolds, tangent vectors, smooth maps, tangent bundle and vector bundles in general, vector fields and integral curves, Sard’s Theorem on the measure of critical values, embedding theorem, transversality, degree theory, the Lefshetz Fixed Point Theorem, Euler characteristic, Ehresmann’s theorem that proper submersions are locally trivial fibrations
2) Differential geometry: Lie derivatives, integrable distributions and the Frobenius Theorem, differential forms, integration and  Stokes’ Theorem, deRham cohomology, including the Mayer-Vietoris sequence, Poincare duality, Thom classes, degree theory and Euler characteristic revisited from the viewpoint of deRham cohomology, Riemannian metrics, gradients, volume forms,  and the interpretation of the classical integral theorems as aspects of Stokes’ Theorem for differential forms
3) Algebraic topology: Basic concepts of homotopy theory, fundamental group and covering spaces, singular homology and cohomology theory, axioms of homology theory, Mayer-Vietoris sequence, calculation of homology and cohomology of standard spaces, cell complexes and cellular homology, deRham’s theorem on the isomorphism of deRham differential –form cohomology and singular cohomology with real coefficient

• Guillemin and Pollack, Differential Topology
• Hatcher, Algebraic Topology (available free as an on line download at http://www.math.cornell.edu/~hatcher/AT/ATpage.html or in paper form from Cambridge University Press)
• Notes (UCLA) by Prof. Peter Petersen available on line http://www.math.ucla.edu/~petersen/
• Spivak, Differential Geometry, vol I, third edition

• Abraham, R., Marsden, J., and Ratiu T. (1988). Manifolds, Tensor Analysis and Applications, New York: Springer Verlag.
• Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry
• Greenberg, M. & Harper, J. (1981). Algebraic Topology, A Course, Reading, Mass.: Benjamin/Cummings Pub. Co.
• Milnor, J. (1965). Topology from the Differential Viewpoint, Charlottesville, University Press of Virginia.
• Warner, F., (1983). Foundations of differentiable manifolds and Lie groups, Springer.

Course material- Mathematics 220ABC.

Topics- Model theory: Chapters 1, 2, and 3 of Model Theory by Chang and Keisler; recursion theory: chapters 1 through 5, sections 7.1-7.5, 9.1-9.4, 11.1-11.4, 14.1-14.5, 14.7. 14.8 through page 326 of Theory of Recursive Functions and Effective Computability by Hartley Rogers Jr.; incompleteness and undecidability: Godel's incompleteness and undecidability results for sufficiently strong theories; the undecidability of predicate logic. Set Theory: Chapters 1,3,4,5 and 6.1-6.4 of Set Theory: An Introduction to Independence Proofs, by Kenneth Kunen; transfinite induction; ordinals and cardinals; cardinal arithematic; constructible sets.

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 Adavanced 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

In addition to the references above:
• 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.