Qualifying Exams 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.

 Basic Examination (for M.A and PhD)

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.
Linear algebra:

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, Gram-Schmidt 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, row-reduced form, LU decomposition.

Suggested References:

For Analysis and Multivariate Calculus:

1.  T. Tao, Analysis I and II
2. T. Gamelin and R. Greene, Introduction to Topology, Chapter 1
3. C. H. Edwards, Advanced Calculus of Several Variables, Chapters I-III

Each of the following texts also covers much of the analytic material on the syllabus of the Basic Exam:

4. T. Apostol, Mathematical Analysis
5.  M.  Rosenlicht, Introduction to Analysis
6.  W.  Rudin, Principles of Mathematical Analysis

For Linear Algebra:

1.      Linear Algebra and Lecture Notes, by Peter Petersen (available at http://www.math.ucla.edu/~petersen/)
2.      Serge Lang, Linear Algebra
3.      K. Hoffman, Linear Algebra
4.      M. Marcus and H. Minc, Introduction to Linear Algebra
5.       Schaum's Outline of 3000 Solved Problems of Linear Algebra (a good source of exercises).

The following texts may be useful for Basic Exam preparation for students with an interest or background in Applied Mathematics:

1.      D. Serre, Matrices: Theory and Applications
2.      A. Ralston and P. Rabinowitz, "A First Course in Numerical Analysis", 2nd edition, Chapters 9 & 10
3.      K. Atkinson, "An Introduction to Numerical Analysis", 2nd edition,Chapters 7, 8 & 9.
4.      M. Marcus and H. Minc, Introduction to Linear Algebra
5.      Schaum's Outline of 3000 Solved Problems of Linear Algebra (a good source of exercises).
 ALGEBRA

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

• Hungerford, Algebra.
• Lang, Algebra, (2nd edition).
• Jacobson, Basic Algebra I.
• Rotman, An Introduction to the Theory of Groups, Graduate Texts in Mathematics: Springer.
• van der Waerden, Algebra (Vol. 1.) (7th Edition), [Ch. 1-8].
 ANALYSIS

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

 • Folland, G.B. (1984). Real Analysis, New York, Wiley. • Roydan, H.L. (1969). Real Analysis, New York, MacMillan • Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition). • Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory, Integration and Hilbert Spaces, Princeton University Press. • Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker. • Ahlfors, L. (1979). Complex Analysis, New York, McGraw Hill (3rd edition). • Gamelin, T.W. (2001). Complex Analysis, New York, Springer. • Stein, E.M. and Sharkarchi, R. (2003). Complex Analysis, Princeton University Press • Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition). • Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.

 APPLIED DIFFERENTIAL EQUATIONS

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.

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
 GEOMETRY/TOPOLOGY

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
Main References:
• 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.

 LOGIC (Fall only)

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.

 NUMERICAL ANALYSIS (COMPUTATIONAL MATHEMATICS)

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.

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.