Graduate Exams

Most of the following topics are normally covered in the course Math 502a.

Direct Methods for Linear systems. Gaussian Elimination and LU Factorization, Banded Systems, Symmetric Matrices, Perturbation Theory and Error Analysis

Matrix Eigenvalue Problems. Canonical Forms, Perturbation Theory, Jacobi Methods, The Power Method (including Inverse and Rayleigh Quotient iterations), Eigenvalues of Condensed Matrices (including unitary elementary transformations, reduction to Hessenberg form, QR algorithm), Singular Value Decomposition (SVD)

Linear Least Squares Problems. The Method of Normal Equations, Least Squares and the SVD (including pseudoinverse solutions), Orthogonal Decompositions

Iterative Methods for Linear Systems. Stationary Iterative Methods (Jacobi and Gauss-Seidel), Successive Overrelaxtion Methods (including convergence analysis), The Conjugate Gradient Method
Preconditioned Methods

References:

• G. Dahlquist and A. Bjorck, Numerical Methods, SIAM, 2003
• L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM,1997
• J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997
• W. Cheney and D. Kincaid, Numerical Analysis, Brooks/Cole, 1996
• E.K. Blum, Numerical Analysis, Addison-Wesley, 1972

Foundations of probability: Equally likely outcomes, principles of counting, permutations, combinations. Principle of inclusion/exclusion. Conditional probability. Independence. Random variables, distributions, joint distribu- tions (continuous and discrete), functions of random variables and vectors.

Properties of Random Variables: Expectation, moments, generating func- tions (for distributions of integer-valued random variables and general se- quences), moment generating functions, characteristic functions (excluding continuity theorem and Bochner’s theorem). Basic inequalities (Cauchy- Schwarz, Chebyshev, H ̈older, Jensen, Markov, power mean/Lyapunov).

Computations: Indicators. Covariance, correlation, covariance matrix. Conditional distribution and density, conditional expectation, conditional variance, the law of total variance. Sums of independent random variables, convolutions. Main families of discrete and continuous distributions (beta, binomial, Cauchy, exponential, gamma, geometric, negative binomial, nor- mal, Poisson, uniform) and relations among them. Multivariate normal distribution.

Limit Theorems: Convergence in probability, in Lp, and in distribution. Law of large numbers, Central Limit Theorem. Poisson approximation.

Special models: Simple random walk, reflection principle, gambler’s ruin.

References

• G. R. Grimmett and D. R. Strizaker, Probability and Random Processes A. Klenke, Probability Theory, especially Chapters 2–8
• S. Ross, A First Course in Probability, especially Chapters 1–8 A. N. Shiryayev, Probability, especially Chapters I and IV

Foundations of probability: Axioms of probability, distribution func- tion, generating σ-fields, Kolmogorov’s extension theorem. Principle of inclusion/exclusion. Conditional probability and independence. Ran- dom variables, distributions, joint distributions (continuous and dis- crete), functions of random variables and vectors.

Properties of Random Variables: Probability generating functions. Ex- pectation, moments. Moment generating functions. Characteristic functions, inversion and continuity theorems. Basic inequalities (Cauchy- Schwarz, Chebyshev, H ̈older, Jensen, Markov, power mean/Lyapunov).

Computations: Conditional distribution and density, conditional ex- pectation given a σ-field, conditional variance, the law of total vari- ance. Main families of discrete and continuous distributions (binomial, Cauchy, exponential, gamma, geometric, normal, Poisson, uniform) and relations among them. Multivariate normal distribution. Sums of in- dependent random variables, convolutions.

Limit Theorems: Modes of convergence (a.s., in probability, in Lp, and in distribution) and relations among them. Theorems of Slutsky and Mann-Wald. Delta method. Convergence of expected values and mo- ments. Borel-Cantelli lemmas. Weak and strong laws of large num- bers, convergence of random series, Kolmogorov’s inequality. Weak convergence, tightness; Helly-Bray and Portmanteau theorems; multidi- mensional weak convergence and characteristic functions. The classical Central Limit Theorem, Lindeberg’s condition. Poisson approximation.

Students should have a good background in linear algebra, including the basic canonical forms; these topics are covered in our undergraduate course Math 471.

Groups: Review of elementary group theory, isomorphism theorems, group actions, orbits, stabilizers, simplicity of An, Sylows theorems, direct prod- ucts and direct sums, semi-direct products and extensions of a group by an abelian group, Fundamental Theorem of Abelian Groups, solvable groups.

Fields: Relative dimensions, automorphisms, splitting fields, isomorphism extension theorem, sep- arable extensions, Galois correspondence, Funda- mental Theorem of Galois Theory, principal element theorem, traces and norms, radical extensions, finite fields, cyclotomic extensions, inseparable extensions, algebraic closure.

Commutative Algebra: Localization, integral extensions, unique factor- ization domains, Eisenstein criterion, principal ideal domains, Noetherian rings, Hilbert basis theorem, varieties, Zariski topology, Hilbert Nullstellen- satz.

Modules: Irreducible modules, torsion modules, free modules, projective modules, modules over PIDs, chain conditions, tensor products, exact se- quences. Noncommutative Rings: Artinian rings, Jacobson radical, Artin- Wedderburn theorem, Maschke’s theorem, Skolem- Noether theorem, divi- sion rings, Wedderburns theorem on finite division rings.

References:

• D. Rotman, An introduction to the theory of groups
• S. Lang, Algebra
• T. Hungerford, Algebra
• T.Y. Lam, Lectures on modules and rings
• M. Atiyah and I.G. MacDonald, Introduction to commutative algebra
• D. Dummitt and R. Foote, Abstract algebra

Elementary properties of holomorphic functions: Power series representation, integral representation (Cauchy’s theorem for ”nice” domains). Cauchy-Riemann equations. Taylor series, Cauchy integral formula, classification of isolated singularities, meromorphic functions. Liouville’s theorem. The elementary holomorphic functions (rational functions, the exponential and logarithm functions, trigonometric functions, powers and roots).

The residue theorem and its applications: Evaluating integrals by the methods of residues, counting zeros and poles. Rouche’s theorem, open mapping theorem, inverse and implicit function theorems. Methods for computing residues. Harmonic functions: Mean value property and maximum principle for harmonic and analytic functions. Realization of a real harmonic function as the real part of an analytic function (construction of the conjugate harmonic function in a simply connected domain). Poisson integral formula. Schwarz’s lemma.

Limits of analytic functions: Properties carried over by uniform convergence of compact subsets, various hypotheses under which one may deduce uniform convergence on compact subsets, normal families. Conformal mapping: Local mapping properties of analytic functions, the elementary mappings (Mobius transformations, exp(z), log(z), etc.), Riemann mapping theorem.

Analytic continuation: Reflection across analytic boundaries (Schwarz reflection principle), conformal mapping of polygons to the disk, Picard’s theorem.

References:

• L.V. Ahlfors, Complex Analysis
• J.B. Conway, Functions of One Complex Variable
• W. Rudin, Real and Complex Analysis

Measures: Sigma-rings, sigma fields. Set functions and measures. Outer measure. Construction of measures on Rn. Variation of signed measures. Hahn decomposition theorem. Absolute continuity. Mutually singular measures. Product measures. Regular measures. Measurable functions. Signed and complex measures.

Integration: Definition and basic properties of integrable functions over an abstract measure space. The Riemann integral and its relation to the Lebesgue integral. Lebesgue’s dominated convergence theorem and related results. Radon-Nikodym theorem. Fubini’s theorem. Convolution. The n-dimensional Lebesgue integral. Polar coordinates.

Convergence: Almost everywhere convergence, uniform convergence, almost uniform convergence, convergence in measure and in mean. Egoroff’s theorem. Lusin’s theorem.

Differentiation: Lebesgue differentiation theorem. Maximal function. Vitali covering lemma. Bounded variation. Absolutely continuous functions. Fundamental theorem of calculus.

Metric spaces: Topological properties, convergence, compactness, completeness, continuity of functions.

References:

• G.B. Folland, Real Analysis: Modern techniques and their applications
• P. Halmos, Measure Theory
• W. Rudin, Real and Complex Analysis

Differentiable manifolds: definition, submanifolds, smooth maps, tangent and cotangent bundles.

Differential forms: exterior algebra, integration, Stokes’ theorem, de Rham cohomology. Lie derivatives: of forms and vector fields.

Differential topology: regular values, Sard’s theorem, degree of a map, and index of a vector field.

“Classical” differential geometry: local theory of surfaces, 1st and 2nd fundamental forms, Gauss-Bonnet formula.

Homotopy theory: definition of homotopy, homotopy equivalences, fundamental groups (change of base point, functoriality, Van Kampen theorem, examples such as the fundamental group of the circle), covering spaces (lifting properties, universal cover, regular (or Galois) covers, relation to π1), higher homotopy groups.

Singular homology theory: definition of the homology groups, functoriality, relative homology, excision, Mayer-Vietoris sequences, reduced homology, connection between H1 and the fundamental group, homology of classical spaces.

References:

• M. Berger and B. Gostiaux: Differential Geometry: Manifolds Curves and Surfaces
• A. Hatcher: Algebraic Topology
• I.M. Singer and J.A. Thorpe: Lecture Notes on Elementary Topology and Geometry
• H. Hopf: Differential Geometry in the Large, Springer Lecture Notes in Mathematics, V. 1000
• M.J. Greenberg and J.R. Harper: Lectures on Algebraic Topology
• J.W. Vick: Homology Theory
• W.S. Massey: Algebraic Topology: An Introduction
• I. Madsen and J. Tornehave: From Calculus to Cohomology

Distributions: Parametric models, families of discrete and continuous distributions, exponential families, multivariate normal distribution, derived distributions from normal samples including t, chi-squared, and F; mixtures.

Probability: Jensen, correlation, Holder, Markov and Chebyshev inequalities; order statistics, quartiles, percentiles, probability integral transformation and its inverse, modes of convergence, limit theorems, Slutsky theorems, delta method, variance stabilizing transformations.

Point estimation: method of moments, maximum likelihood, unbiased estimation, Bayes estimation, comparison of estimators, optimality, Fisher information, Cramer Rao inequality, asymptotic efficiency, sufficiency, completeness, Rao Blackwell and Lehman Scheffe theorems

References:

• G. Casella and R.L. Berger, Statistical Inference
• T.S. Ferguson, A Course in Large Sample Theory
• E.L. Lehmann, Theory of Point Estimation

First order equations: Method of characteristics for fully nonlinear, quasilinear, and linear cases. The Cauchy problem.

Laplace equation: Harmonic and Subharmonic functions. Mean value property. Harnack principle. Maximum principle. Liouville’s theorem. Poisson formula. Green’s function.

Heat equation: Cauchy problem. Energy equality. Maximum principle. Nonhomogeneous heat equation. Backward uniqueness.

Wave equation: D’Alamert’s formula. Spherical means. Energy equality. Duhamel’s principle. Domain of dependence.

Sobolev spaces: Weak derivatives. Embedding theorems (Gagliardo-Nirenberg and Morrey). Rellich compactness theorem. Trace theorem. H-1 space. Rademacher’s theorem.

References:

• L.C. Evans, Partial Differential Equations
• G.B. Folland, Introduction to Partial Differential Equations
• F. John, Partial Differential Equations

Existence, uniqueness and dependence of initial data. Continuation of solutions. Linear systems, periodic linear systems, Floquet’s Theorem, stability of critical points, and peri- odic orbits. Grobman-Hartman theorem. Lyapunov functions. Two dimensional systems, classification of elementary critical points, Poincar ́e-Bendixson Theorem. Invariant sets and manifolds. Stable Manifold Theorem.

References:

• L. Barreira and C. Valls: Ordinary Differential Equations: Qualitative Theory (main reference)
• E.A. Coddington and N. Levinson: Theory of Ordinary Differential Equations
• J. Hale: Ordinary Differential Equations
• C. Chicone: Ordinary Differential Equations with Applications
• P. Hartman: Ordinary Differential Equations