Courses are listed in numerical order. The letters in parentheses after the name of each course mean:
An asterisk (*) before the name of a course indicates that it is statistics-related.
Calculus of finite differences. Interpolation. Numeric integration. Optimization. Systems of algebraic equations. Systems of ordinary differential equations.
Prerequisites: 217, 309, and some programming experience, e.g., Math 1201, CS 136G, CS 265, CS 101, or permission.
Computer arithmetic and algebra, error propagation, discretization and sampling, approximation of functions, algorithms of mathematical analysis, Fast Fourier Tranforms
Prerequisites: Math 1201, 309, and 318.
Properties of space curves and surfaces. Prerequisite: Math 309 and 318, or the equivalent.
Order statistic, Pitman randomization, rank-order test, runs test, goodness-of-fit tests. Prerequisite: Math 420 or 493.
Limits and infinite series; limits of sequences, convergence of series. Convergence test. Series of functions. Taylor series. Fourier series; orthogonality, Bessel's inequality, convergence. Applications. Weierstrass's approximation theorem. Integration; Stieltjes integrals, mean value theorems, improper integrals, divergent integrals, special functions. Prerequisite: Math 318.
Differential calculus of functions of several variables; partial derivatives, the chain rule, differentials, Taylor's theorem, maxima and minima, Lagrange multipliers. Integral calculus of functions of several variables; multiple integrals, iterated integrals, line and surface integrals, exact differentials, theorems of Gauss, Stokes and Green, change of variables in integration. The implicit function theorem. Prerequisite: Math 411.
This is a course in the theory of calculus with a radically different style from the 100-300 level calculus courses. The emphasis is on proving theorems, not on calculation techniques or applications of calculus to other quantitative disciplines. The homework and exams will be heavily slanted toward providing proofs of various statements with very few "routine calculation" problems. It's not assumed that students have had prior experience with devising proofs. To the contrary, one of the goals of the course is to help students develop proof skills. Although the lectures will try to provide examples illustrating various theoretical ideas, the bulk of class time will be the proverbial repetition of definition, theorem statement, proof. Prerequisite: Math 233.
An introduction to the theory of P.D.E.'s and their solution. Elliptic, parabolic, and hyperbolic equations. Heat and wave equations. Separation of variables, Fourier series, Bessel functions, orthogonal polynomials. Numerical methods: relaxation, method of characteristics, finite elements, finite differences. Prerequisite: Math 217 and 309 or permission of instructor.
Analytic functions, line integrals, the Cauchy integral formula, power series, residues, poles, conformal mapping, and applications. Prerequisite: SSM 317, Math 318, or Math 411.
A course in set theory, metric spaces, and topological spaces, intended to prepare students for subsequent courses in analysis and topology. Introduction to set theory, transfinite methods, cardinal and ordinal numbers; metric spaces, Euclidean spaces, spaces of continuous functions; complete spaces, Baire Category theorem, contraction mapping theorem; compactness, separability, bases and subbases, Lindelof's theorem; introduction to topological spaces. Prerequisite: Math 412.
Compactness, separation and connectedness in topological spaces, with emphasis on their significance in metric spaces; product and quotient spaces; approximation theorems; Tychonoff's theorem, Ascoli's theorem, Tietze's extension theorem, Stone-Cech compactification. Although credit in Mathematics 417 is not contingent upon completion of Mathematics 418, the former course by itself will not give a clear picture, even of metric spaces, since compactness and connectedness are not adequately discussed until the latter course. Prerequisite: Math 417.
A first course in the design and analysis of experiments from the point of view of regression. Factorial, randomized block, split-plot, latin square, and similar designs are covered. Prerequisite: Math 320 or the equivalent.
Elementary properties of complex numbers. Limits, continuity and differentiability of complex valued functions. Line integrals. Cauchy's theorem and applications. Expansions in power series. Local behavior of regular functions. Isolated singularities. The point at infinity. Elementary functions. Introduction to conformal mapping. Calculus of residues and its applications. Prerequisite: Math 417-418, or permission of the Department.
Normal families. Riemann mapping theorem. Poisson integral. Jensen's theorem. Analytic continuation. Univalent functions. Integral and meromorphic functions. Elliptic functions. Prerequisite: Math 5021, or permission of the Department.
Linear equations and matrices: row-reduction, matrix multiplication, matrix inversion. Vector spaces: bases, coordinates, row equivalence. Linear transformations: isomorphism, matrix representations, similarity. Determinants: product formula, Cayley-Hamilton theorem, eigenvectors, eigenvalues. Inner product spaces: linear functionals and adjoints, unitary and normal operators, spectral theorem. Prerequisite: Math 318, or permission of the Department.
Integers: Euclidean algorithm, unique factorization, congruence. Groups: matrix groups, permutation groups, subgroups, cosets, Lagrange's theorem, Cayley's theorem, homomorphism, normal subgroups, quotient groups, Sylow's existence theorem. Rings: matrix rings, polynomial rings, homomorphism, ideals, maximal and prime ideals, field of quotients, principal ideal domains. Fields: characteristic, dimension, geometric constructions, splitting a polynomial, finite fields. Prerequisite: Math 5029, or similar course.
Groups: Actions of groups, class formula, symmetric groups, Sylow theorems, Permutation groups, Abelian groups, Jordon-Holder Theorem. Rings and Modules: Polynomial rings, localisation, Noetherian rings, Hilbert basis theorem, Principal ideal domains, Dedekind domains, Unique factorisation domains, Gauss' lemma, Integral extensions, Noether normalisation theorem. Fields: Extension of fields, finite fields, Galois theory, Kummer theory. More ring theory: Hilbert Nullstellensatz, Primary decomposition (and if time permits: Filtration and completion of modules, Hilbert polynomial, valuation theory.) Prerequisite: Math 430, or permission of the Department.
Multilinear Algebra: Matrices, bilinear forms, tensor product, symmetric algebra, exterior algebra, semi-simple rings, Wedderburn's theorem, Representation of finite groups, Clifford Algebra. Homological Algebra and Category theory: Categories, Functors, Adjoint functors, Injective modules, Complexes, Derived functors, Koszul complex. Prerequisite: Math 5031, or permission of the Department.
An introduction to algebraic structure: groups, rings, integral domains, division rings, fields. Applications to curcuit design and analysis, algebraic coding theory, atomic physics. Prerequisite: Math 118 and 309 or 5029, or permission of the instructor.
Life-table analysis and testing, mortality and failure rates, Kaplan-Meier or product-limit estimators, hypothesis testing and estimation in the presence of random arrivals and departures, and the Cox proportional hazards model. Used in medical research, industrial planning, and the insurance industry. Some topics covered on the actuarial examination. Prerequisite: Math 320 and 309 or equivalents.
This is a quick introduction to the theory of Algebraic Curves. We will do some generalities on Projective varieties and then do the theory of algebraic curves in more depth. We plan to study divisors on curves, Riemann-Roch Theorem, Construction of Jacobians and possibly Theta functions.
Introduction to linear optimization theory from the point of view of linear algebra. Linear and quadratic programming, decision theory, dynamic programming, network analysis. The LINDO computer package introduced and used. Along with a course in numerical analysis, useful in preparing for the third examination of the Society of Actuaries. Prerequisites: Math 233 and 309.
Unified treatment of those statistical methods having their basis in linear algebra. General linear hypothesis, joint confidence regions, experimental design models, variance components models. Prerequisite: A course in linear algebra, such as Math 309 or 5029, and a course in statistics that includes regression, such as Math 320.
Differentiability, mean value theorem and chain rule for mappings of R^{n} into R^{m}. Special properties of C^{¥} functions and mappings, tangent vectors at a point of R^{n}. Inverse function theorem, implicit function theorem and theorem on rank. Definition and examples of differentiable manifolds and submanifolds. Tangent space to a manifold, immersions and embeddings. Vector fields and one-parameter groups, existence theorem for ordinary differential equations. Lie groups as an example of a manifold, action of a groups on a manifold. Frobenius' theorem. Covectors and covector fields, bilinear forms on a manifold. Partitions of unity and applications. Prerequisite: Math 412, 418, and 5029, or permission of the Department.
Tensor fields and their behavior under mappings. Tensor product and exterior product. The calculus of differential forms. Stokes' Theorem for manifolds with boundary. Covariant differentiation on Riemannian manifolds, parallel transport, geodesics. Examples from classical differential geometry and Lie groups. Riemannian curvature and symmetric spaces. Further topics from the geometry of Riemannian manifolds and Lie Groups. Prerequisite: Math 5041.
Set functions and the construction of measures. The Hahn-Jordan decomposition. Lebesgue and Lebesgue--Stieltjes measure. Definition of the integral and convergence theorems. Product measures and Fubini's theorem. Convergence in the space measurable functions: pointwise convergence, convergence in measure, convergence in L^p. Egoroff's theorem. The Riesz--Fischer theorem L^p spaces and the Riesz representation theorem. Prerequisite: Math 417--418, or permission of the Department.
Topological groups and Haar measure. Topological spaces. Hahn-Banach theorem. Weak topologies. The closed graph theorem. The Banach--Steinhaus theorem. Locally convex spaces and the Krein--Milman theorem. Measures on locally compact spaces. Prerequisite: Math 5051.
Practice and theory of statistical computation. First half of course is a thorough introduction to statistical software, emphasizing SAS. Second half covers the computational matrix algebra underlying most statistical models, computational aspects of maximum likelihood, and the resampling methods known as the jackknife and bootstrap. Prerequisite Math 320 and 493 (or 493 may be taken concurrently)
Theory and application of mathematical probability. Prerequisite: Math 318, or permission of instructor.
Parametric and non parametric significance and hypothesis testing; order statistics; theory of estimation; theory of runs, sampling schemes, analysis of variance, sequential analysis. Prerequisite: Math 493, or permission of the Department.
Random walks, Markov chains, Gaussian processes, and empirical processes. Prerequisite: Math 318 and 493, or permission of the instructor.
Prerequisite: Senior standing, a distinguished performance in upper level mathematics courses, and permission of the Chair of the Undergraduate Committee.
500 level courses are offered each year depending on the interests of students and faculty. Listed below are a few examples of the contents of these special courses.
Register for section corresponding to supervising instructor.
PrerequisitesSenior standing and permission of the instructor. Credit variable, max 6 units
An introductory graduate level course. Probability spaces; derivation and transformation of probability distributions; generating functions and characteristic functions; law of large numbers, central limit theorem; exponential family; sufficiency, uniformly minimum variance unbiased estimators, Rao-Blackwell theorem, information inequality; maximum likelihood estimation; estimating equation; Bayesian estimation; minimax estimation; basics of decision theory. Prerequisite: Math 4111-4121 or the equivalent, or permission of instructor.
Continuation of Math 5061. Bayes estimates, minimaxity, admissibility; maximum likelihood estimation, consistency, asymptotic efficiency; confidence regions; Neyman-Pearson theory of hypothesis testing, uniformly most powerful tests; likelihood ratio tests and large-sample approximation; decision theory. Prereq: Math 5061 or permission of instructor.
Existence, uniqueness, and regularity for solutions of partial differential equations. Elliptic, parabolic, and hyperbolic equations. Boundary value problems. Distributions. Pseudodifferential operators and Fourier integral operators. Functional analysis techniques. Fixed point theorems. Double and single layer potentials.
Operators on Hilberts spaces. Hardy and Bergman spaces. Spectral theorem. Toeplitz operators. C*-algebras and von Neumann algebras. Ext and K-theory
The topics covered here vary from year to year. We treat the Fourier transform, convergence techniques, Hardy spaces, singular integrals, pseudodifferential operators, interpolation theory, domains with Lipschitz boundary, A^{p} weights, Lusin integrals,and other modern topics.
Discrete Fourier transforms: "fast" factored implementations, real-valued versions such as Hartley, sine, and cosine transforms, and implementation issues such as accuracy and memory requirements.
Local trigonometric bases: existence and construction or orthonormal bases for L^{2}(R) consisting of compactly-supported smooth functions. Discrete local trigonometric transforms and their implementations.
Discrete wavelet transforms: multiresolution analysis, conjugate quadrature filters, Mallat's algorithm, Daubechies' filters. Phase shifts, frequency localization, and periodization. The discrete wavelet transform and its inverse. Implementation issues such as memory requirements and boundary artifacts.
Discrete wavelet packets: definitions, frequency localization, libraries of orthonormal bases, information cost functions, and the best-basis transform. Separable mulitdimensional wavelet packets, basis labeling, anisotropy, and algorithmic complexity.
Time-frequency analysis: the idealized time-frequency plane, graph tilings and arbitrary tilings. Phase shifts and deviation from linear phase, and implementations with wavelet packets and local trigonometric functions.
Applications: Transform coding image compression, fast approximate factor analysis, speech analysis and recognition, statistical "de-noising", and functional calculi.
Prerequisites: We shall refer to theorems from advanced calculus at the level of Math 411 and Math 412, and linear algebra at the level of Math 309. Familiarity with a computer programming language is recommended.
Quasiconformal mapping in the plane: Quasiconformal maps are generalizations of conformal maps. They are important nowadays in many subjects, such as complex analysis, harmonic analysis, topology, and partial differential equaltions. Course contents included a number of equivalent definitions of q.c. maps, convergence and distortion theorems, boundary behavior, quasicircles, Beltrami equations, and optimal smoothness theorems.
Potential theory: For domains in Euclidean space of dimension at least two, we studied, among other things, subharmonic functions, the Dirichlet problem, harmonic measure, logarithmic and Newtonian capacity, connections with analytic functions and connections with probability theory.
Discrete source and channel model, definition of information rate and channel capacity, coding theorems for sources and channels, encoding and decoding of data for transmission over noisy channels. Corequisite: ESE 520.
Interpolation by bounded analytic functions on the disk and bidisk. Operator theory aspects of Nevanlinna-Pick interpolation.
An introduction to C*-algebras, von Neumann algebras, and related objects from the point of view of quantization. The unifying theme is that of finding Hilbert space analogs (quantum groups, operator modules, Lipschitz algebras, etc.) of classical constructions (topological groups, Banach bundles, metric spaces, etc).
The basics about algebraic groups and their representations, using selected topics from books of Fulton ("Representation Theory"), Borel ("Algebraic Groups"), and Humphreys ("Algebraic Groups").
This course covers complex tori and their line bundles, Appel-Humbert Theorem, Theta functions and Theta groups, addition formula and multiplication formula for Theta functions, embeddings of abelian varieties into projective spaces via Theta functions, and moduli spaces of abelian varieties.
Prerequisites: Math 5031-2, 5041-2, and a little knowledge of algebraic geometry. Or, permission of the instructor.
This course covers sheaf theory, complex vector bundles, Chern classes, elliptic operator theory, Hodge theory on compact complex manifolds, Hodge-Riemann bilenar relations on Kaehler manifolds, Kodaira-Nakano vanishing theorem, and Kodaira embedding theorem.
A first course on Riemannian geometry. Nonpositive curvature in geometry and group theory.
An introduction to stochastic calculus on manifolds.
Probability theory based on measure theory. Strong law of large numbers, central limit theorem, martingales, applications of martingales, Markov processes, Brownian motion.
We study the Bass-Serre theory of groups acting on (simplicial) trees. In particular, we cover Part I of Serre's book "Trees." We then investigate the more general case of groups acting isometrically on Λ-trees, focusing on the case Λ = \mathbb R .
This is an introduction to the function theory of several complex variables. Comparisons and contrasts are drawn with the one-variable theory. The idea of domain of holomorphy, pseudoconvexity, and related ideas is developed. The Levi problem is studied. The Cauchy-Riemann equations, sheaves, and other tools for solving the Levi problem are treated. Zeros of holomorphic functions, holomorphic mappings, invariant metrics, and other more advanced topics are treated as time permits.
Prerequisites:Math 5021-5022, or permission of instructor.
This course will develop methods of numerical analysis and signal processing with origins in classical harmonic analysis. Though applied to quantized data, these methods take advantage of properties of the underlying continuous functions.
Credit variable, max 3 units. Contact department directly for details on faculty/sections and enrollment.
Register for section corresponding to supervising instructor. Credit variable, max 3 units.
Techniques of mathematical instruction for prospective teaching assistants.
Prerequisites:Permission of Instructor
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