Graduate Courses

110.601-602 Algebra
An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for Number Theory, Algebraic Geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras. Prerequisites 110.403-404.

110.604 Strategies for Computer-Assisted Mathematics Instruction
This course is designed to introduce teaching assistants to the Maple program and to explore strategies using Maple in the teaching of undergraduate mathematics. It may be required as a part of their normal duties. Others may enroll only with permission of instructor.

110.605 Real Variables
Measure and integration on abstract and locally compact spaces (extension of measures, decompositions of measures, product measures, the Lebesgue integral, differentiation, L^p-spaces); introduction to functional analysis; integration on groups; Fourier transforms. Prerequisites: 110.405, 110.413, or equivalent.

110.607 Complex Variables
Analytic functionsof one complex variable. Topics include Mittag-Leffler Theorem, Weierstrass factorization theorem, elliptic functions, Riemann-Roch theorem, Picard theorem, and Nevanlinna theory. prerequisites: 110.311 and 110.405.

110.611-612 Several Complex Variables
Domains of holomorphy and pseudoconvexity, Levi pseudoconvexity. The Weierstrass preparation and division theorems, properties of the local ring of germs of holomorphic functions, complex analytic varieties, the Ruckert Nullstellensatz. Sheaves and cohomology, coherent analytic sheaves, Oka's coherence theorem, Dolbeault cohomology. Additional topics such as Chow's theorem, L^2 cohomology, integral formulas, Cartan's Theorems A and B, compact complex manifolds. Prerequisites: 110.412, 110.413. Recommended: 110.605-606.

110.615-616 Algebraic Topology
Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincar&ecute; and Alexander duality theorems. Prerequisites: 110.403, 110.413.

110.617-618 Number Theory
Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta-functions and connections with Hecke's treatment, semi-simple algebras over local and number fields, adele geometry. Prerequisites: 110.403-404.

110.619-620 Lie Groups and Lie Algebras
Lie groups and Lie algebras, classification of complex semi-simple Lie algebras, compact forms, representations and Weyl formulas, symmetric Riemannian spaces. Prerequisite: 110.404.

110.631-632 Partial Dffferential Equations
An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalevski theorem, Laplace's equation, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles. The second term focuses on special topics such as second order elliptic theory. Prerequisites: 110.605-606.

110.635-636 Microlocal Analysis
Microlocal analysis is the geometric study of singularities of solutions of partial differential equations. The course will begin by introducing the geometric theory of (Schwartz) distributions: Fourier transform and Sobolev spaces, pseudo-differential operators, wave front set of a distribution, elliptic operators, Lagrangean distributions, oscillatory integrals, method of stationary phase, Fourier integral operators. The second semester will develop the theory and apply it to special topics such as asymptotics of eigenvalues/eigenfunctions of the Laplace operator on a Riemann manifold, linear and non-linear wave equation asymptotics of quantum systems, Bochner-Riesz means, maximal theorems. Prerequisites: 110.605-606. Recommended: 110.631.

110.643-644 Algebraic Geometry
Affine varieties and commutative algebra. Hilbert's theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch Theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, schemes, connections with complex analytic geometry and topology. Prerequisites: 110.601-602.

110.645-646 Riemannian Geometry
Differential manifolds, vector fields, flows, Frobenius' theorem. Differential forms, DeRham's theorem, vector bundles, connections, curvature, Chern classes, Cartan structure equations, Riemannian manifolds, Bianchi identities, geodesics, exponential maps. Other topics as time permits, e.g. harmonic forms and Hodge's theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems. Prerequisites: 110.405, 110.413.

110.711-712 Quantum Cohomology
Quantum cohomology is the currently fashionable name for a way of organizing recent new insights into a large class of nonlinear global problems in the calculus of variations. Work of Gromov, Floer, Witten, and others on an infinite-dimensional version of Morse theory leads to new kinds of invariants for geometrical structures, with applications including the enumeration of rational curves on classical algebraic varieties, new invariants of four-manifolds, proof of Arnold's conjecture on stationary points of dynamical systems, as well as to questions motivated by quantum field theory. Although the results are formulated in the language of algebraic topology, the theory of elliptic differential equations is the main technical tool. Prerequisites: some knowledge of ordinary cohomology and some familiarity with either PDE or differential geometry.

110.721-722 Homotopy Theory
Homotopy groups, fiber spaces, fiber bundles, Hurewicz isomorphism theorem, local coefficients, spectral sequences, cohomology operations, obstruction theory, Postnikov systems. Prerequisites: 110.615-616.

110.723-724 Topics in Automorphic Functions

110.725-726 Topics in Analysis

110.727-728 Topics in Algebraic Topology

110.729-730 Topics in Several Complex Variables

110.733-734 Topics in Algebraic Number Theory

110.735-736 Topics in Hodge Theory

110.737-738 Topics in Algebraic Geometry

110.741-742 Topics in Partial Differential Equations

110.751-751 Topics in Group Representations

110.761-762 JAMI Seminar
This course focuses on the topic chosen for the Japan-U.S. Mathematics Institute (JAMI) Workshop and Conference held each year in March/April.The topic for 1996-97 is "Elliptic curves and Arithmetic Geometry." Students will be required to read and work through current research papers and attend seminar talks.

110.799 Thesis Research

110.800 Independent Study, Graduate

Cross-Listed

171-710 Lie Groups and Quantum Mechanics