# Graduate Program

## Courses

#### Courses currently being offered:

Check the course schedules/descriptions available via the Registrar's Office for the official schedules for the widest range of terms for which such information is available.

Below you will find a list of all graduate courses that have been offered.**NOTE: Not all of these courses are offered in any given year.**

- MTH 403 THEORY OF PROBABILITY
Characteristic functions; the central limit theorem; infinitely divisible laws; random walk on groups.

*Prerequisites: MTH 471**Last Offered: Spring 2019* - MTH 436 ALGEBRA I
Rings and modules, group theory, fields and Galois theory.

*Prerequisites: MTH 237 or equivalent. Undergrads must have permission of instructor.**Last Offered: Fall 2018* - MTH 437 ALGEBRA II
Multilinear algebra, quadratic forms, simple and semi-simple rings and modules.

*Prerequisites: MTH 436. Permission of instructor required for undergraduates.**Last Offered: Spring 2019* - MTH 440 GENERAL TOPOLOGY
Continuity; compactness, connectedness, metrizability; product spaces.

*Prerequisites: MTH 265 or equivalent. Permission of instructor required for undergraduates.**Last Offered: Fall 2018* - MTH 443 ALGEBRAIC TOPOLOGY
The combinatorial structure of complexes and the homology of polyhedra; applications of algebraic techniques in topology to classification of surfaces, fixed point theory, and analysis.

*Prerequisites: MTH 436 and MTH 440. Permission of instructor required for undergraduates.**Last Offered: Fall 2018* - MTH 448 COMPUTATIONAL TOPOLOGY
Computational topology is an emerging field of study at the intersection of mathematics and computer science, devoted to the study of efficient algorithms for topological problems, especially those that arise in other areas of computing. Topics to be covered: algorithms based on higher dimensional topological structures as low dimensional data structure algorithms such as graph algorithms; topology of cell complexes, some graph theory algorithms, homotopy, covering spaces, simplicial homology, persistent homology of large data sets, discrete Morse theory, discrete differential geometry, and normal surface theory. Computing topics may include algorithms for computing topological invariants, graphics and geometry processing, mesh generation, curve and surface reconstruction, VLSI routing, motion planning, manifold learning, clustering, image processing, and combinatorial optimization.

*Prerequisites: MTH 236 and (MTH 265 or MTH 240).**Last Offered: Fall 2016* - MTH 453 DIFFERENTIABLE MANIFOLDS
Differentiable manifolds, mappings and embeddings, exterior differential forms, affine connections, curvature and torsion. Riemannian geometry, introduction to Lie groups and Lie algebras.

*Prerequisites: MTH 265 or equivalent. Permission of instructor required for undergraduates.**Last Offered: Spring 2019* - MTH 463 DIFFERENTIAL EQUATIONS
Classical PDE's, including the heat and wave equations, with both quantitative and qualitative analysis.

*Prerequisites: MTH 263 or equivalent.**Last Offered: Fall 2018* - MTH 467 THEORY ANALYTIC FUNCTIONS
Cauchy theorems, Taylor and Laurent series, residues, conformal mapping, analytic continuation, product theorems.

*Prerequisites: MTH 265 or equivalent**Last Offered: Spring 2019* - MTH 471 REAL ANALYSIS
Lebesgue measure on the line; measure spaces; integration; convergence theorems; Radon-Nikodym theorem; differentiation; Fubini's theorem; function spaces.

*Prerequisites: MTH 265 or equivalent.**Last Offered: Fall 2018* - MTH 472 FUNCTIONAL ANALYSIS
Banach spaces; dual spaces; Riesz representation theorem; Hilbert spaces; Fourier series; projective and unitary operators; spectral analysis of completely continuous self-adjoint operators. Applications.

*Prerequisites: MTH 471**Last Offered: Spring 2019* - MTH 503 THEORY OF PROBABILITY
Characteristic functions; the central limit theorem; infinitely divisible laws; random walk on groups.

*Prerequisites: MTH 471**Last Offered: Fall 2014* - MTH 504 STOCHASTIC PROCESSES
Probability spaces, conditional expectation. Stochastic processes, separability, limit theorems. Random walk. Markov processes, invariant measures, semigroups. Probabilistic treatment of potential theory, compactifications.

*Last Offered: Spring 2018* - MTH 506 ADV. TOPICS IN NUMBER THEORY
- MTH 530 ELLIPTIC CURVES
- MTH 531 TOP IN ALG. NUMBER THEORY
Valuations, ideal theory, divisors. Class number, unit theorem. Geometric applications.

*Last Offered: Spring 2019* - MTH 535 COMMUTATIVE ALGEBRA
Field theory, valuations, local rings, affine schemes. Applications to number theory and geometry.

*Last Offered: Fall 2017* - MTH 537 HOMOLOGY
Projective and injective modules, complexes and resolutions, derived functors, including Ext and Tor, the homology and cohomology theory of groups and algebras, applications to the extension problem, etc.

*Prerequisites: MTH 436**Last Offered: Fall 2018* - MTH 538 TOPICS IN ALGEBRAIC GEOMETRY
Spaces with structure sheaf, schemes, cohomology of schemes, applications to algebraic curves and algebraic groups.

*Last Offered: Spring 2016* - MTH 539 TOPICS IN ALGEBRAIC GEOM II
- MTH 546 COHOMOLOGY
Brown's representability theorem, CW-spectra, and the Atiyah-Hirzebruch and Adams spectral sequence. The classical spectra BO, BU, their connective spectra, and the Thom spectra. Applications to the classical problems of Hopf invariant, vector fields on spheres, and geometric dimension of vector bundles.

*Last Offered: Fall 2011* - MTH 547 TOPICS IN DIFFERENTIAL GEOM.
*No description**Last Offered: Fall 2010* - MTH 548 LIE GROUPS AND ALGEBRA
Structure theory of finite dimensional Lie algebras, root-weight systems, Dynkin diagrams, classification of semi-simple Lie algebras and Lie groups and applications. If time permits further topics include p-adic Lie algebras and pro-p groups, finite simple groups of Lie type and knot invariants of Lie type.

*Last Offered: Fall 2017* - MTH 549 TOPICS IN ALGEBRAIC TOPOLOGY
I TOPICS: The course will cover the classical theory of fiber/fibre bundles and their associated principal G-bundles with a focus on vector bundles and characteristic classes. If time permits some discussion and applications of K-theory will also be covered. These topics are relatively classical (1950’s and 60’s) but fundamental to much current work in topology, geometry and modern physics. Tentative syllabus below: PREREQs: Prior intro course in algebraic topology and at least simultaneously taking a manifolds course.

*Last Offered: Spring 2018* - MTH 550 TOPICS IN TOPOLOGY
- MTH 555 TOPICS IN ADV. DIFF. GEOMETRY
Moving frames, connections, bundles; Gauss-Bonnet theorem and generalizations; theorems of Chern-Lashof; geodesics, Jacobi fields, index theorem.

*Last Offered: Fall 2012* - MTH 557 TOPICS IN DIFFERENTIAL GEOMETRY
Subject matter to be selected from among advanced copies of current interest in differential geometry and geometric analysis.

*Last Offered: Spring 2019* - MTH 562 FOURIER ANALYSIS
*No description* - MTH 565 TOPICS IN PARTIAL DIFFERENTIAL EQUATION
Linear partial differential operators with constant coefficients. Elementary solutions; elliptic, hypo-elliptic, and hyperbolic operators.

*Prerequisites: MTH 564**Last Offered: Fall 2018* - MTH 568 TOPICS IN NUMBER THEORY
This course starts with the definitions and introductory theory of modular forms, presents an overview of some of the classic papers on the subject, and focuses in on some of the recent advances. Particular topics chosen each year are left up to the individual instructor.

*Last Offered: Fall 2018* - MTH 569 TOPICS IN ANALYTIC NUMBER THEORY
Selected topics in non-multiplicative analytic number theory considered on a seminar basis.

*Last Offered: Fall 2017* - MTH 570 TOPICS IN ERGODIC THEORY & ARITHMETIC GEOMETRY
An introduction to the probabilistic viewpoint in dynamical systems, and the more recent equidistribution results in arithmetic geometry. After brief overview of ergodic theory and dynamical systems, the course will center on arithmetic heights in dimension 1. The emphasis will be on the intersection of these topics: Arithmetic dynamics.

*Last Offered: Fall 2015* - MTH 578 TOPICS IN HARMONIC ANALYSIS
*No description**Last Offered: Spring 2018* - MTH 999B IN-ABSENTIA ABROAD
*No description**Last Offered: Spring 2019*