First Year Graduate Courses

The courses described below are part of the first year graduate program. The program undergoes regular reevaluation and change, so the list of topics is only approximate. Each year, a number of more advanced courses are offered, with the subjects varying from year to year.

312. Analysis I - Measure & Integration. Measure theory, including Lebesgue and Hausdorff measures, integration, convergence theorems, the Radon-Nikodym theorem, and differentiation theory. Normed linear spaces, Lp spaces, completeness, duality, and the Riesz representation theorem. Fourier series, the Poisson summation formula, and the boundedness of the Hilbert transform. PQ: Math 262, 270, 272, and 274.

313. Analysis II - Functional Analysis. Frechet spaces, spaces of smooth functions, weak topologies and weak convergence, distributions and Fourier analysis, including mollifiers, convolution, the Paley-Wiener theorem, and local solvability of constant coefficient PDE. Sobolev spaces and the embedding theorems. Operator theory, including compact and bounded operators, integral operators, spectral theory and Fredholm operators. Applications to representation theory of compact groups (the Peter-Weyl theorem) and an introduction to the calculus of variations. PQ: Math 312.

314. Analysis III - Complex Variables. A review of the basic theory of one complex variable: Cauchy's theorem, the Cauchy-Riemann equations, power series expansions, the maximum modulus principle, classification of singularities, and the residue theorem. Normal families, conformal mapping and the Riemann mapping theorem. Prescribing zeros and poles of meromorphic functions. Harmonic functions and the Dirichlet problem. Introduction to Riemann surfaces. Negative curvature and Picard's Big Theorem. According to the inclinations of the instructor, further topics may include: holomorphic functions of several variables (e.g. Hartogs' Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces. PQ: Math 313.

317. Topology and Geometry I - Smooth Manifolds. Definition of manifolds, tangent and cotangent bundles, vector bundles. Inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem. Vector fields and flows, Frobenius' theorem, differential forms and the associated formalism of pullback, wedge product, integration, etc. Cohomology via differential forms, and computational tools, e.g. the Poincaré lemma and the Mayer-Vietoris sequence. The degree of a map between compact oriented manifolds. Lie groups and Lie algebras. PQ: Math 261-2-3.

318. Topology and Geometry II - Differential Geometry. Riemannian metrics, connections and curvature on vectorbundles, the Levi-Civita connection, and the multiple interpretations of curvature. Geodesics and the associated variational formalism (formulas for the 1st and 2nd variation of length), the exponential map, completeness, and the influence of curvature on the structure of a manifold (positive versus negative curvature). The Gauss-Bonnet theorem and possibly the Hodge Theorem. PQ: Math 317.

319. Topology and Geometry III - Basic Homology. The fundamental group, covering space theory and Van Kampen's theorem (with a discussion of free and amalgamated products of groups). CW complexes, higher homotopy groups, cellular and singular cohomology, the Eilenberg-Steenrod axioms, computational tools including Mayer-Vietoris, cup products, Poincaré duality, and the Lefschetz fixed point theorem. Homotopy exact sequence of a fibration and the Hurewicz isomorphism theorem. Remarks on characteristic classes. PQ: Math 318.

325. Algebra I - Group theory. Linear groups, semisimple algebras and modules, and group representations. PQ: Math 254-5-6.

326. Algebra II - Commutative Rings and Homology. Noetherian rings and modules, the Hilbert basis theorem. Integral extensions, the going-up theorem. Localisation, exactness of localisation. Finitely generated algebras over a field, varieties, the Noether normalisation lemma, Hilbert's Nullstellensatz, dimension. Discussion of the dictionary between commutative algebra and algebraic geometry. Other possible topics include: Kahler differentials, smoothness, completions, power series rings, the p-adic numbers. Ext and Tor. Dedekind domains. The spectrum of a commutative ring and the sheaf associated to a module. PQ: Math 325.

327. Algebra III - Topics in Algebra. According to the inclinations of the instructor, this course may cover: Galois theory, algebraic number theory, algebraic curves, multilinear algebra (tensor, symmetric and exterior algebras), Lie algebras, homological algebra and/or the cohomology of groups. PQ: Math 326.