Courses
First Year 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 - Real Variables. Measure theory, including the Radon-Nikodym theorem and differentiation theory. Lp spaces. The Riesz representation theorem. Fourier series. Hilbert transform. PQ: graduate student status or instructor consent.
313. Analysis II - Functional Analysis. Basic principles of functional analysis. Distribution theory. Fourier transform. Sobolev and other classical function spaces. Bounded operators, compact operators, spectral theory, Fredholm theory. Applications to partial differential equations: boundary value problems and variational principles. PQ: Math 312.
314. Analysis III - Complex Analysis and Topics in Analysis. Complex Analysis and Topics in Analysis. Basic complex analysis. Riemann mapping theorem including continuity up to the boundary. Picard theorems. Riemann surfaces. Further topics in analysis. 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.