Not available at this time

Maximum credit, 6.

Not available at this time

Not available at this time

The main goal of this class is to develop a number of tools from functional analysis and to show how they are used in various contexts by considering concrete examples (from partial differential equations, probability, dynamical systems, etc...). Among the topics covered in this class are: (i) Banach and Hilbert spaces, Linear functionals, Dual Spaces, Hahn-Banach Theorem. (ii) Linear operators (bounded and unbounded), open mapping and closed graph theorem, spectrum, Banach algebras, functional calculus. (iii) Spectral theory for compact operators, Fredholm theory, positive operators.

An introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, vector spaces, rings, and homomorphism between them). Emphasis will be placed on being able to compute these invariants, not just on their definitions and associated theorems. Prerequisites: Math 671 and Math 611

This class will cover both fundamentals of algebraic number theory and various advanced topics, including: rings of integers, prime decomposition in number fields, ideal class groups, units, class field theory, cyclotomic fields, Iwasawa theory. Prerequisite: Math 612 or equivalent

Maximum credit, 6.

Inverse and implicit functions theorems, rank of a map. Regular and critical values. Sard's theorem. Differentiable manifolds, submanifolds, embeddings, immersions and diffeomorphisms. Tangent space and bundle, differential of a map. Partitions of unity, orientation, transversality embed-dings in Rn. Vector fields, local flows. Lie bracket, Frobenius theorem. Lie groups, matrix Lie groups, left invariant vector fields, tensor fields, differential form, integration, closed and exact forms. DeRham's cohomology, vector bundles, connections, curvature.