Course Overview: Groups like the integers, the circle, and general linear groups (over R or C) share a number of properties naturally captured by the notion of a topological group. Providing a unified framework for these groups and properties was an important achievement of 20th century mathematics, and in this course we shall develop this framework.
Highlights will include the existence and uniqueness of Haar integrals for locally compact topological groups, the topology of dual groups, and the existence of characters in various topological groups. Throughout, the course will use the tools of analysis to tie together the topology and algebra, getting at superficially more algebraic facts by analytic means.
Course Synopsis: [7 lectures] Definition of topological groups. Examples and non-examples, and basic topo- logical properties. Subgroups. Quotient groups. The Open Mapping Theorem.
[4 lectures] Complete regularity of topological groups. Continuous partitions of unity and Fubini’s Theorem. Existence and uniqueness of Haar integrals.
[5 lectures] The Peter-Weyl Theorem for compact topological groups. Dual groups of topo- logical groups. Local compactness of the dual of a locally compact topological group.
