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: [6 lectures] Definition of topological groups. Examples and non-examples. Quotient groups. Subgroups. Compactness and local compactness. Non-functional separation axioms. The Open Mapping Theorem.

[5 lectures] Complete regularity of topological groups. Continuous partitions of unity and Fubini’s Theorem. Existence and uniqueness of Haar integrals.

[5 lectures] Dual groups of topological groups. Local compactness of the dual of a locally compact topological group. Peter-Weyl Theorem for locally compact Abelian topological groups.