Topological Groups (2019-2020)

2019-2020
Lecturer(s): 
Prof. Tom Sanders
General Prerequisites: 

A5 Topology and A4 Integration essential.

Course Term: 
Trinity
Course Overview: 

Groups like the integers, the torus, and $\text{GL}_n$ 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 of Haar measure for (not necessarily Abelian) locally com-pact Hausdorff topological groups, Pontryagin duality, and the structure theorem for locally compact Hausdorff Abelian topological groups. Throughout, the course will use the tools of analysis to tie together the topology and algebra, getting at superficially more algebraic facts such as the structure theorem through analytic means.

Course Synopsis: 

[4 lectures] Definition of topological groups. Examples including $\mathbb{R}^n$, $S^1$, the Cantor group, and $\text{GL}_n(\mathbb{R})$, and non-examples.

Quotient groups; subgroups; isomorphism theorems; local compactness; separation axioms.

[6 lectures] Covering numbers. Regular Borel measures. Existence of Haar measure assuming the Caratheodory Extension Theorem. Uniqueness of Haar measure. Examples.

[6 lectures] Abelian groups and their duals, annihilators and the relationship between subgroups and quotient groups, and between direct and complete direct sums. Pontryagin duality. Dual properties \emph{e.g.} discrete and compact; torsion-free-discrete and connected-compact; \emph{etc}. *** Time permitting the dual of the rationals. ***

Structure theorem for locally compact Abelian groups, and also for compactly generated locally compact Abelian groups. Examples. **Time permitting, discussion of the structure theorem for locally compact groups.**

Reading List: 
  1. [Fol95] G. B. Folland. A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.
  2. [Rud90] W. Rudin. Fourier analysis on groups. Wiley Classics Library. John Wiley & Sons Inc., New York, 1990. Reprint of the 1962 original, A Wiley-Interscience Publication.

    Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.