The Other Pure Seminar (TOPS): Functional Analysis (1 of 2)

Speaker: Joshua Celeste

Title: Functional Analysis

Abstract: Finite dimensional linear algebra is relatively simple. However, for most modern problems of interest (Differential Equations, Quantum Mechanics etc.), the natural vector spaces we encounter are all infinite dimensional (and thankfully equipped with a norm). It is precisely the tools in functional analysis that were built to deal with these sorts of problems, in which we can obtain many generalisations and extensions of finite dimensional results and encounter new objects as well.

In these series of talks I will provide a brief introduction into linear operators on Hilbert spaces, their spectrum and spectral decomposition, as well as some new concepts which only appear in infinite dimensions, such as the Fredholm index. Culminating in a discussion of a baby index theorem due to Fritz Noether if time permits.