Manan Jain, Joshua J Abraham and company organizes an online seminar on abstract harmonic analysis following the beloved textbook by Folland. We shall meet every Sunday at 9 AM.
| Date | Topic | Speaker |
|---|---|---|
| 9 Aug | Schur’s lemma for unitary representations | Joshua |
| 16 Aug | Borel functional calculus for the proof of Schur’s lemma | Rupadarshi |
| 23 Aug | Asymptotic Schur orthogonality | Manan |
We revive the seminar by shaking things up!
| Date | Topic | Speaker |
|---|---|---|
| 22 Sept | Does every group have an invariant metric, on the Birkhoff-Kakutani theorem | Manan |
| 26 Sept | Yet another talk on amenability | Manan |
| A zoo of obstructions to invariant metrics | Rupadarshi | |
| Which definition of amenability is your favourite? | Rupadarshi | |
| [math/0606794v1] Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces | Manan | |
| Ergodic theorem for metric groups | Rupadarshi | |
Schur’s lemma for unitary representations
Joshua kickstarted our harmonic analysis seminar with the proof of Schur’s lemma for unitary representations, following Folland’s textbook.
Borel functional calculus for the proof of Schur’s lemma
I had already read Gelfand-Naimer theorem from Dietmar for the previous seminar on harmonic analysis.
I looked at Folland’s text for the construction of Borel functional calculus, which was needed in the proof of Schur’s lemma.
And then gave a talk on it. The following are the notes from the talk.
Asymptotic Schur orthogonality
Manan talked about the topics he had read in the past few months. This became a (sorta) sequel to his previous talk on the Gelfand-Raikov theorem.
I took the following notes.
Does every group have an invariant metric?
Abstract
The study of topological groups naturally leads one to the question in the title. In this talk, we will prove the Birkhoff-Kakutani theorem, which shows that the metrisability of a Hausdorff group is equivalent to first countability, which in turn is equivalent to the existence of an invariant metric generating its topology. The proof will illustrate dyadic decompositions in a very general framework. The only formal prerequisites for the talk are basic analysis and group theory.
Manan shared the following notes: The Birkhoff-Kakutani Theorem.
Yet another talk on amenability
Abstract
The notion of amenability can be traced back to Lebesgue’s musings about his description of the integral, which snowballed into the Banach-Tarski paradox, and has now exploded into a vast area of investigation. In this talk, we will discuss this historical backdrop and some equivalent formulations of amenability for groups. The talk will assume some familiarity with basic measure theory and functional analysis. No special knowledge of group theory is needed.