On, around, and beyond Ryll-Nardzewski's theorem III

Date

Jul 3, 2018

Time

3:00 PM - 4:00 PM

Speaker

Maxime Gheysens

Affiliation

TU Dresden, Institut für Geometrie

Language

en

Main Topic

Mathematik

Other Topics

Mathematik

Host

Prof. Dr. A. Thom

Description

12.06. / 19.06. / 03.07.2018 The lectures will focus on a cornerstone theorem of functional analysis, Ryll-Nardzewski's theorem. This powerful result asserts the existence of a fixed point for a very broad class of actions on convex sets. I. Ryll-Nardzewski's theorem. After recalling some facts about group actions and functional analysis, we will state and prove Ryll-Nardzewski's theorem. II. Applications. The second lecture will focus on applications of Ryll-Nardzewski's theorem. We will cover in details the existence of Haar measures for compact groups and the unbounded vs. fixed-point dichotomy for isometric actions on reflexive Banach spaces. Other consequences (for instance, for von Neumann algebras or for compactifications of groups) will be mentionned, according to the interest of the audience. III. Sibling theorems. We will explain three interesting related theorems: a fixed-point theorem for $\mathrm{L}^1$-spaces (an unexpected application of Ryll-Nardzewski's theorem), a fixed-point theorem for actions on convex cones (a generalisation thereof), and the Day--Rickert characterization of amenability (which shows that Ryll-Nardzewski's theorem is somehow optimal).

Links

• Graduate Lectures in Mathematics

Last modified: Jun 5, 2018, 4:46:34 PM