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

date
19.06.2018
time
03:00 PM - 04:00 PM
speaker
Maxime Gheysens
affiliation
TU Dresden, Institut für Geometrie
language
en
main topic
Mathematics: general
host
Prof. Dr. A. Thom
abstract

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).

Last update: 30.05.2018 11:51.

venue

TUD Willers-Bau (WIL A 120)
Zellescher Weg 12-14
01069 Dresden
homepage
https://navigator.tu-dresden.de/etplan/wil/00

organizer

TUD Mathematik
Willersbau, Zellescher Weg 12-14
01069 Dresden
telefon
49-351-463 33376
homepage
http://tu-dresden.de/mathematik
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