Operator Theoretic Aspects of Ergodic Theory
lunchtime seminar // 10 January 2023
Jochen Glück (Wuppertal):Ergodic theory meets partial differential equations: The Halmos-von Neumann theorem and the shape of a drum

The aim of this talk is to present and compare two classical - but quite different, at first glance - topics from the realms of ergodic theory and partial differential equations:

  1. The Halmos-von Neumann theorem in ergodic theory says that certain dynamical systems are uniquely determined up to isomorphism by their spectral properties.
  2. The - still wide open - problem under which conditions one can "hear the shape of a drum" in PDE theory asks: when does the spectrum of the Laplace operator (say, with Dirichlet boundary conditions) on a bounded domain in Rd determine the domain uniquely up to congruence.
We discuss that a connection between these two, apparently rather different, topics can be found by using the language of positive operators: both the conclusion of the Halmos-von Neumann theorem and the desired conclusion of the "shape of a drum" problem can be rephrased as the existence of a positivity preserving linear operator that intertwines certain dynamical systems The question thus arises whether insights from the proof of the Halmos-von Neumann theorem can be useful to gain a better understanding of the "shape of a drum" problem.

Organized by Tanja Eisner, Bálint Farkas, Markus Haase, Henrik Kreidler, Rainer Nagel

The talks take place on Tuesdays at 12:30 CET simultaneously in
  • Kiel: Raum EG.029, Math. Seminar, Heinrich-Hecht-Platz 6
  • Wuppertal: G15.25, Gaußstraße 20
  • Tübingen: Raum C6P05, Auf der Morgenstelle 10
  • zoom: please register to get the link
If you are interested in participating, please write an e-mail to otet (at) math.uni-wuppertal.de or to kreidler (at) uni-wuppertal.de