Mean convergence of ergodic averages and continuous model theory

CDSNS Colloquium
Monday, February 15, 2016 - 11:00am
1 hour (actually 50 minutes)
Skiles 005
University of Texas at San Antonio
The Mean Ergodic Theorem of von Neumann proves the existence of limits of (time) averages for any cyclic group K = {U^n : n \in Z} acting on some Hilbert space H via powers of a unitary transformation U.  Subsequent generalizations apply to so-called _multiple_ ergodic averages when Z is replaced by an arbitrary amenable group G, provided the image group K is nilpotent (Walsh's ergodic 2014 theorem for Z; generalization to G amenable by Zorin-Kranich).  In this talk we survey a framework for mean convergence of polynomial group actions based on continuous model theory.  We prove mean convergence of unitary polynomial Z-actions, and discuss how the full framework accomodates the most recent results mentioned above and allows generaling them.