Mean convergence of ergodic averages and continuous model theory

Series: 
CDSNS Colloquium
Monday, February 15, 2016 - 11:00
1 hour (actually 50 minutes)
Location: 
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.