May 2015 - The National Science Foundation awards a grant to Dr. Jose Iovino & Dr. Eduardo Dueñez for their project entitled “Model Theory and Ergodic Theorems”.
Ergodic theory is a branch of mathematics concerned with the behavior of dynamical systems when they are allowed to run over long time intervals. One of the fundamental results of ergodic theory is the Mean Ergodic Theorem of von Neumann (1932), which amounts to an abstract statement about the limiting average state of a conservative system. In this project we seek to apply concepts and techniques of model theory, a branch of mathematical logic, to establish results on ergodic averages and ergodic recurrence by reinterpreting and extending research results discovered in the decades following von Neumann. The model-theoretic viewpoint should help illuminate and potentially unveil connections between ergodic theory and other branches of mathematics.
One of the project's goals is casting recent results on convergence of multiple ergodic averages into a suitable model-theoretic framework that allows combining analytic arguments with the theory of types, forking calculus, and ordinal ranks. An important concept in the proposed research is a ranking of the complexity of "polynomial actions" of a group G on some measure space X, or rather of the induced (polynomial) actions of G on various topological spaces (say, of functions on X). This requires developing an abstract algebraic framework that extends Leibman's theory of polynomial mappings between groups. Ancillary anticipated products of the model-theoretic approach include results on metastable convergence (akin to a weak form of uniformity in situations in which uniform convergence is absent), on convergence of averages on ultraproducts of polynomial-ergodic systems, and on polynomial-ergodic actions of ordinal (transfinite) rank (generalized Leibman degree).Connections and applications to combinatorics, Ramsey theory, and number theory will also be studied.