Description:

Title: Følner's Theorem and Model Theory

Abstract: In 1954, Følner proved that if G is any abelian group and A is a subset of positive upper Banach density, then the difference set A-A "almost contains" an open neighborhood U in the Bohr topology on G in the sense that U\(A-A) has upper Banach density 0. An analogous result for countable amenable groups was proved in 2009 by Bergelson, Beiglböck, and Fish using methods from ergodic theory. In this talk, I will present a new proof of Følner's Theorem valid for an arbitrary (discrete) amenable group. The argument uses 1980's results in topological dynamics along with relatively classical functional analysis. However, the overall strategy is directly inspired by model theory, and the proof can be viewed as an application of local stable group theory in continuous logic. Time permitting, I will also discuss stronger forms of Følner's Theorem under additional model-theoretic tameness assumptions (namely, stability and bounded VC-dimension).