In 2013, Ben Yaacov showed that Shelah's result on definability of local types for stable formulas can be seen as a corollary of Grothendieck's "double-limit" theorem, which characterizes relatively weakly compact sets in certain Banach spaces. Moreover, this theorem applies equally well to continuous logic, and is also valid for formulas that are only stable with respect to a fixed model. In this talk, we examine how this connection applies to stable group theory. We develop the foundations of stable group theory directly from topological dynamics and, in particular, results of Ellis and Nerurkar on weakly almost periodic flows (which are based on the same work of Grothendieck). Moreover, we extend the main ingredients of stable group theory to invariant formulas that are only stable "in a model" and, more generally, to the purely combinatorial setting of an arbitrary group G and an arbitrary left-invariant Boolean algebra of subsets that are "stable in G". Finally, we discuss some applications to additive combinatorics and combinatorial number theory.
Please send an email to r.zoghi@gmail.com to join this lecture.