Logic, dynamics and Ramsey theory

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The program aims at applications of logic to various branches of mathematics such as topological dynamics, operator algebras or Ramsey theory. The long-term objectives are:******I. Find new techniques in the study of countable Borel equivalence relation, with connections to geometric group theory and symbolic dynamics.***II. Study the universality question for locally compact amenable groups***III. Find new examples of separable C*-algebras that come from model theory and, in particular from continuous logic.***IV. Study combinatorial problems in Ramsey theory using recent techniques developed in the study of Ramsey spaces.******I(a). The theory of countable Borel equivalence relation is strictly connected with geometric group theory and the study of actions of countable groups on measure spaces. One of the central themes in the subject is the notion of treeability (and strong treeability), developed (among others) in the theory of cost. Recent breakthroughs by Conley, Marks and Tucker-Drob have lead to proofs of strong treeability of an important class of groups (surface groups). Our program concentrates on pursing this path and studying other classes of groups motivated by geometric groups theory (such as hyperbolic groups).******I(b) The conjugacy problem in symbolic dynamics can be viewed as a countable equivalence relation. Several authors (Gao, Jackson, Seward, Thomas) have asked about the complexity of this equivalence relation for various classes of groups. Within this program we plan to study the conjugacy relation of several classes of minimal subshifts.******II. The question of the existence of universal objects in various categories leads either to interesting new constructions, or to the development of structural theory. The study of the existence of universal amenable discrete groups resulted in deep structural results on the growth of Foelner sequences (in the work of Erschler). Within this program we plan to focus on locally compact amenable groups.******III. The applications of logic to operator algebras is a rapidly developing subject that uses methods from both set theory and model theory. During his ICM address in Seoul, Farah proposed several new directions for applications of logic to the study of separable C*-algebras. Within this program we plan to find applications of continuous model theory in building new examples of separable C*-algebras.******IV. Recent breakthroughs in Ramsey theory have lead to proofs of several long-standing conjectures on density versions of Ramsey theorems for trees. This includes the work of Dodos, Kanellopoulos, Karagiannis and Tyros and is motivated by the framework of Ramsey spaces developed recently by Todorcevic. We plan to extend and apply these techniques to work on multi-dimensional versions of classical density Ramsey theorems, as well as to extend their methods to other Ramsey spaces.**