The PI intends to conduct research on the structure of Polish groups and applications of Descriptive Set Theory and Ramsey theory in Functional and Harmonic Analysis. Recently, a number of techniques originating in abstract harmonic analysis have been adopted to the setting of non-locally compact Polish groups, thus providing a rich tool set for furthering the understanding of the topological groups arising internally in descriptive set theory and the model theory of countable first-order structures. The PI is engaged is studying various aspect of these tools, both applying the resulting structure theory of Polish groups to problems in functional analysis, as witnessed in his recent collaboration with V. Ferenczi on the existence of optimal norms on Banach spaces, and calibrating properties of countable first-order structures M with topological properties of their automorphism group Aut(M). In addition, the PI is studying the block Ramsey theory originating in the work of W.T. Gowers, which has led to significant results in the geometric theory of Banach spaces, in particular, a classification program for separable Banach spaces based on infinite-dimensional Ramsey theory. As has become clearer recently, the underlying Ramsey theoretical results are best understood in terms of infinite games of perfect information with very strong winning strategies and to which usual determinacy does not immediately apply.
The research described in the proposal is largely inter-disciplinary, connecting a variety of fields from mathematical logic to functional and harmonic analysis. The PI aims to further the continuous integration of descriptive set theory with other branches of mathematics and thus provide new venues for a field traditionally having benefited enormously from deep work in purer areas of set theory. In part through the PI's work, eventually, these results will hopefully trickle down to applications to rather tame problems in analysis, i.e., not a priori involving set theory or logic.
