Rosendal proposes to pursue a general study of Polish groups from the point of view of descriptive set theory, model theory and topological dynamics. In the project Rosendal will investigate both algebraic and topological properties of these groups, such as the small index property, the Bergman property, phenomena of automatic continuity of homomorphisms and extreme amenability. The main problems investigated are to a great extent motivated by the model theoretical problem of reconstructing a countable structure from its group of automorphisms. One of the principal tools used in this connection is the automatic continuity of isomorphisms of automorphism groups, which motivated the study of the broader phenomenon of automatic continuity of arbitrary homomorphisms between classes of Polish groups. Rosendal also plans to apply these ideas to study the topological dynamics of uncountable discrete groups most notably in connection with the fixed point on metric compacta property. The study of Polish groups using the very diverse methods of several fields seems likely to promote the further integration of separate knowledge and deeper understanding of the objects considered. In a separate project, Rosendal intends to continue his work with V. Ferenczi on a question of G. Godefroy concerning the number of non-isomorphic subspaces of a non-Hilbertian Banach space. The methods that have proven useful at this moment have been highly set theoretical in nature and have reduced the problem to the case of minimal spaces. Probably more analytical tools will be of greater utility for the next steps. In connection with this, Rosendal plans to revisit Gowers' determinacy theorem and by using set theoretical tools extend its applications beyond its nominal reach of analytic sets. This should provide tools in Banach space theory not obtainable by classical geometric considerations. Also Rosendal will pursue another ongoing project with B.D. Miller of classifying Borel transformations up to a descriptive notion of Kakutani equivalence. This is a completely parallel project to the theory of Kakutani equivalence in ergodic theory, but due to the nature of the objects, the methods used are completely descriptive set theoretical and go back to works of Glimm and Effros in operator algebra. By stressing the interconnections of mathematical logic with other domains of mathematics, Rosendal hopes to enrich both logic itself and provide new insight into objects outside of mathematical logic. The main applications of his research will be in functional analysis (Banach space theory), topological groups, and ergodic theory.
