UNIVERSITY OF ILLINOIS
The PI intends to work on three different topics involving descriptive set theory and its applications in Banach space theory and topological dynamics. Firstly, he intends to further elaborate on the structure theory of large Polish group and, in particular, automorphism groups of countable first order structures. The PI's main interest in this domain is the existence of generic representations of countable discrete groups in such automorphism groups and its uses in investigating the dynamics and algebraic structure of the automorphism group.
Secondly, the PI will be working on the classical problem of J.P.R. Christensen of whether any universally measurable homomorphism between Polish groups is continuous. Again, this problem is tightly connected with the geometry and dynamical properties of Polish groups and very likely to leads to new techniques and concepts of value outside of the study of automatic continuity.
Finally, Rosendal will be continuing his collaboration with V. Ferenczi on the rough classification of separable Banach spaces, a programme initiated by W. T. Gowers in the course of the solution to the homogeneous space problem. This programme heavily relies on Ramsey theoretical and game theoretical methods from descriptive set theory and thus enriches both descriptive set theory by developing new techniques and functional analysis by contributing to one of the most central problems of Banach space theory.
In recent years, there has been a growing collaboration and sharing of ideas between researchers working in dynamical systems and descriptive set theory. Initially, this was largely fostered by work on Borel equivalence relations but has lately also come to include topological dynamics. The principal objectives of descriptive set theory consists in studying the definable subsets of the real line, showing that these are much better behaved than arbitrary subsets. This has developed into a large structure theory of such sets which gives a solid framework within which most of mathematical analysis takes place, and as such, the use of descriptive set theoretical techniques in parts of functional analysis is continuously expanding.