Aspects of Polish group dynamics

The project will mostly focus on symmetry groups of certain mathematical objects. Two examples of such objects are: a space with a measure and a generic generalized curve. The interest in these objects and their symmetries originates in classical mechanics and in geometric topology, respectively, but they will be investigated using methods coming from a variety of mathematical fields. The interaction of these fields is of independent mathematical interest. The project involves graduate student training. The project will tackle problems on the boundary of Descriptive Set Theory, or more broadly, Logic, Ergodic Theory, and Topology. A precise analysis will be made of closed groups generated by measure preserving transformations. Topological dynamics of the homeomorphism group of a certain canonical continuum, called the pseudoarc, will be investigated. Combinatorial properties related to Ramsey theory of a class of maps of $n$-dimensional simplex will be explored. These diverse themes will be tied together by a point of view coming from Mathematical Logic. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.