CAREER: Mapping class groups, diffeomorphism groups, and moduli spaces

Topology is the study of spaces and their fundamental properties. Related to this project, there are two foundational principles. First, a given space can be fruitfully probed by understanding its symmetries. Second, if one wants to study some collection of spaces (for a simple example, think of polygons in the plane), then it can be helpful to turn this collection of spaces into a single space (often called a moduli space) whose properties are illuminating. For the educational component, the PI will organize topology workshops with the aim of introducing graduate students to active areas of research and giving them tools to contribute to these areas. The PI will orchestrate summer directed reading programs that will help prepare undergraduate students for research and broaden participation. Finally, the PI will continue his outreach to high school students through math circles. Support from this grant will help increase teacher involvement through events hosted on campus. This project is concerned with group actions on manifolds and moduli spaces. Regarding group actions, the focus is on determining when a group of isotopy classes of homeomorphisms can be realized as a group of homeomorphisms. This problem, known as Nielsen realization, dates back to the work of Nielsen in the early 1900s and has connections to dynamics, foliation theory, and the geometry of manifolds and fiber bundles. The moduli spaces of interest in this proposal relate to low-dimensional topology and nonpositive curvature. A central goal is to compute new homological and homotopical invariants of the moduli spaces of interest. The specific research project are as follows. (1) Solve Nielsen realization problems for 3- and 4-dimensional manifolds. (2) Study finite groups actions on aspherical manifolds with exotic smooth structures using rigidity results related to the Borel conjecture. (3) Show non-triviality of twist tori in the homology of finite covers of moduli space by reducing the problem to tropical geometry and combinatorics. (4) Construct new characteristic classes of surface bundles using the curve complex. (5) Compute homotopy groups of embedding spaces for high dimensional hyperbolic manifolds, building on work of Farrell-Jones. 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.