Reflection groups, moduli spaces and mapping class groups of 4-manifolds

The goal of this project is to understand fundamental relationships between symmetries arising in different parts of mathematics. An example of such a (still mysterious) connection is between certain 4-dimensional spaces central in physics and collections of reflective symmetries of negatively curved spaces. This project will also have the broader impact of training many PhD students and postdocs. The project is meant to develop a theory of mapping class groups of closed 4-dimensional manifolds by bringing in viewpoints, methods and ideas from the 2-dimensional case. This goal is only one piece of a broader research program of the principal investigator relating the topology of moduli spaces, the monodromy of fiber bundles, the theory of arithmetic/reflection groups and 4-manifold topology. 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.