# Research

## Reseach Summary

My research interests are at the interface of algebra and mathematical physics. More specifically, I am interested in mathematical structures arising in three-dimensional gravity, Chern-Simons theory and quantum geometry. These interest of these research subjects is that they tie together a variety of topics from algebra, geometry and topology:

### Algebra

- Poisson-Lie groups
- Hopf algebras and quantum groups
- representation theory of Lie groups and quantum groups
- (braided) monoidal categories and higher categories
- braid groups and mapping class groups
- Fuchsian groups

### Geometry

- Moduli spaces of flat connections
- Poisson and symplectic geometry
- Teichmüller theory
- differential geometry in three dimensions
- hyperbolic geometry in two and three dimensions

### Topology

- link and knot invariants
- quantum invariants of manifolds contructed from representation theory
- topological quantum field theories