My research area is Arithmetic Algebraic Geometry. I am particularly interested in algebraic geometric problems which originate from the Langlands program or the theory of Shimura varieties. There are relations to other areas, for instance to number theory and representation theory.
Specifically, I am working on
- Moduli spaces of abelian varieties in positive characteristics, Shimura varieties
- Affine Deligne-Lusztig varieties, Bruhat-Tits buildings
- Local models of Shimura varieties, varieties of quiver representations
- Affine Grassmannians und affine flag manifolds and perverse sheaves on these spaces, singularities of Schubert varieties
- Affine Weyl groups and Iwahori-Hecke algebras
Some of these questions lead to concrete combinatorial problems which can be investigated using computer programs. For instance, I have written programs which can do computations involving affine Weyl groups, the Bruhat order, and galleries in Bruhat-Tits buildings, and Kazhdan-Lusztig polynomials.