"Geometric Methods in Lie Theory"
This project is relevant to Lie theory, a well-established domain in pure mathematics which involves algebra and geometry. The geometric methods pop up with a rich variety: complex algebraic geometry, Schubert calculus, rational points on varieties, affine Grassmannian, Bruhat-Tits buildings, Berkovich spaces…
The aim of this project is to share ideas among specialists of these subjects. These themes have deep interplays: for example, the multiplicative Horn problem concerning orbits of compact Lie groups is related to Geometric Invariant Theory, quantum Schubert calculus, affine Grassmannian and semistability in infinite dimension. Moreover, the problem of conjugacy in infinite dimensional Lie algebras as studied by Gille et al. is relevant for both items 1 and 2 below. Another example is the work of DeJong-He-Starr on rationality questions on homogeneous spaces which is of interest for quantum Schubert calculus and Galois cohomology.
The project is divided into the four axes below.
- 1. Points of algebraic groups, Bruhat-Tits buildings and Berkovich spaces
- 2. Affine Grassmannians and Kac-Moody groups
- 3. (Quantum) cohomology of homogeneous spaces
- 4. Geometric problems related to (co)-adjoint actions
Main fields of research
- Algebraic geometry
- Algebraic groups
- Berkovich spaces