# Lecture series by J. Kamnitzer

The lecture series *"Geometric Satake, MV polytopes, and quiver varieties"* by **Joel Kamnitzer** will take place on Thursdays, November 6, 13, 27 and December 4, 11, at 14:15 in Room MAA110, EPFL.

**Abstract:**

The representation theory of reductive groups is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology.

One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geometry of spaces called affine Grassmannians with the representation theory of reductive groups. This correspondence was originally developed from the viewpoint of the geometric Langlands program, but it has many other interesting applications. For example, this theory gives rise to a natural basis for irreducible representation. This basis is labelled by certain varieties called Mirkovic-Vilonen cycles.

On the other hand, there exists another natural basis for irreducible representations, known as Lusztig's (semi)-canonical basis. This basis is constructed through Lusztig's nilpotent varieties, which parametrize representations of preprojective algebras. This leads to the question of the relationship between these two bases.

In my lectures, I will explain the construction of these two bases. I will then describe a combinatorial link between them using the theory of MV polytopes. Finally, I will explain ongoing work to give a more geometric foundation to this combinatorial link.