# Emmy-Noether-Seminar

**Hochschild cohomology of partial flag varieties**

Maxim Smirnov (Augsburg)

Abstract: The Hochschild-Kostant-Rosenberg decomposition gives a description of the Hochschild cohomology of a smooth projective variety in terms of the sheaf cohomology of exterior powers of the tangent bundle. In all but a few cases it is a non-trivial task to compute this decomposition, and understand the extra algebraic structure which exists on Hochschild cohomology. I will give a general introduction to Hochschild cohomology and this decomposition, and how this problem can be studied for partial flag varieties (i.e. varieties of the form $G/P$ for $G$ a semisimple algebraic group and $P$ a parabolic subgroup), in particular for the case of maximal parabolic subgroups (i.e. Grassmannians in type A and their analogues in other types). This is joint work in progress with Pieter Belmans.