# MATHEMATICS COLLOQUIUM - Yuri Berest (Cornell University)

**Abstract:** The set of all representations of a Lie algebra a in a finite-dimensional Lie algebra g has a natural structure of an affine scheme, called the representation scheme Repg(a). The representation functor Repg is not “exact" and can be derived in the sense of non-abelian homological algebra. In this talk, I will explain the construction and properties of derived representation schemes of Lie algebras and their homology (which we call representation homology). As a main example, we will consider the derived schemes associated with classical commuting schemes of complex reductive Lie algebras. We will present a general conjecture about the structure of these derived schemes and discuss its implications. Time permitting, we will also discuss a topological version of representation homology and its relation to higher order Hochschild homology and rational homotopy theory.