Title: Attaching algebraic varieties from representations

Abstract: Given a category of representations of some algebraic (geometric objects), there are many ways to attach algebraic varieties, so that the representations can be regarded as quasi-coherent sheaves on it, and then using the algebraic geometric method in turn to construct invariants of the representations and to stratify the category of representations. There are many different ways to construct the algebraic varieties. In this talk I will review the cohomological construction from representations of finite groups finite groups and Lie algebras. Taking these as a motivation, I will describe the constructing cohomological construction of algebraic varieties for representations of vertex operator algebras, which can be thought as the Koszul dual of another construction called the associated varieties. There are other ways to attach algebraic varieties such as moduli spaces of stability conditions, which will not be covered in this talk. All these constructions can be thought as part of (derived) non-commutative algebraic geometry.