Combinatorial Models of Perverse Sheaves

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The principal investigator will study combinatorial models of categories of perverse sheaves on spaces which carry important combinatorial and representation theoretic information, mainly flag varieties and toric varieties. The models use data obtained from natural symmetries on these spaces via the moment map. These methods will be applied to problems in representation theory, including Lusztig's conjecture on modular representations and Beilinson, Ginzburg and Soergel's theory of Koszul duality for Lie algebra representations. The PI and his collaborators will develop computer algebra software to do computations with these models. This project involves geometry related to representation theory. Representation theory studies ways to present a given algebraic object, usually a group of symmetries, as a collection of linear transformations. It has applications to many fields, from number theory to mathematical physics. An important thread in modern representation theory involves extracting algebraic information from the geometry of an associated space, which is generally also intrinsically beautiful in its own right.