DMS Algebra Seminar Pretalk

Speaker: Hang Huang

Title: Derived Grassmannians, derived Schur functors, and their applications

Abstract: In this talk, we will revisit Grothendieck's theory of Grassmannians and flag schemes and the theory of Schur and Weyl module functors studied in $GL_n$-representation theory from the perspective of derived algebraic geometry (DAG). We will explain how to use the DAG framework to extend these theories from modules to complexes, and the numerous theoretical benefits of doing so. Next, we will show how these two new theories are connected by a derived generalization of the Borel--Bott--Weil theorem. Finally, we will discuss how this framework broadens the application range of classical theories and sheds new light on many classical problems, including the study of derived categories of singular schemes, and of Hilbert schemes and compactified Jacobians of integral curves, as well as their applications to a variety of recent research topics including Hecke correspondences for surfaces and two-dimensional categories. The talk will be based on papers arXiv:2202.11636 and arXiv:2212.10488.