Department of Mathematics - Hong Kong Geometry Colloquium - The Geometry of Grassmannians of Complexes and Their Applications

Derived Algebraic Geometry (DAG), pioneered by Toën, Vezzosi, Lurie, and others, enhances classical algebraic geometry by incorporating homotopy theory concepts and ideas. Using the DAG framework, we can extend Grothendieck’s theory of Grassmannians of quasi-coherent sheaves to those of complexes. This extension is crucial for uncovering the inherent structures of Grassmannian fibrations and for unifying and extending various formulas. Additionally, it is valuable for constructing and studying moduli spaces and their wall-crossings.

In this talk, I will outline the construction and properties of derived Grassmannians of complexes, and reveal structural results of their Chow groups, K-theory, and derived categories. This includes a unifying formula that simultaneously extends existing formulas for Grassmannian bundles, blowups, standard flips, projectivizations, and Grassmannian flips. Furthermore, I will discuss the application of this framework in geometric representation theory. This talk is based on a series of my previous papers and ongoing collaborations with Weiping Li and Yu Zhao.