Grant Details

Enumerative Geometry deals with counting all sorts of geometric objects on algebraic varieties, with wide applications to various fields of mathematics and theoretical physics (most notably quantum field theory). In recent years, new meanings of the word “counting” have arisen, wherein instead of trying to calculate explicit numbers, one seeks to understand “higher” structures that underlie the geometry: various filtrations and algebra actions on (co)homology groups of the spaces involved, lifts of such structures to K-theory groups and derived categories, and from here to connections to categorified link invariants and diagrammatic categories. For example, a successful realization of such a program in recent years was the interpretation of Donaldson-Thomas invariants via cohomological / K-theoretic / categorified Hall algebras, which continues to produce beautiful mathematics. A recurring feature in studying such “higher structures” pertaining to algebraic spaces is the appearance of quantum groups and vertex algebras, that provide symmetries for the geometry in question, and they are ubiquitous in the study of cohomological/K-theoretic/categorified structures. The purpose of this project is for the PI’s from MIT and the University of Pisa to share their complementary knowledge in various applications of the theories of quantum groups and vertex algebras to the “higher counting” in enumerative geometry, with the goal of uncovering deeper connections between fields and discovering new geometric features of algebraic spaces.