The Hecke algebra and its generalizations are central objects in modern representation theory. They naturally appear in number theory, representation theory, algebraic geometry, and even low-dimensional topology. A categorification of the Hecke algebra was used to define a new topological invariant of knots and links, known as HOMFLY-PT homology. However, it is extremely difficult to compute this invariant from the definition. The project is focused on understanding the algebraic, geometric, and combinatorial structure of link homology and categorified Hecke algebras, with the goal of unifying, deepening, and clarifying connections between these concepts.
Recent progress strongly indicates a connection between the HOMFLY-PT homology and algebraic geometry of the Hilbert scheme of points on the plane, a central object in modern algebraic geometry and geometric representation theory. In this collaborative project, the investigators plan to compare and unify different approaches to the study of this connection and to develop the fundamental understanding of the relation between the category of Soergel bimodules and the Hilbert scheme. They also plan to provide an algebro-geometric construction of HOMFLY-PT homology and to understand its relation to the combinatorics of Macdonald polynomials.