Research

My research interests are categorification, low-dimensional topology, and representation theory. I am particularly interested in homology theories in low-dimensional topology and their categorical and representation-theoretic foundations. I am driven by the rich structures afforded by link homology theories such as Khovanov homology and HOMFLY–PT homology, and their deep connections with geometric representation theory and algebraic geometry.

Link homology and how to compute it

Khovanov–Rozansky homology assigns to each link a bigraded homology built from Soergel bimodules and Rouquier complexes. A recurring theme in my work is making these invariants computable and uncovering the higher algebraic structure they carry — for example the computation of homology of torus links via categorified Young symmetrizers, $y$-ification and its link to Hilbert schemes, and stable $\mathfrak{gl}_N$ homology of torus knots.

Hecke categories, traces, and centers

Much of my recent work concerns the categorical trace and Drinfeld center of the Hecke category of Soergel bimodules, and skein-theoretic descriptions of singular Soergel bimodules. These structures organize link homology operations and clarify the symmetries present in Khovanov–Rozansky homology.

Categorical diagonalization and idempotents

With collaborators I have developed a theory of categorical diagonalization — a categorical analogue of diagonalizing a linear operator — together with techniques for constructing categorical idempotents in triangulated monoidal categories. These tools apply to the full twist and underlie several of the link-homology computations above.

Handleslides and Kirby colors

A more recent direction studies handleslides in Khovanov homology and the construction of a Kirby color, including the role of the annular Bar–Natan category, with an eye toward four-manifold invariants and a categorification of the Turaev–Viro TQFT.

For the precise statements and references, see the research statement and the list of papers.