I am broadly interested in geometry and topology. My research focuses on categorification in quantum topology, especially knot- and link homology theories and their deep relationships to (higher) representation theory and mathematical physics. All my papers are available on the mathematics arXiv. They are also listed by Google Scholar, ORCiD and ResearchGate.
11) Evaluations of annular Khovanov-Rozansky homology
We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.
With Eugene Gorsky.
10) Khovanov homology and categorification of skein modules
For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated gl(2) skein algebra and verify an analogue of a positivity conjecture of Fock–Goncharov and Turston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman-Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda–Przytycki–Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them.
With Hoel Queffelec.
9) Extremal weight projectors II
In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley–Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N > 1, with a view towards categorifying the corresponding torus skein algebras via Khovanov–Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subalgebra, and a categorification of power-sum symmetric polynomials.
With Hoel Queffelec.
8) Exponential growth of colored HOMFLY-PT homology
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies.
7) Rational links and DT invariants of quivers
We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links-quivers correspondence of Kucharski–Reineke–Stošić–Sułkowski as well as the LMOV conjecture hold for rational links. Along the way, we extend the links-quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson-Thomas invariants.
With Marko Stošić.
International Mathematical Research Notices
6) Functoriality of colored link homologies
We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
With Michael Ehrig and Daniel Tubbenhauer.
Proceedings of the London Mathematical Society 117-5 (2018), 996--1040
5) q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links
We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.
Journal of Pure and Applied Algebra 223-4 (2019), 1434--1439
4) Extremal weight projectors
We introduce a quotient of the affine Temperley-Lieb category that encodes all weight preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein module of the torus.
With Hoel Queffelec.
Mathematical Research Letters 6-25 (2018), 1911-1936
3) Deformations of colored sl(N) link homologies via foams
We prove a conjectured decomposition of deformed sl(N) link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored sl(N) link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.
With David Rose.
Geometry & Topology 20-6 (2016), 3431--3517
2) Super q-Howe duality and web categories
We use super q–Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of gl(N)–modules (and, more generally, gl(N|M)–modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation-theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY–PT polynomials.
With Daniel Tubbenhauer and Pedro Vaz.
Algebraic & Geometric Topology 17-6 (2017), 3703--3749
1) Categorified sl(N) invariants of colored rational tangles
We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links and discuss consequences for the corresponding decategorified invariants.
Algebraic & Geometric Topology 16-1 (2016), 427--482