Research
My main areas of research are representation theory of Lie algebras, categorification, and its applications to low dimensional topology. I am also interested in representation theoretic questions about affine Hecke algebras, Lie superalgebras, and quantum groups. Various quantum invariants of knots and 3-manifolds such as the Jones polynomial and Witten-Reshetikhin-Turaev invariant possess important integrality properties. The idea of categorification is that these integral structures should arise naturally as Euler characteristics of homological invariants of these objects. These homological invariants contain more structure than the classical invariants and their functoriality should lead to invariants of higher dimensional objects such as a cobordism between two knots or a 4-manifold when viewed as a cobordism between two 3-manifolds. The challenge is to find concrete examples which give powerful topological invariants.
Recently I have become interested in quantum computation and specifically applications of TQFTs towards this subject.
Papers
Density and unitarity of the Burau representation from a non-semisimple TQFT. Joint with Nathan Geer, Aaron Lauda, and and Bertand Patureau-Mirand. arXiv:2402.13242
Symmetries of equivariant Khovanov-Rozansky homology. Joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner. arXiv:2306.10729
A braid group action on an A-infinity category for zigzag algebras. Joint with Ben Cooper and You Qi. arXiv:2305.02824
Symmetries of gl(N) foams. Joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner. arXiv:2212.10106
Non-semisimple Levin-Wen models and Hermitian TQFTs from quantum (super)groups. Joint with Nathan Geer, Aaron Lauda, and and Bertand Patureau-Mirand. arXiv:2208.114566
A categorification of the colored Jones polynomial at a root of unity. Joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner. arXiv:2111.13195
A Lie theoretic categorification of the coloured Jones polynomial. Accepted (JPAA) Joint with Catharina Stroppel. arXiv:2109.12889
On some p-differential graded link homologies II. Accepted (Algeb. Geom. Topology) Joint with You Qi. arXiv:2108.10722
Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs. Joint with Nathan Geer, Aaron Lauda, and and Bertand Patureau-Mirand. Accepted (Annals of Physics) arXiv:2108.10798
A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules. Joint with Nathan Geer, Aaron Lauda, and and Bertand Patureau-Mirand. Accepted (Letters in Math. Physics) arXiv:2108.09242
A braid group action on a p-DG homotopy category. Joint with You Qi and Yasuyoshi Yonezawa. Accepted (Journal of Algebra) arXiv:2012.15181
On some p-differential graded link homologies. Joint with You Qi. Accepted (Forum of Math. Pi) arXiv:2009.06498
Curved Rickard complexes and link homologies. Joint with Sabin Cautis and Aaron Lauda. Accepted (Crelle) arXiv:1901.08195
p-DG cyclotomic nilHecke algebras II. Joint with You Qi. Accepted (Memoirs of the AMS) arXiv:1811.04372
Preprojective Analogue of the Cone Construction. Joint with Ben Cooper. Accepted (Homol. Homotopy Appl.) arXiv:1804.09068
Braid group actions from categorical symmetric Howe duality on deformed Webster algebras. Joint with Mikhail Khovanov, Aaron Lauda, and Yasuyoshi Yonezawa. Accepted (Trans. Groups) arXiv:1802.05358
p-DG cyclotomic nilHecke algebras. Joint with Mikhail Khovanov and You Qi. Accepted (Memoirs of the AMS) arXiv:1711.07159
The Elliptic Hall algebra and the deformed Khovanov Heisenberg category. Joint with Sabin Cautis, Aaron Lauda, Anthony Licata, and Peter Samuelson. Accepted (Selecta). arXiv:1609.03506
The Soergel category and the redotted Webster algebra. Joint with Mikhail Khovanov. Accepted (JPAA) arXiv:1605.02678
Categorification at prime roots of unity and hopfological finiteness. Joint with You Qi. Accepted (AMS Contemporary Mathematics, Perspectives on Categorification). arXiv:1509.00438
W-algebras from Heisenberg categories. Joint with Sabin Cautis, Aaron Lauda, and Anthony Licata. Accepted (Journal of the Inst. of Math. of Jussieu). arXiv:1501.00589.
A categorification of twisted Heisenberg algebras. Joint with David Hill. Accepted (Advances in Math). arXiv:1501.00283
A categorification of the Burau representation at prime roots of unity. Joint with You Qi. Accepted (Selecta). arXiv:1312.7692.
A categorification of the positive half of quantum gl(m|1). Joint with Mikhail Khovanov. Accepted (Trans. of the AMS). arXiv:1406.1676.
On a categorical boson-fermion correspondence. Joint with Sabin Cautis. Accepted (CMP). arXiv:1403.6019.
Braid group actions via categorified Heisenberg complexes. Joint with Sabin Cautis and Anthony Licata. Compositio Math. (150) (2014), 105--142. arXiv:1207.5245.
Categorified Jones-Wenzl Projectors: a comparison. Joint with Catharina Stroppel. To appear in AMS Contemporary Mathematics volume entitled Perspectives in Representation Theory. arXiv:1105.3038.
Categorifying fractional Euler characteristics, Jones-Wenzl projector and 3j-symbols. Joint with Igor Frenkel and Catharina Stroppel. Quantum Topology Volume 3, Issue 2, 2012, pp. 181-253. arXiv:1007.4680.
The Khovanov-Lauda 2-category and categorifications of a level two quantum sl(n) representation. Joint with David Hill. Special issue `Categorification in Representation Theory’, Int. J. Math. Math. Sci. 2010 Art. Id 892387. arXiv:0910.2496.
Degenerate Affine Hecke-Clifford Algebras and Type Q Lie Superalgebras. Joint with David Hill and Jonathan Kujawa. Math. Z., 268 (2011), no. 3-4, pp. 1091-1158. arXiv:0904.0499.
Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum sl(k). arXiv:0804.2034.
Category O and sl(k) link invariants. PhD Thesis Yale University. math.QA/0701045.
Notes
An invitation to categorification. Joint with Aaron Lauda. Notices of the AMS, Volume 69, Number 1, (2022).