Research

Publications.

1. Two-dimensional topological quantum field theories of rank two over Dedekind domains (with Fabian Espinoza and Mee Seong Im), to appear in Contemp. Math., arXiv:2502.04502.

2. Entropy, cocycles, and their diagrammatics (with Mee Seong Im), arXiv:2409.08462.

3. Foams with flat connections and algebraic K-theory (with David Gepner, Mee Seong Im and Nitu Kitchloo), arXiv:2405.14465.

4. Foam cobordism and the Sah-Arnoux-Fathi invariant (with Mee Seong Im), to appear in Arnold Math., arXiv:2403.06030.

5. Boolean TQFTs with accumulating defects, sofic systems, and automata for infinite words (with Paul Gustafson and Mee Seong Im), Lett. Math. Phys. 114 (2024), no. 135, 1–35, arXiv:2312.17033.

6. On the universal pairing for 2-complexes (with Vyacheslav Krushkal and John Nicholson), to appear in Bulletin of the London Math. Soc., arXiv:2312.07429.

7. From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions (with Mee Seong Im), Contemp. Math. 817 (2025), 179–220, arXiv:2309.00708.

8. A topological theory of unoriented SL(4) foams (with Jozef H. Przytycki, Louis-Hadrien Robert and Marithania Silvero), Mediterr. J. Math. 21 (2024), no. 2, Paper No. 62, 1–33, arXiv:2307.00674.

9. Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters (with Mee Seong Im and Victor Ostrik), Theory Appl. Categ. 44 (2025), no. 2, 15–83, arXiv:2303.02696.

10. Automata and one-dimensional TQFTs with defects (with Paul Gustafson, Mee Seong Im, Remy Kaldawy and Zachary Lihn), Lett. Math. Phys. 113 (2023), no. 93, 1–38, arXiv:2301.00700.

11. One-dimensional theories with defects: the linear case (with Mee Seong Im), Contemp. Math. 791 (2024), 105–146, arXiv:2210.00371.

12. Odd two-variable Soergel bimodules and Rouquier complexes (with Krzysztof Putyra and Pedro Vaz), Contemp. Math. 791 (2024), 205–227, arXiv:2205.12794.

13. Topological theories and automata (with Mee Seong Im), arXiv:2202.13398.

14. Categorical lifting of the Jones polynomial: a survey (with Robert Lipshitz), Bull. Amer. Math. Soc. 60 (2023), no. 4, 483–506, arXiv:2202.02473.

15. Monoidal categories, representation gap and cryptography (with Maithreya Sitaraman and Daniel Tubbenhauer), Trans. Amer. Math. Soc. Ser. B 11 (2024), 329–395, arXiv:2201.01805.

16. Foams, iterated wreath products, field extensions and Sylvester sums (with Mee Seong Im), arXiv:2107.07845.

17. Anchored foams and annular homology (with Rostislav Akhmechet), Algebr. Geom. Topol. 23 (2023), no. 7, 3129–3204, arXiv:2105.00921.

18. Planar diagrammatics of self-adjoint functors and recognizable tree series (with Robert Laugwitz), Pure Appl. Math. Q. 19 (2023), no. 5, 2409–2499, arXiv:2104.01417.

19. Two-dimensional topological theories, rational functions and their tensor envelopes (with Victor Ostrik and Yakov Kononov), Selecta Math. 28 (2022), no. 4, Paper No. 71, 1–68, arXiv:2011.14758.

20. Conical SL(3) foams (with Louis-Hadrien Robert), J. Comb. Algebra 6 (2022), no. 1-2, 79–108, arXiv:2011.11077.

21. Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series, J. Pure Appl. Algebra 228 (2024), no. 11, 107689, arXiv:2010.05730.

22. Evaluating thin flat surfaces (with You Qi and Lev Rozansky), Communications in Mathematical Physics 385 (2021) no. 3, 1835–1870, arXiv:2009.01384.

23. Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category (with Radmila Sazdanovic), Fund. Math. 264 (2024), no. 1, 1–20, arXiv:2007.11640.

24. Universal construction of topological theories in two dimensions, arXiv:2007.03361.

25. Link homology and Frobenius extensions II (with Louis-Hadrien Robert), Fundamenta Mathematicae 256 (2022), no. 1, 1–46, arXiv:2005.08048.

26. A deformation of Robert-Wagner foam evaluation and link homology (with Nitu Kitchloo), Contemp. Math. 791 (2024) 147–204, arXiv:2004.14197.

27. Diagrammatic categorification of the Chebyshev polynomials of the second kind (with Radmila Sazdanovic), Journal of Pure and Applied Algebra 225 (2021) no. 6, 106592, arXiv:2003.11664.

28. A faithful braid group action on the stable category of tricomplexes (with You Qi), SIGMA (Symm. Integr. Geom. Methods Appl.) 16 (2020), no. 019, 1–32, arXiv:1911.02503.

29. Foam evaluation and Kronheimer--Mrowka theories (with Louis-Hadrien Robert), Advances in Mathematics 376 (2021), 107433, arXiv:1808.09662.

30. Braid group actions from categorical symmetric Howe duality on deformed Webster algebras, (with Aaron D. Lauda, Joshua Sussan, and Yasuyoshi Yonezawa), Transformation groups 27 (2022), 919–982, arXiv:1802.05358.

31. Towards functor exponentiation, (with Yin Tian), J. Algebra 505 (2018), 125–149, arXiv:1712.02208.

32. p-DG cyclotomic nilHecke algebras (with You Qi and Joshua Sussan), Mem. Amer. Math. Soc. 293 (2024), no. 1462, v+94, arXiv:1711.07159.

33. How to categorify the ring of integers localized at two, (with Yin Tian), Quantum Topology 10 (2019), no. 4, 723–775, arXiv:1702.07466.

34. The Soergel category and the redotted Webster algebra (with Joshua Sussan), Journal of Pure and Applied Algebra 222 (2018), no. 8, 1957–2000, arXiv:1605.02678.

35. Linearization and categorification, arXiv:1603.08223.

36. A categorification of the positive half of quantum gl(m|1) (with Joshua Sussan), Transactions of the AMS 369 (2017), no. 3, 1627–1664, arXiv:1406.1676.

37. Positive half of the Witt algebra acts on triply graded link homology (with Lev Rozansky), Quantum Topology 7 (2016), no. 4, 737–795, arXiv:1305.1642.

38. An approach to categorification of small quantum groups (with You Qi), Quantum Topology 6 (2015), no. 2, 185–311, arXiv:1208.0616.

39. The odd nilHecke algebra and its diagrammatics (with Alexander Ellis and Aaron Lauda), Int. Math. Research Notices 2014 (2014), no. 4, 991–1062, arXiv:1111.1320.

40. The Hopf algebra of odd symmetric functions (with Alexander Ellis), Advances in Math. 231 (2012), no. 2, 965–999, arXiv:1107.5610.

41. Categorification of the polynomial ring (with Radmila Sazdanovic), Fundamenta Mathematicae 230 (2015), no. 3, 251–280, arXiv:1101.0293.

42. Heisenberg algebra and a graphical calculus, Fundamenta Mathematicae 225 (2014), 169–210, arXiv:1009.3295.

43. Categorifications from planar diagrammatics, Japanese J. of Mathematics 5 (2010), 153–181, arXiv:1008.5084.

44. A categorification of the Casimir of quantum sl(2) (with Anna Beliakova and Aaron Lauda), Advances in Math. 230 (2012), no. 3, 1442–1501, arXiv:1008.0370.

45. How to categorify one-half of quantum gl(1|2), Banach Center Publications 103, Pt.3: Knots in Poland III (2014), 211–232, arXiv:1007.3517.

46. Extended graphical calculus for categorified quantum sl(2) (with Aaron Lauda, Marco Mackaay, and Marko Stosic), Memoirs of the AMS 219 (2012), no. 1029, 1–87, arXiv:1006.2866.

47. Diagrammatics for Soergel categories (with Ben Elias), Int. J. Math. Math. Sci. (2010) Article 978635, arXiv:0902.4700.

48. A diagrammatic approach to categorification of quantum groups III (with Aaron Lauda), Quantum Topology 1 (2010), no. 1, 1–92, arXiv:0807.3250.

49. A diagrammatic approach to categorification of quantum groups II (with Aaron Lauda), Trans. Amer. Math. Soc. 363 (2011), 2685–2700, arXiv:0804.2080.

50. Notes on link homology (with Marta Asaeda), Low-Dimensional Topology, ed. T. Mrowka and P. Ozsvath, IAS/Park City Mathematics Series, (2009) 139–196, arXiv:0804.1279.

51. A diagrammatic approach to categorification of quantum groups I (with Aaron Lauda), Representation Theory 13 (2009), 309–347, arXiv:0803.4121.

52. A brief review of abelian categorifications (with Volodymyr Mazorchuk and Catharina Stroppel), Theory and Applications of Categories 22 (2009), no. 19, 479–508, arXiv:math.RT/0702746.

53. Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial (with Lev Rozansky), Journal of Gökova Geometry Topology 1 (2007), 116–214, arXiv:math.QA/0701333.

54. An invariant of tangle cobordisms via subquotients of arc rings (with Yanfeng Chen), Fundamenta Mathematicae 225 (2014) no. 0, 23–44, arXiv:math.QA/0610054.

55. Braid cobordisms, triangulated categories, and flag varieties (with Richard Thomas), Homology, Homotopy and Applications 9 (2007), 19–94, arXiv:math.QA/0609335.

56. A categorification of integral Specht modules (with Volodymyr Mazorchuk and Catharina Stroppel), Proceedings of the AMS 136 (2008), no. 4, 1163–1169, arXiv:math.RT/0607630.

57. Link homology and categorification, Proceedings of the ICM-2006, Madrid 2 (2007), 989–999, arXiv:math.QA/0605339.

58. A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products (with Igor Frenkel and Catharina Stroppel), Selecta Mathematica 12 (2006), no. 3-4, 379–431, arXiv:math.QA/0511467.

59. Triply-graded link homology and Hochschild homology of Soergel bimodules, International Journal of Math. 18 (2007), no. 8, 869–885, arXiv:math.GT/0510265.

60. Hopfological algebra and categorification at a root of unity: the first steps, Communications in Contemporary Math. 25 (2016), no. 3, 1640006, arXiv:math.QA/0509083.

61. Matrix factorizations and link homology II (with Lev Rozansky), Geometry and Topology, 12 (2008), 1387–1425, arXiv:math.QA/0505056.

62. Link homology and Frobenius extensions, Fundamenta Mathematicae, 190 (2006), 179–190, arXiv:math.QA/0411447.

63. Topological Landau-Ginzburg models on a world-sheet foam (with Lev Rozansky), Adv. Theor. Math. Phys. 11 (2007), no. 2, 233–259, arXiv:hep-th/0404189.

64. Matrix factorizations and link homology (with Lev Rozansky), Fundamenta Mathematicae, 199 (2008), 1–91, arXiv:math.QA/0401268.

65. Homological realization of Nakajima varieties and Weyl group actions (with Igor Frenkel and Olivier Schiffmann), Compositio Mathematica 141 (2005), no. 6, 1504–1530, arXiv:math.QA/0311485.

66. sl(3) link homology I, Algebr. Geom. Topol. 4 (2004) 1045–1081 (electronic), arXiv:math.QA/0304375.

67. Categorifications of the colored Jones polynomial, J. Knot theory and its Ramifications 14 (2005) no. 1, 111–130, arXiv:math.QA/0302060.

68. An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), 315–327, arXiv:math.QA/0207264.

69. Categorification of some level two representations of sl(n) (with Stella Huerfano), J. Knot theory and its Ramifications 15 (2006) no. 6, 695–713, arXiv:math.QA/0204333.

70. Crossingless matchings and the cohomology of (n,n) Springer varieties, Communications in Contemporary Math. 6 (2004) no. 2, 561–577, arXiv:math.QA/0202110.

71. Patterns in knot cohomology I, Experimental mathematics 12 (2003) no. 3, 365–374, arXiv:math.QA/0201306.

72. A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741 (electronic), arXiv:math.QA/0103190.

73. Quivers, Floer cohomology, and braid group actions (with Paul Seidel), J. Amer. Math. Soc. 15 (2002), no. 1, 203–271 (electronic), arXiv:math.QA/0006056.

74. A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
    (with Joseph Bernstein and Igor Frenkel), Selecta Math. (N.S.) 5 (1999), no. 2, 199–241, arXiv:math.QA/0002087.

75. A category for the adjoint representation (with Stella Huerfano),  J. Algebra 246 (2001), no. 2, 514–542, arXiv:math.QA/0002060.

76. A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426, arXiv:math.QA/9908171.

77. NilCoxeter algebras categorify the Weyl algebra, Comm. Algebra 29 (2001), no. 11, 5033–5052, arXiv:math.RT/9906166.

78. Web bases for sl(3) are not dual canonical (with Greg Kuperberg), Pacific J. Math. 188 (1999), no. 1, 129–153, arXiv:q-alg/9712046.

79. Kazhdan-Lusztig polynomials and canonical basis (with Igor Frenkel and Alexander Kirillov Jr), Transform. Groups 3 (1998), no. 4, 321–336, arXiv:q-alg/9709042.

80. Graphical calculus, canonical bases and Kazhdan-Lusztig theory, PhD thesis, Yale University, May 1997.

81. Canonical bases in tensor products and graphical calculus for Uq(sl2) (with Igor Frenkel), Duke Math. J. 87 (1997) no. 3, 409–480.

82. Doodle groups, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2297–2315.

83. Real K(π,1) arrangements from finite root systems, Math. Res. Lett. 3 (1996), no. 2, 261–274.

84. Some remarks on Tabachnikov's invariants of plane curves. J. Knot Theory Ramifications 5 (1996), no. 6, 849–857.

85. Representations of tensor categories and Dynkin diagrams (with Pavel Etingof), Internat. Math. Res. Notices 1995 (1995) no. 5, 235–247, arXiv:hep-th/9408078.

86. The set of slice doodles is NP-complete, preprint 1995.

87. Surfaces in three-space and braided algebras, preprint 1995.

88. New geometrical constructions in low-dimensional topology, preprint 1990.