Research Description: My main interest in mathematics is on higher representation theory, algebraic geometry, and their applications to low-dimensional topology. Currently, I'm working on categorification of small quantum groups-a special kind of finite dimensional Hopf algebra, and trying to construct a four-dimensional topological quantum field theory out of them. This is the original motivation of categorification initiated by Louis Crane and Igor Frenkel. The picture on the top-right is a categorical interpretation of one of the so called quantum Serre relations for the small quantum group sl(3).
I'm also looking into the algebro-geometric meaning behind the small quantum groups and their applications in topological field theories.
More information about my past experience is contained in my CV. Further details on my earlier work, ongoing projects, and future plans can be found in my research statement.
Work partially supported by the Simons Foundation Collaboration Grants for Mathematicians, 2022-2028 and the National Science Foundation DMS-2401376, 2024-2027.
Published and accepted works:
[1] Hopfological Algebra, Compositio Mathematica, Volume 150, Issue 01, 2014, pp. 1-45, arXiv:1205.1814.
[2] An approach to categorification of some small quantum groups, joint with Mikhail Khovanov, Quantum Topology, Volume 6, Issue 2, 2015, pp. 185-311, arXiv:1208.0616.
[3] An approach to categorification of some small quantum groups II, joint with Ben Elias, Advances in Mathematics, Volume 288, 2016, pp. 81-151. arXiv:1302.5478.
[4] A categorification of the Burau representation at a prime root of unity, joint with Josh Sussan, Selecta Mathematica, Volume 22, Issue 3, 2016, pp. 1157-1193, arXiv:1312.7692.
[5] A categorification of quantum sl(2) at prime roots of unity, joint with Ben Elias, Advances in Mathematics, Volume 299, 2016, pp. 863-930, arXiv:1503.05114.
[6] The differential graded odd nilHecke algebra, joint with Alexander P. Ellis, Communications in Mathematical Physics, Volume 344, Issue 1, 2016, pp. 275-331. arXiv:1504.01712.
[7] Categorification at prime roots of unity and hopfological finiteness, joint with Joshua Sussan, AMS Contemporary Mathematics, Volume 683, pp. 261-286, 2017, arXiv:1509.00438.
[8] The center of small quantum groups I: the principal block in type A, joint with Anna Lachowska, International Mathematics Research Notices, Volume 2018, Issue 20, pp. 6349-6405, 2018, arXiv:1604.07380.
[9] A categorification of a quantum Frobenius map, Journal of the Institute of Mathematics of Jussieu, Volume 18, Number 5, pp. 899-939, 2019, arXiv:1607.02117.
[10] The center of small quantum groups II: singular blocks, joint with Anna Lachowska, Proceedings of the London Mathematical Society, Volume 118, pp. 513-544, 2019, arXiv:1703.02457.
[11] Morphism spaces in stable categories of Frobenius algebras, Communications in Algebra, Volume 47, Number 8, pp. 3239-3249, 2019, arXiv:1801.07838.
[12] A faithful braid group action on the stable category of tricomplexes, joint with Mikhail Khovanov, SIGMA 16 (2020), 019, 32 pages, arXiv:1911.02503.
[13] Evaluating thin flat surfaces, joint with Mikhail Khovanov and Lev Rozansky, Communications in Mathematical Physics, Volume 385, pp. 1835-1870, 2021, arXiv:2009.01384.
[14] Remarks on the derived center of small quantum groups, joint with Anna Lachowska, Selecta Mathematica, Volume 27, Article Number 68, 40 pages, 2021, arXiv:1912.08783.
[15] A categorification of cyclotomic integers, joint with Robert Laugwitz, Quantum Topolology, Volume 13, Issue 3, 2022, pp. 539-577. arXiv:1804.01478.
[16] A braid group action on a p-DG homotopy category, joint with Joshua Sussan and Yasuyoshi Yonezawa, Journal of Algebra, Volume 598, 15 May 2022, Pages 470-517. arXiv:2012.15181.
[17] On some p-differential graded link homologies, joint with Joshua Sussan, Forum of Mathematics, Pi, 10, E26. DOI:10.1017/fmp.2022.19, 2022. arXiv:2009.06498.
[18] On some p-differential graded link homologies II, joint with Joshua Sussan, Algebraic & Geometric Topology Volume 23, Issue 7, 2023, 3357-3394. arXiv:2108.10722.
[19] Actions of sl(2) on algebras appearing in categorification, joint with Ben Elias, Quantum Topology, Volume 14, Number 4 2023, pp. 733-806. arXiv:2103.00048.
[20] A Rickard equivalence for hopfological homotopy categories,Journal of Pure and Applied Algebra, Volume 227, Issue 5, May 2023, Article Number 107252. arXiv:2204.14220.
[21] Categorifying Hecke algebras at prime roots of unity, part I, joint with Ben Elias, Transactions of the AMS, Volume 376 2023, pp. 7691-7742. arXiv:2005.03128.
[22] p-DG cyclotomic nilHecke algebras, joint with Mikhail Khovanov and Joshua Sussan, Memoirs of the American Mathematical Society, Volume 293, Number 1462, 2024. arXiv:1711.07159.
[23] p-DG cyclotomic nilHecke algebras II, joint with Joshua Sussan, Memoirs of the American Mathematical Society, Volume 293, Number 1463, 2024. arXiv:1811.04372.
[24] Symmetries of gl(N)-foams, joint with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner, Quantum Topology, DOI:10.4171/QT/215, published online on May 3, 2024, arXiv:2212.10106.
[25] A braid group action on an A-infinity category for zigzag algebras, joint with Benjamin Cooper and Joshua Sussan, accepted by AMS Contemporary Mathematics, 2024, arXiv:2305.02824. Book Chapters:
[1] Connections to Link Invariants, joint with Johannes Flake. In Introduction to Soergel bimodules, edited by Ben Elias, Shotaro Makisumi, Ulrich Thiel and Geordie Williamson. RSME Springer Series, Volume 5, pp. 421-440, Springer, Cham. 2020. Preprints:
[1] A categorification of the colored Jones polynomial at a root of unity, joint with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner, 2021, arXiv:2111.13195.
[2] Symmetries of equivariant Khovanov-Rozansky homology, joint with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner, 2023, arXiv:2306.10729. Unpublished notes:
[1] On the axioms of module algebras over Hopf algebras, 2022, available here. Work in progress:
[1] A categorification of the colored Jones polynomial at a root of unity, part II, joint with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner, in preparation.
[2] Categorifying Hecke algebras at prime roots of unity, part II, joint with Ben Elias, in preparation.
[3] Hopfological algebra and algebraic K-theory, in preparation. MPhil Thesis: This is a printer-friendly version of my MPhil Thesis, written under the guidance of Professor Guowu Meng at Hong Kong University of Science and Technology. DOI: 10.14711/thesis-b1023323. PhD Thesis: This is an updated version of my PhD Thesis, supervised by Professor Mikhail Khovanov at Columbia University. DOI:10.7916/D8G73M28. A shortened version of the work is published as publication [1] in Compos. Math. All comments are welcome! Some Talks:
[1] On Grothendieck-Riemann-Roch, handwritten notes for a talk given at a summer graduate student seminar organized by A. Johan de Jong and me on this theorem, 2009.
[2] On Mukai's derived equivalences of abelian varieties, handwritten notes for a talk given at A. Johan de Jong's graduate student seminar on classical litterature in algebraic geometry, Apr, 2010.
[3] On matrix factorizations and hypersurface singularities, handwritten notes for a talk given at A. Johan de Jong's graduate student seminar on derived categories, Mar, 2011.
[4] On generic Picard numbers of surfaces, handwritten notes for a talk given at A. Johan de Jong's graduate student seminar on Hodge theory, Nov, 2011.
[5] Two talks on categorification of small quantum groups, handwritten notes for two talks given at USC, Sept, 2012.
[6] A talk on categorification of small quantum groups given at Yale, Sept, 2012.
[7] A talk on categorification of small quantum groups given at MIT, Oct, 2012.
[8] A talk on categorification of small quantum groups given at SUNY Buffalo, Nov, 2012.
[9] A talk on categorification of small quantum groups in the conference "Hecke algebras in number theory and categorification", May, 2013.
[10] A talk on categorification of small quantum groups given at UC Davis, Mar, 2014.
[11] A talk on categorification of small quantum groups given at HKUST, Mar, 2014. Handwritten notes.
[12] A talk on categorification of small quantum sl(2) given at CRM, a workshop on "Categorification and geometric representation theory" June, 2014. Handwritten notes.
[13] A talk on categorification of small quantum groups given at the CBMS conference at North Carolina State University, July, 2014.
[14] A talk on categorification of small quantum groups given at the representation theory seminar at CUNY graduate center, Sept., 2014.
[15] A talk on "A New Year's Resolution", Spring Eastern AMS Sectional Meeting at Georgetown University, Mar., 2015.
[16] A talk on categorification of small quantum groups given at the algebra seminar at University of Oregon, Mar., 2015.
[17] A talk on "A New Year's Resolution", Joint Meeting of AMS-EMS-SPM, Porto, Portugal, Jun., 2015.
[18] A talk on categorification of small quantum groups, Colloquium, CUNY Medgar Evers College, Oct., 2015.
[19] Five lectures on categorification at prime roots of unity, IMPJ, Paris, France, Nov., 2015. Some hand written notes are available here.
[20] A talk on categorification at prime roots of unity, a conference on "Geometric and Categorical Representation Theory", Mooloolaba, Australia, Dec., 2015.
[21] Two talks on categorification at prime roots of unity, Kyoto University and RIMS, Kyoto, Japan, Feb., 2016.
[22] A talk on the center of small quantum groups, Yau Mathematical Sciences Center, Tsinghua University, Sanya, China, Mar., 2016. Hand written notes are available here.
[23] A talk on categorification of small quantum sl(2), Informal Mathematical Physics Semniar, Columbia, Apr., 2016.
[24] Three lectures on categorification at prime roots of unity, Workshop: Categorification, Mathematical Institute, University of Bonn, Germany, May 2016. Some hand written notes are available here.
[25] A talk on the center of small quantum groups, Geometric Methods in Representation Theory Seminar, University of North Carolina, Chapel Hill, Sep. 2016.
[26] A talk on the center of small quantum groups, Algebra Seminar, University of Virginia, Sep. 2016.
[27] A talk on Categorification at prime roots of unity, Hausdorff Institute, University of Bonn, Dec. 2016.
[28] A colloquium talk on categorification at prime roots of unity at Hong Kong University of Science and Technology, Jan. 2017.
[29] A colloquium talk on categorification at prime roots of unity at UC Santa Cruz, Feb. 2017.
[30] A talk on categorification at prime roots of unity at the Lie Groups/Quantum Math Seminar Rutgers, Mar. 2017.
[31] A talk on categorification at prime roots of unity at "Perspectives of Mathematics in the 21st Century," a Conference in Celebration of the 90th Anniversary of Mathematics Disciplines of Tsinghua University", Tsinghua, Beijing Apr. 2017.
[32] A talk on categorification at prime roots of unity at Guanghua Forum, Fudan University, Shanghai, Dec. 2017.
[33] A talk on the center of small quantum groups, Winter Young Algebraic Geometer Workshop, Southern University of Science and Technology, Shenzhen, Dec. 2017.
[34] A talk on the center of small quantum groups, Lie Theory Seminar, UC Riverside, Apr. 2018.
[35] A talk on the center of small quantum groups, Colloquium, CUNY Medgar Evers College, May, 2018.
[36] A talk on categorification of cyclotomic rings, Categorification and Higher Representation Theory Conference, Mittag-Leffler Institute, Stockholm, Sweden, Jul., 2018.
[37] Two lectures on categorification at prime roots of unity, Summer School and Workshop on Categorification of Quantum Three-Manifold Invariants, University of Southern California, Jul., 2018. Handwritten notes.
[38] A talk on a tensor product categorification at prime roots of unity, Yau Mathematical Sciences Center, Tsinghua University, Sanya, China,
Oct., 2018.
[39] A 45-minute address on "Categorification at prime roots of unity", International Congress of Chinese Mathematicians, June 2019, Beijing, China.
[40] Four lectures on Hopfological algebra and categorification at prime roots of unity, Conference on "Quiver Hecke algebra and its applications to topology", Nagoya, Japan, July 2019.
[41] A mini-course on categorification at prime roots of unity, QUACKS conference (Quantum groups, Categorification, Knot invariants, and Soergel bimodules), online meeting due to Covid19, information (lecture videos and notes) available here.
Aug., 2020.
Updated 07/17/2024
by You Qi.
