Research

My research focuses on noncommutative geometry and it also involves homological algebra, representation theory, and higher category theory. In particular I am interested in applying the theory of dg-category to problems in algebra, geometry, and topology. One of my goals is to bridge the categorical and analytical approaches to noncommutative geometry.

I am also interested in related areas such as geometric representation theory and homological mirror symmetry.

Here is my research statement.

I write reviews for MathSciNet and Zentralblatt Math.

Published and Accepted Papers

1. Coherent sheaves, superconnections, and Riemann-Roch-Grothendieck, with Jean-Michel Bismut and Shu Shen, published as a monograph Progress in Mathematics 347, Birkhauser, arXiv:2102.08129.

2. Descent of dg cohesive modules for open covers on complex manifolds, Eur. J. Math., 9 (2023), no. 3, article 77, arXiv:1804.00993.

3. Tensor products of infinite dimensional modules in the BGG category of a quantized simple Lie algebra of type ADE, Int. Electron. J. Algebra, no. 35(2024), 108-120, a preliminary version.

4. Twisted complexes and simplicial homotopies, Eur. J. Math., 7 (2021), no. 3, 1102–1123, arXiv:1905.07460.

5. A proof of the Baum-Connes conjecture for real semisimple Lie groups with coefficients on flag varieties, Methods Funct. Anal. Topology, 26 (2020), no. 4, 389–404, arXiv:1211.4544.

6. Tensor-closed objects in the BGG category of a quantized semisimple Lie algebra, Int. Electron. J. Algebra, no. 29 (2021), 175-186, arXiv:1910.02326.

7. Scalar extensions of categorical resolutions of singularities, J. Pure Appl. Algebra, 222 (2018), no. 7, 1565–1578, arXiv:1701.05687.

8. Explicit homotopy limits of dg-categories and twisted complexes, with Jonathan Block and Julian V.S. Holstein, Homology Homotopy Appl.,19 (2017), no. 2, 343-371, arXiv:1511.08659.

9. The full exceptional collections of categorical resolutions of curves, J. Pure Appl. Algebra, 220 (2016), no. 9, 3332–3344, arXiv:1506.02991.

10. Twisted complexes on a ringed space as a dg-enhancement of perfect complexes, Eur. J. Math., 2 (2016), no. 3, 716–759, arXiv:1504.05055.

11. The noncommutative Poisson bracket and the deformation of the family algebras, J. Math. Phys., 56 (2015), no. 7, 071703, arXiv: 1211.5865.

Preprints

1. Deformations of cohesive modules on compact complex manifolds, arXiv:2308.01266.

2. The descent of twisted perfect complexes on a space with soft structure sheaf, arXiv:1605.07111.

3.  Covariant Weil algebras,  arXiv:1211.3552.

Miscellaneous

1. These are the slides and video of my talk at the 2022 Online New York City Noncommutative Geometry Seminar.

2. These are the slides and video of my talk at the 2020 Online Noncommutative Geometry Seminar of WUSTL/TAMU.

3. This is an expository writing on A∞-quasi inverses of dg-natural transformations: arXiv:1903.01639.

4. This is the slides of my talk at the 2018 AMS Fall Western Sectional Meeting, Special Session on Homological Aspects of Noncommutative Algebra and Geometry.

5. These are slides 1 and 2 of two talks I gave at the Algebra Seminar of Kent State University in Fall 2019 on BGG category and tensor problems.