Research

My research is in homological algebra, Hopf algebras, and representation theory of finite-dimensional algebras.

In particular, I study various cohomology structures and their properties: finite generation of cohomology ring, Gerstenhaber structure of Hochschild cohomology, stable cohomology rings of finite-dimensional (Hopf) algebras and related algebraic objects; classification of Hopf algebras (over a field of positive characteristic), Hopf actions on algebras, as well as some homological conjectures in representation theory and commutative algebra.

Recently, together with my collaborators, we introduced a notion of quantum symmetric equivalence and studied it for various (noncommutative) algebras such as (N-Koszul) Artin-Schelter regular algebras and superpotential algebras. The chart below briefly summarizes some of my current research projects and interests. If you are interested in continuing any of the work here, or starting a new collaboration, please feel free to contact me!

My research is currently supported by the NSF DMS-2201146 Grant Award.

An overall picture of my research topics

Navy research topics? Contact me!

Publications and Preprints:

Also see my arXiv Listings and my Google Scholar page.

Published and Accepted Articles:

1. Quantum-symmetric equivalence for superpotential algebras, joint with H. Huang, K.B. Varshaw, P. Veerapen, and X. Wang, International Mathematics Research Notices, 2026 (2026), issue 17.
2. On a minimal free resolution of the residue field over a local ring of codepth 3 of class T, joint with O. Veliche, to appear in J. of Commutative Algebra (2026).
3. On the finite generation of the cohomology of bosonizations, joint with N. Andruskiewitsch, D. Jaklitsch, A. Oswald, J. Plavnik, A. Shepler, and X. Wang, J. Algebra, special issue on the occasion of YJABR 60th anniversary, 691 (2026), 741-780.
4. Quantum-symmetric equivalence is a graded Morita invariant, joint with H. Huang, K.B. Varshaw, P. Veerapen, and X. Wang, Proc. Amer. Math. Soc., 153 (2025), 1389-1409.
5. Twisting Manin's universal quantum groups and comodule algebras, joint with H. Huang, C. Ure, K.B. Varshaw, P. Veerapen, and X. Wang, Advances in Math., 445 (2024), 109651.
6. Iterated mapping cones on the Koszul complex and their application to complete intersection rings, joint with O. Veliche, J. of Algebra and Its Applications, 24 (2025), no. 10, 2550234.
7. A cogroupoid associated to preregular forms, joint with H. Huang, C. Ure, K.B. Varshaw, P. Veerapen, and X. Wang, J. Noncommut. Geom., 18 (2024), no. 4, 1453-1483.
8. Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation, joint with H. Huang, C. Ure, K.B. Varshaw, P. Veerapen, and X. Wang, Transformation Groups, 29 (2024), 1459-1500.
9. Tensor representations for the Drinfeld double of the Taft algebra, joint with G. Benkart, R. Biswal, E. Kirkman, and J. Zhu, J. Algebra, 606 (2022), 764-797.
10. McKay matrices for finite-dimensional Hopf algebras, joint with G. Benkart, R. Biswal, E. Kirkman, and J. Zhu, Canadian J. of Math., 74 (2022), no. 3, 686-731.
11. A truncated minimal free resolution of the residue field, joint with O. Veliche, In: Miller, C., Striuli, J., Witt, E.E. (eds) Women in Commutative Algebra, Proceedings of the 2019 WICA Workshop, Association for Women in Mathematics Series, vol. 29, p. 399-437, Springer, 2021.
12. New approaches to finite generation of cohomology rings, joint with X. Wang and S. Witherspoon, J. Algebra, 587 (2021), 390-428.
13. Gerstenhaber structure of a class of special biserial algebras, joint with J. Meinel, B. Pauwels, M. Redondo, and A. Solotar, J. Algebra, 580 (2021), 264-298.
14. Finite generation of some cohomology rings via twisted tensor product and Anick resolutions, joint with X. Wang and S. Witherspoon, J. Pure Appl. Algebra, 223 (2019), no. 1, 316–339.
15. Preprojective algebras of tree-type quivers, joint with G. Todorov and S. Zhu, Communications in Algebra, 47 (2019), no. 1, 252–275.
16. Dominant dimension and tilting modules, joint with I. Reiten, G. Todorov, and S. Zhu, Mathematische Zeitschrift, 292 (2019), no. 3-4, 947–973.
17. Primitive deformations of quantum $p$-groups, joint with L. Wang and X. Wang, Algebra & Representation Theory, 22 (2019), no. 4, 837–865.
18. Pointed $p^3$-dimensional Hopf algebras in positive characteristic, joint with X. Wang, Algebra Colloquium, 25 (2018), no. 3, 399–436.
19. Gerstenhaber brackets on Hochschild cohomology of twisted tensor products, joint with L. Grimley and S. Witherspoon, J. Noncommut. Geom., 11 (2017), no. 4, 1351–1379.
20. Classification of connected Hopf algebras of dimension $p^3$ I, joint with L. Wang and X. Wang, J. Algebra, 424 (2015), 473–505.
21. Finite generation of the cohomology of some skew group algebras, joint with S. Witherspoon, Algebra and Number Theory, 8 (2014), no. 7, 1647-1657; Erratum, 12 (2018), no. 2, 491–492.
22. Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras, J. Pure Appl. Algebra, 217 (2013), no. 10, 1967–1979.

Submitted Preprints:

1. Finite generation of Tate cohomology of symmetric Hopf algebras.
2. Tate-Hochschild cohomology ring of a group algebra.

In Preparation:

1. Centralizer algebra over the Drinfeld double of the Taft algebra, joint with E. Kirkman and J. Zhu.
2. Projective quantum-symmetric equivalence, joint with H. Huang, K.B. Varshaw, P. Veerapen, and X. Wang.
3. Stable 2-cocycles and stable equivalences of Morita-Takeuchi type, joint with H. Huang, K.B. Varshaw, P. Veerapen, and X. Wang.

Recent Talks:

Invited Speaker, "TBD," Colloquium Seminar, Hood College, Nov. 2026.
Invited Speaker, "Finite generation of cohomology for smash products and bosonizations," Special Session on Noncommutative Algebra, AMS Fall Eastern Sectional Meeting, George Washington University, Washington DC, Oct. 2026.

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