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, I am also interested in questions related to other algebraic objects (for examples, artin algebras, Lie algebras, Gorenstein rings, quivers, preprojective algebras,...) and their module categories. 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 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. A cogroupoid associated to preregular forms, joint with H. Huang, C. Ure, K.B. Varshaw, P. Veerapen, and X. Wang, to appear in J. Noncomm. Geometry (2023).
2. 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 (2022), https://doi.org/10.1007/s00031-022-09779-9.
3. 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.
4. New approaches to finite generation of cohomology rings, joint with X. Wang and S. Witherspoon, J. Algebra, 587 (2021), 390-428.
5. 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, 2022.
6. 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.
7. 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.
8. 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.
9. Dominant dimension and tilting modules, joint with I. Reiten, G. Todorov, and S. Zhu, Mathematische Zeitschrift, 292 (2019), no. 3-4, 947–973.
10. Preprojective algebras of tree-type quivers, joint with G. Todorov and S. Zhu, Communications in Algebra, 47 (2019), no. 1, 252–275.
11. Pointed $p^3$-dimensional Hopf algebras in positive characteristic, joint with X. Wang, Algebra Colloquium, 25 (2018), no. 3, 399–436.
12. Primitive deformations of quantum $p$-groups, joint with L. Wang and X. Wang, Algebra & Representation Theory, 22 (2019), no. 4, 837–865.
13. 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.
14. Classification of connected Hopf algebras of dimension $p^3$ I, joint with L. Wang and X. Wang, J. Algebra, 424 (2015), 473–505.
15. 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.
16. Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras, J. Pure Appl. Algebra, 217 (2013), no. 10, 1967–1979.

Submitted Preprints:

1. Iterated mapping cones on the Koszul complex and their application to complete intersection rings, joint with O. Veliche.
2. Twisting Manin’s universal quantum groups and comodule algebras, joint with H. Huang, C. Ure, K.B. Varshaw, P. Veerapen, and X. Wang.
3. Finite generation of Tate cohomology of symmetric Hopf algebras.
4. 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. A minimal free resolution of the residue field over a local ring of codepth 3, joint with O. Veliche.
3. Bass numbers and their relation to the Koszul complex, joint with L. Christensen and O. Veliche.

Recent Talks:

Invited Speaker, "TBA," Special Session on Homological Techniques in Noncommutative Algebra, Joint Mathematics Meetings, San Francisco, CA, Jan. 2024.
Invited Speaker, "Deforming graded quantum groups and solutions to quantum Yang-Baxter equation," Maurice Auslander Distinguished Lectures and International Conference, Woods Hole, MA, April 2023.
Invited Speaker, "Research directions on Hopf algebras," Combinatorics & Algebra Seminar, George Washington University, Feb. 2023.
Invited Speaker, "On deformations of quantum groups," Flash Talks in Representation Theory III (international, virtual), hosted by the Norwegian University of Science and Technology (NTNU), Jan. 2023.
Invited Speaker, "Tensor representations of finite-dimensional Hopf algebras," Workshop in Noncommutative Geometry and Noncommutative Invariant Theory, Banff International Research Station, Alberta, Canada Sept. 2022.
Invited Speaker, "Quantum symmetries through the lens of linear algebra," Algebra Seminar, joint seminar between Bowling Green State University and University of Toledo, virtual, Mar. 2022.
Invited Speaker, "Tensor representations via a diagrammatic algorithm," Special Session on Rep. Theory of Algebras and Related Combinatorics, AMS Spring Southeastern Sectional Meeting #1175, University of Virginia, Mar. 2022 (cancelled due to COVID-19).
Invited Speaker, "Twisting pairs and cocycle twists of graded Hopf algebras," Special Session on Noncommutative Algebra and Noncommutative Invariant Theory, Joint Mathematics Meetings, Seattle, WA, Jan. 2022 (cancelled due to COVID-19).
Invited Speaker, "An attempt to character theory for finite-dimensional Hopf algebras," Special Session on Cohomology, Representation Theory, and Lie Theory, AMS Fall Southeastern Sectional Meeting, \#1173, virtual (formerly at the University of South Alabama, Mobile, AL), Nov. 2021.
Invited Speaker, "Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation," Online Noncommutative Algebra Seminar Series (international, virtual), Nov. 2021.
Invited Speaker, "Building resolutions from Koszul complex," Seattle Noncommutative Algebra Day (SNAD, virtual), May 2021.
Invited Speaker, "Quantum symmetries through the lens of linear algebra," FD Seminar (international, virtual), May 2021.
Invited Speaker, "Minimal free resolutions and Golodness of a local ring," Special Session on Commutative Algebra, AMS Spring Eastern Sectional Meeting #1165, virtual (formerly at Brown University), Mar. 2021.
Invited Speaker, "Toward Golodness of a local ring," Rep. Theory & Related Topics Seminar, Northeastern University, Boston, Feb. 2021.
Speaker, "On minimal free resolution of a local ring," Combinatorics Algebra & Topology (CAT) Seminar, U.S. Naval Academy, Feb. 2021.
Invited Speaker, "McKay matrices for finite-dimensional Hopf algebras," Special Session on Geometry and Repr. Theory of Quantum Algebras and Related Topics, AMS Fall Western Sectional Meeting #1162, virtual (formerly at University of Utah), Oct. 2020.
Invited Speaker, "Encoding quantum symmetries in linear algebra," NARC Seminar, U.S. Naval Academy, Oct. 2020.
Invited Speaker, "Finite generation of cohomology and support variety theory," Special Session on Connections Between Noncommutative Algebra and Algebraic Geometry, AMS Fall Central Sectional Meeting #1150, University of Wisconsin-Madison, Sept. 2019.
Invited Speaker, "On the existence of some tilting modules," Conference on Homological Methods and Tilting Theory of Finite Dimensional Algebras, in honor of Gordana Todorov’s 70th birthday, University of Iowa, Iowa City, Aug. 2019.
Invited Speaker, "On the existence of permanent cocycles via twisted tensor product and Anick resolutions," Mini-workshop on Cohomology of Hopf algebras and tensor categories, Mathematisches Forschungsinstitut Oberwolfach, Germany, Mar. 2019.