I work in an area of mathematics called noncommutative algebra, specifically in noncommutative algebraic geometry, noncommutative ring theory, noncommutative invariant theory, and quantum symmetries.
Noncommutative algebra
In noncommutative algebra, we study mathematical structures (called rings) whose multiplication operation is not necessarily commutative. Such mathematical structures are actually ubiquitous.
If you have taken linear algebra, then you have seen that the set of all 2 ✕ 2 matrices with entries in R has a natural multiplication, but for two matrices A and B, the products AB and BA are different, in general. One way to understand this phenomenon is that A and B can be viewed as linear transformations R2 → R2, where multiplication of matrices corresponds to composition of functions. And, of course, the order in which you compose functions matters (your day turns out very differently if you first put on your shoes and then your socks vs. first putting on your socks and then your shoes).
Noncommutative rings also arise naturally in quantum mechanics (the position and momentum operators do not commute), conformal field theory (where symmetries are captured by so-called quantum groups), the study of Lie algebras (the universal enveloping algebra a Lie algebra is not necessarily commutative), and the study of differential operators (multiplication by t and differentiation d/dt do not commute).
I study noncommutative rings using both ring-theoretic and homological techniques. I also study Hopf algebras and weak Hopf algebras, and their actions on rings (this is part of what is known as quantum symmetry). If you’d like to learn more about some of my favorite parts of noncommutative algebra, I can recommend the following:
- An Invitation to Noncommutative Algebra by Chelsea Walton
- Symmetries of Algebras: A Guide for Newcomers a forthcoming book by Chelsea Walton
- Invariant Theory of Artin–Schelter Regular Algebras: A Survey by Ellen Kirkman
- Artin–Schelter Regular Algebras by Daniel Rogalski
- An Introduction to Noncommutative Projective Geometry by Daniel Rogalski
I also enjoy thinking about certain problems in combinatorics (in particular, about capsets in F3n).
Capsets
Consider n-dimensional affine space F3n over the field with three elements. A capset in F3n is a collection of points in which no three points are collinear. Finding the largest possible capset in F3n is a very difficult problem (in fact, the exact answer is not known when n > 6). One interesting and useful fact is: although the definition of a capset is geometric, there is an equivalent arithmetic definition. A subset of F3n is a capset if and only if it contains no three-term arithmetic progressions. Hence, techniques of combinatorial number theory can (and have) been used to study capsets.
I learned about capsets as an undergraduate at Lafayette College’s REU. Since then, I have advised several student projects of my own related to capsets. It turns out that the affine geometric space F34 can be visualized using the deck of cards from the game SET. My REU mentor Liz McMahon (together with her family) wrote a great book on the mathematics of SET.
- The Joy of SET by Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon.
My Erdős number is 3 (Robert Won ➔ Michael Tait ➔ Fan Chung ➔ Paul Erdős).
Preprints
| Actions of Taft algebras on noetherian down-up algebras with Simon Crawford and Jason Gaddis. | arXiv |
| Ozone groups of Artin–Schelter regular algebras satisfying a polynomial identity with Kenneth Chan, Jason Gaddis, and James J. Zhang. | arXiv |
| Homological regularities and concavities with Ellen Kirkman and James J. Zhang. | arXiv |
Refereed Publications
| Symmetries of algebras captured by actions of weak Hopf algebras with Fabio Calderón, Hongdi Huang, and Elizabeth Wicks. Algebras and Representation Theory (Accepted). | arXiv |
| Ozone groups and centers of skew polynomial rings with Kenneth Chan, Jason Gaddis, and James J. Zhang. International Mathematics Research Notices, Volume 2024, Issue 7, 5689–5727 (2024). | Journal arXiv |
| Weight modules over Bell–Rogalski algebras with Jason Gaddis and Daniele Rosso. Journal of Algebra, Volume 633, 270–297 (2023). | Journal arXiv |
| Weighted homological regularities with Ellen Kirkman and James J. Zhang. Transactions of the American Mathematical Society, 376, 7407–7445 (2023). | Journal arXiv |
| Universal quantum semigroupoids with Hongdi Huang, Chelsea Walton, and Elizabeth Wicks. Journal of Pure and Applied Algebra, Volume 227, Issue 2 (2023). | Journal arXiv |
| Pointed Hopf actions on quantum generalized Weyl algebras with Jason Gaddis. Journal of Algebra, Volume 601, 312–331 (2022). | Journal arXiv |
| Reflexive hull discriminants and applications with Kenneth Chan, Jason Gaddis, and James J. Zhang. Selecta Mathematica, New Series, 28, Article 40 (2022). | Journal arXiv |
| Degree bounds for Hopf actions on Artin–Schelter regular algebras with Ellen Kirkman and James J. Zhang. Advances in Mathematics, Vol. 397, Article 108197 (2022). | Journal arXiv |
| Algebraic structures in comodule categories over weak bialgebras with Chelsea Walton and Elizabeth Wicks. Communications in Algebra, 50:7, 2877–2910 (2022) | Journal arXiv |
| Semisimple reflection Hopf algebras of dimension sixteen with Luigi Ferraro, Ellen Kirkman, and W. Frank Moore. Algebras and Representation Theory, 25, 615–647 (2022). | Journal arXiv |
| A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras with Daniel Rogalski and James J. Zhang. Algebra & Number Theory, Vol. 15, No. 4, 971–997 (2021). | Journal arXiv |
| Improved bounds on sizes of generalized caps in AG(n, q) with Michael Tait. SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, 521–531 (2021). | Journal arXiv |
| Simple Z-graded domains of Gelfand–Kirillov dimension two with Luigi Ferraro and Jason Gaddis. Journal of Algebra, Volume 562, 433–465 (2020). | Journal arXiv |
| Three infinite families of reflection Hopf algebras with Luigi Ferraro, Ellen Kirkman, and W. Frank Moore. Journal of Pure and Applied Algebra, Volume 224, Issue 8 (2020). | Journal arXiv |
| Fixed rings of generalized Weyl algebras with Jason Gaddis. Journal of Algebra, Volume 536, 149–169 (2019). | Journal arXiv |
| Sidon sets and 2-caps in F3n with Yixuan Huang and Michael Tait. Involve, a Journal of Mathematics, Vol. 12, No. 6, 995–1003 (2019). | Journal arXiv C++ code |
| Auslander’s Theorem for permutation actions on noncommutative algebras with Jason Gaddis, Ellen Kirkman, and W. Frank Moore. Proceedings of the American Mathematical Society 147, 1881–1896 (2019). | Journal arXiv |
| Discriminants of Taft algebra smash products and applications with Jason Gaddis and Daniel Yee. Algebras and Representation Theory, Volume 22, Issue 4, 785–799 (2019). | Journal arXiv |
| The noncommutative schemes of generalized Weyl algebras. Journal of Algebra, Volume 506, 322–349 (2018). | Journal arXiv |
| The Picard group of the graded module category of a generalized Weyl algebra. Journal of Algebra, Volume 493, 89–134 (2018). | Journal arXiv |
| A structure theorem for product sets in extra special groups with Thang Pham, Michael Tait, and Le Anh Vinh. Journal of Number Theory, Volume 184, 461–472 (2018). | Journal arXiv |
| Partitions of AG(4,3) into maximal caps with Michael Follett, Kyle Kalail, Elizabeth McMahon, and Catherine Pelland. Discrete Mathematics, Volume 337, 1–8 (2014). | Journal arXiv |
| The graded module category of a generalized Weyl algebra. Ph.D. thesis, University of California San Diego (2016). | eScholarship |
Invited Talks
| Ozone groups of PI Artin–Schelter regular algebras AMS Southeastern Sectional Meeting, Savannah, GA, October 2024 | TBA |
| Ozone groups of PI Artin–Schelter regular algebras AMS Central Sectional Meeting, San Antonio, TX, September 2024 | Slides |
| Symmetries captured by weak Hopf algebra actions AMS Western Sectional Meeting, Fresno, CA, May 2023 | Slides |
| Symmetries captured by weak Hopf algebra actions Seattle Noncommutative Algebra Day, Seattle, WA, March 2023 | Slides |
| The card game SET, finite affine geometry, and combinatorial number theory US Naval Academy Combinatorics, Algebra, and Topology Seminar, Annapolis, MD, October 2022 | Slides |
| PI skew polynomial rings and their centers Workshop on Noncommutative Geometry and Noncommutative Invariant Theory Banff International Research Station, Banff, Canada, September 2022 | Slides |
| Pointed Hopf actions on generalized Weyl algebras Joint Mathematics Meetings, Seattle, WA, January 2022 | |
| Universal quantum semigroupoids Seattle Noncommutative Algebra Day, Online, May 2021 | Slides |
| Algebraic structures in comodule categories over weak bialgebras AMS Central Sectional Meeting, Online, April 2021 | Slides |
| Degree bounds for Hopf actions on Artin–Schelter regular algebras AMS Eastern Sectional Meeting, Online, March 2021 | Slides |
| Degree bounds for Hopf actions on Artin–Schelter regular algebras AMS–MAA Joint Mathematics Meetings, Online, January 2021 | Slides Abstract |
| Algebraic structures in comodule categories over weak bialgebras AMS Western Sectional Meeting, Online, October 2020 | Slides Abstract |
| A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras AMS Western Sectional Meeting, Riverside, CA, November 2019 | Slides Abstract |
| A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras Conference on Operad Theory and Related Topics, Qufu, China, October 2019 | Slides |
| A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras Shanghai University of Finance and Economics Seminar, Shanghai, China, October 2019 | |
| A translation principle for generalized Weyl algebras AMS Central Sectional Meeting, Madison, WI, September 2019 | Slides |
| Algebraic structures in comodule categories over weak Hopf algebras Seattle Noncommutative Algebra Day, Seattle, WA, July 2019 | |
| The card game SET, finite affine geometry, and combinatorial number theory Portland State University Colloquium, Portland, OR, May 2019 | Slides |
| A translation principle for generalized Weyl algebras AMS Central/Western Sectional Meeting, Honolulu, HI, March 2019 | Slides Abstract |
| Z-graded noncommutative algebraic geometry University of Washington Algebra/Algebraic Geometry Seminar, Seattle, WA, February 2018 | Slides |
| Simple Z-graded domains of Gelfand–Kirillov dimension 2 AMS–MAA Joint Mathematics Meetings, San Diego, CA, January 2018 | Slides Abstract |
| Z-graded noncommutative algebraic geometry Miami University Colloquium, Oxford, OH, November 2017 | Slides |
| Noncommutative invariant theory and Auslander’s Theorem Miami University Algebra Seminar, Oxford, OH, November 2017 | Slides |
| Discriminants of Taft algebra smash products and applications AMS Central Sectional Meeting, Denton, TX, September 2017 | Slides Abstract |
| Auslander’s Theorem for permutation actions on noncommutative algebras AMS Western Sectional Meeting, Pullman, WA, April 2017 | Slides Abstract |
| The noncommutative schemes of generalized Weyl algebras AMS Western Sectional Meeting, Denver, CO, October 2016 | Slides Abstract |
| The noncommutative schemes of generalized Weyl algebras AMS Eastern Sectional Meeting, Brunswick, ME, September 2016 | Slides Abstract |
| The category of graded modules of a generalized Weyl algebra AMS Central Sectional Meeting, East Lansing, MI, March 2015 | Slides Abstract |
Other Talks
| Symmetries in noncommutative algebra George Washington University Graduate Student Seminar, Washington, DC, March 2024 | |
| The card game SET and (affine) linear algebra Soka University of America, MATH 217 (Linear Algebra), Online, May 2023 | Slides |
| The mathematics of the card game SET Soka University of America, MATH 160 (Liberal Arts Mathematics), Online, December 2022 | Slides |
| The card game SET, finite affine geometry, and combinatorial number theory George Washington University Combinatorics and Algebra Seminar, Washington, DC, November 2022 | Slides |
| Quantum symmetry and noncommutative algebra George Washington University Graduate Student Seminar, Washington, DC, December 2021 | |
| The geometry of the card game SET Loyola Marymount University Math Club, Los Angeles, CA, September 2019 | Slides Photo |
| SET and AG(4,3) Wake Forest Combinatorics Seminar, Winston-Salem, NC, October 2016 | Slides |
| Categories of graded modules: What they are and what you can do with them GradSWANTAG III, La Jolla, CA, May 2016 | |
| The graded module category of a generalized Weyl algebra UC San Diego Final Defense, La Jolla, CA, May 2016 | Slides Abstract |
| The category of graded modules of a generalized Weyl algebra AMS–MAA Joint Mathematics Meetings, Seattle, WA, January 2016 | Slides Abstract |
| Z-graded noncommutative projective geometry UC San Diego Algebra Seminar, La Jolla, CA, November 2015 | Slides Pre-talk Abstract |
| What is noncommutative algebraic geometry? UC San Diego Graduate Algebraic Geometry Seminar, La Jolla, CA, August 2015 | Slides |
| A (gentle) introduction to Hopf algebras UC San Diego Informal Noncommutative Algebra Seminar, La Jolla, CA, June 2015 | |
| SET and AG(4,3) UC San Diego Food For Thought Seminar, La Jolla, CA, February 2015 | Slides Abstract |
| Graded modules over generalized Weyl algebras UC San Diego Advancement to Candidacy, La Jolla, CA, December 2014 | Slides |
| SET and disjoint complete caps in AG(4,3) AMS–MAA Joint Mathematics Meetings, New Orleans, LA, January 2011 | Abstract |
‘I will take the Ring,’ he said, ‘though I do not know the way.’
― The Fellowship of the Ring, J.R.R. Tolkien