Research

I work in an area of mathematics called noncommutative algebra, specifically in noncommutative algebraic geometry, and noncommutative invariant theory. I also enjoy thinking about certain problems in combinatorics (in particular, about capsets in F3n).

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 2x2 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 R2R2, 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. If you'd like to learn more about noncommutative algebra, I can recommend the following:

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.

My Erdős number is 3 (via two paths: Michael Tait→Fan Chung→Paul Erdős and Thang Pham→Arie Bialostocki→Paul Erdős).

In a past life, I did some research in neuroscience. Those papers can be found at the bottom of this page.

Preprints

Improved bounds on sizes of generalized caps in AG(n, q)
with Michael Tait.
arXiv
A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras
with Daniel Rogalski and James J. Zhang.
arXiv
Algebraic structures in comodule categories over weak bialgebras
with Chelsea Walton and Elizabeth Wicks.
arXiv
Semisimple reflection Hopf algebras of dimension sixteen
with Luigi Ferraro, Ellen Kirkman, and W. Frank Moore.
arXiv
Simple Z-graded domains of Gelfand–Kirillov dimension two
with Luigi Ferraro and Jason Gaddis.
arXiv

Publications

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 (Alice) Huang and Michael Tait.
Involve, a Journal of Mathematics, Vol. 12, No. 6, 995–1003. (2019).
Alice is a Wake Forest undergraduate who was awarded a Wake Forest Research Fellowship to work with me during the summer of 2018. She presented a poster at Wake Forest's Undergraduate Research Day.
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
pdf

Invited Talks

A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras
AMS Western Sectional Meeting, Riverside, CA, November 2019
Slides
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
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

The geometry of the card game SET
Loyola Marymount University Math Club, Los Angeles, CA, September 2019
Slides
Photo
Algebraic structures in comodule categories over weak Hopf algebras
Seattle Noncommutative Algebra Day, Seattle, WA, July 2019
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

Non-math Publications

Appelbaum, L.G., Boehler, C.N., Davis, L.A., Won, R.J., Woldorff, M.G. (2014). "The dynamics of proactive and reactive cognitive control processes in the human brain". Journal of Cognitive Neuroscience. 26(5), 1021–1038. Journal
PubMed
Appelbaum, L.G., Boehler, C.N., Won, R.J., Davis, L.A., Woldorff, M.G. (2012). "Strategic Allocation of Attention Reduces Temporally Predictable Stimulus Conflict". Journal of Cognitive Neuroscience. 24(9), 1834–1848. Journal
PubMed

‘I will take the Ring,’ he said, ‘though I do not know the way.’
― J.R.R. Tolkien, The Fellowship of the Ring