Research

On this page are links to papers, preprints, and slides from talks I've given. You can also find all of my preprints on arXiv.

I work in an area of mathematics called noncommutative algebra, specifically in noncommutative algebraic geometry, and noncommutative invariant theory. If you'd like to learn more about some of those topics, I can recommend the following:

I also enjoy thinking about certain problems in combinatorics (in particular, about capsets in F3n). It turns out that the card game SET is related to these affine geometric spaces. To learn more:

  • The Joy of SET by Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon.

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.

Publications

Fixed rings of generalized Weyl algebras
with Jason Gaddis.
Journal of Algebra. (To appear).
arXiv Journal
Sidon sets and 2-caps in F3n
with Yixuan (Alice) Huang and Michael Tait.
Involve, a Journal of Mathematics. (To appear).
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.
arXiv Journal
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, no. 5, 1881–1896. (2019).
arXiv Journal
Discriminants of Taft algebra smash products and applications
with Jason Gaddis and Daniel Yee.
Algebras and Representation Theory. (To appear).
arXiv Journal
The noncommutative schemes of generalized Weyl algebras.
Journal of Algebra 506, 322–349. (2018).
arXiv Journal
The Picard group of the graded module category of a generalized Weyl algebra.
Journal of Algebra 493, 89–134. (2018).
arXiv Journal
A structure theorem for product sets in extra special groups
with Thang Pham, Michael Tait, and Le Anh Vinh.
Journal of Number Theory 184, 461–472. (2018).
arXiv Journal
Partitions of AG(4,3) into maximal caps
with Michael Follett, Kyle Kalail, Elizabeth McMahon, and Catherine Pelland.
Discrete Mathematics 337, 1–8. (2014).
arXiv Journal
The graded module category of a generalized Weyl algebra. Ph.D. thesis, University of California, San Diego (2016). pdf eScholarship

Preprints

Simple Z-graded domains of Gelfand-Kirillov dimension two
with Luigi Ferraro and Jason Gaddis.
arXiv
Three infinite families of reflection Hopf algebras
with Luigi Ferraro, Ellen Kirkman, and W. Frank Moore.
arXiv

Invited Talks

"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
Abstract Slides
"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
Abstract Slides
"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
Abstract Slides
"Auslander's Theorem for permutation actions on noncommutative algebras"
AMS Western Sectional Meeting, Pullman, WA, April 2017
Abstract Slides
"The noncommutative schemes of generalized Weyl algebras"
AMS Western Sectional Meeting, Denver, CO, October 2016
Abstract Slides
"The noncommutative schemes of generalized Weyl algebras"
AMS Eastern Sectional Meeting, Brunswick, ME, September 2016
Abstract Slides
"The category of graded modules of a generalized Weyl algebra"
AMS Central Sectional Meeting, East Lansing, MI, March 2015
Abstract Slides

Other Talks

"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
Abstract Slides
"The category of graded modules of a generalized Weyl algebra"
AMS-MAA Joint Mathematics Meetings, Seattle, WA, January 2016
Abstract Slides
"Z-graded noncommutative projective geometry"
UC San Diego Algebra Seminar, La Jolla, CA, November 2015
Abstract Slides
Pre-talk
"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
Abstract Slides
"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. PubMed Journal
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. PubMed Journal