## 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).

##### 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 (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.

### 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