*[Note: This page is rarely updated, because research is not
the emphasis of my current position.]
*

My research interests include additive number theory, combinatorics, and graph theory. The guiding principle of my research is to aid in the education of undergraduate students, and so problems that can be expressed and explored without a lot of background, but that require perceptive approaches to solve, are of particular interest to me.

### Combinatorics and Number Theory

My doctoral thesis concerned bounds for the size of
large sumfree subsets of higher-dimensional lattices. An adaptation of
the first chapter of the thesis appeared as this paper
in *Acta Arithmetica*.

In Ben Green's review of the paper (for Mathematical Reviews) he pointed out an unpublished note by Peter Cameron which supersedes some of the results in the paper. I was not previously aware of Cameron's work, and after reading it I was able to use material from my work to extend his results to higher dimensions. I have not published these results, but I would be happy to discuss them if the investigation of sumfree sets has somehow led you here.

In the near future, I hope to post slides from a talk I gave about this content (which includes the improved bounds described above).

### The Mathematics of Puzzles

Recently I have been parlaying my interest in abstract logic puzzles into mathematical research about the structures involved in these puzzles. This is a growing area of interest, although the lion's share of attention is directed at sudoku (since it is by far the most popular of these puzzle types).

At the 2012 Joint Mathematics Meetings, I prepared a talk about Futoshiki puzzles for a special session on the mathematics of pencil puzzles. Futoshiki are very similar to sudoku in that both ask the solver to complete a Latin square subject to an additional constraint. (In Futoshiki, the constraint is greater-than relationships imposed on certain pairs of adjacent squares.) The talk I gave is here, and here are a couple of puzzles I prepared for a handout at the session. One of the puzzles from the handout also appeared in the June/July 2012 issue of MAA Focus.

In 2022, I gave a more expansive talk at Brown's Symposium for Undergraduates in the Mathematical Sciences (SUMS), covering Futoshiki as well as other logic puzzle types. Here are the slides, which also contain a not-so-subtly-hidden puzzle...

In general, I am very interested in these constrained-Latin-Square puzzles, both in terms of which constraints lead to the most interesting problems from a mathematical perspective and which lead to the most interesting puzzles from a solving perspective.