Research Interests:
Current areas of interest:
- The study of "sparse" partial latin squares and their completions.
- The complexity class of completing certain classes of partial latin squares.
- The applications of families of mutually orthogonal latin squares to finding decompositions of "almost-complete" graphs.
- Quasirandom graphs, and in particular extending the notion of quasi-randomness to colorings of graphs and hypergraphs.
- Applications of quasirandom graphs to Hadamard matrices and residue classes in large finite fields.
(If you're one of my students and you're curious about what these things are and why we'd care as mathematicians, send me an email! A lot of the questions I'm currently working on are remarkably accessible, and I'd be glad to talk with you about them and lay out some interesting areas for you to explore.)
Papers in progress:
- NP-completeness and completing partial latin squares..
- Completions of sets of partial mutually orthogonal latin squares..
- Quasi-random colorings of graphs and hypergraphs.
Papers written:
- Completing ≤ϵ-dense partial latin squares, published in the Journal of Combinatorial Design, June 2013.
Conference talks:
- Completing ≤ϵ-dense partial latin squares, 17th Coast Combinatorics Conference, University of Victoria, October 2012; 26th Midwest Conference on Combinatorics, Cryptography and Computing, Southern Utah University, March 2012; WilsonFest 2012, California Institute of Technology, March 2012.
Other talks:
- NP-completeness and completing partial latin squares, Caltech Combinatorics Seminar, November 2012.
- Random and quasirandom graphs, Caltech Grad Student Seminar, January 2012.
- Completing sparse partial latin squares, Caltech Combinatorics Seminar, October 2011.
- How to (not) prove the four-color theorem, Canada/USA Mathcamp, July 2012.