I am interested in a range of areas in, and related to, combinatorics. My original background is in the structural theory of finite fields, and
I have worked on various questions related to the existence of primitive normal bases for finite fields. I am also interested in applications of
finite fields to combinatorial designs, and to coding theory and cryptography.
Recently I have been investigating connections between combinatorics and cryptography, via the concept of external difference families and their generalizations.
These have a natural connection to optimal AMD codes. Many of the constructions in this area use cyclotomic methods from finite fields, but there are rich
connections with other areas of combinatorics, finite geometry and group theory. I was recently funded
by a Carnegie Research Incentive Grant to
work on this with my collaborators.
Another strand of my research is viewing combinatorial objects as relational structures, and asking questions about partial orders on these structures. In particular, I
have been investigating the homomorphic image order, a natural analogue of the substructure order, with a focus on the setting of graphs and digraphs.