about research publications miscellaneous
My area of research lies broadly in number theory. While I mainly focus on Galois representations attached to low-dimensional abelian varieties, I have several other distinct research agendas:
The Arithmetic of Orthogonal Polynomials: My work with Farshid Hajir has mainly centered on the irreducibility of certain families of classical polynomials (notably the Legendre Polynomials). But there are interesting questions on the geometry of these polynomials as well -- in particular the algebraic curves and their Jacobians defined by parametric families of polynomials.
Arithmetic in Towers: Farshid and I proved a discriminant formula for the ramification locus in an iterated tower defined by rational functions. While I haven't worked on the Galois theory of towers recently, I have a new line of research concerning the group structure of abelian varieties in towers of finite fields.
Voting Theory: One of my favorite research questions has nothing to do with arithmetic geometry: what is the "fairest" way to vote? What if a voter's preferences are non-linear? (E.g. suppose a voter prefers chocolate to strawberry, chocolate to vanilla, and no preference between vanilla and strawberry.) If that voter is forced to submit a ranked list of preferences, then she is required to make an arbitrary choice between vanilla and strawberry. Sam Hsiao and I proved a existence/uniqueness theorem for certain functions on posets with an application to voting with partially-ordered preferences. I am currently working on generalizing this work.
Miscellaneous Mathematics: I'm always interested in starting new projects and some of my research questions don't fall into any of the above categories. (For instance, I have an ongoing project on the classification of certain Lie algebras over finite fields.) I also very much enjoy working on new projects with students and am always happy to talk research in my office.