2026 REU Seminars will take place Tuesdays from 1-3pm from June 9 through July 28 in 1324 East Hall. All students participating in an REU project will speak at a seminar and are required to attend on the Tuesdays they are in Ann Arbor. Participants should submit their talk title and abstract at least a week in advance of their seminar date.
The moduli space of complex curves parametrizes closed Riemann surfaces and admits a natural compactification, due to Deligne and Mumford, by the inclusion of stable nodal curves. The boundary of this compactification carries a rich combinatorial structure, stratified by the dual graphs of the corresponding degenerations. In this talk, I discuss analogous constructions for the moduli space of (Z/pZ)^2g-covers of curves, focusing on compactification, boundary strata, and the combinatorial data that encode degenerations of such covers.
We study the image size of a rational function f(X) over a finite field F_q. One can get a first approximation to these numbers in terms of monodromy groups of f(X), namely the Galois group of the numerator of f(X)-t over the fields F_q(t) and \overline{F_q}(t), using the value set formula #f(F_q) = c(A,G)q+O(√q), where A and G are the 2 monodromy groups, c(A,G) is a constant dependent on A and G, and O(√q) is an error term. In this talk, we discuss the constant c(A,G) in some specific cases of f(X).