2026 U-M REU Seminars

Devayani Pradhan: Curves & Covers: The Boundary Stratification of The Moduli

Mentor: Sam Payne

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.

Youyi Zhang

Mentor: Reynold Fregoli

Grant Koh

Mentor: Michael Zieve

Chi Dinh: Image Size of Rational Function

Mentor: Michael Zieve

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).

Alena Gurevich

Mentor: Bar Roytman

Edward Lee

Mentor: Bar Roytman

Charlie Jiang

Mentor: Bar Roytman

Yotam Chopra

Mentor: Bar Roytman

Mo Chen and Wesal Hammo

Mentor: Quiyana Murphy

Shivansh Sharma

Mentor: Tony Bloch

Emily Ma

Mentor: Bar Roytman

Zhiwu Wang

Mentor: Alessandro Danelon

Maya Mustata

Mentor: Alessandro Danelon

Benjamin Singer

Mentor: Arun Soor

Grant Hacking

Mentor: Arun Soor

Zachary DeVivo

Mentor: Ian Tobasco

Joseph Gascho

Mentor: Ian Tobasco

Rishi Dadlani

Mentor: Jinho Baik

Shaotian Sun

Mentor: Jinho Baik

Logan Edwards

Mentor: Silas Alben

Aman Makhija

Mentor: Ian Tobasco and Suraj Shankar

Jeremy Lee

Mentor: Patrick Lutz

Gavin Bell

Mentor: Patrick Lutz

Daniel Le

Mentor: Sabina Haque

George Simmons

Mentor: Sabina Haque

Samuel Waid

Mentor: Ralf Spatzier

Jaan Amla Srimurthy

Mentor: Ralf Spatzier

Spencer Danese

Mentor: Stephen DeBacker