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: Chasing the Irrational

Mentor: Reynold Fregoli

I will talk about automatic sequences first and then introduce the number theory problem of Diophantine approximation related to the automatic sequence numbers. After introducing previous results in that aspect, I will talk about our study in multidimensional Diophantine approximation and show our theorem. 

Grant Koh: Small Value Set Polynomials Over Finite Fields

Mentors: Michael Zieve, Javier Santiago

Gomez-Calderon and Madden provided an explicit list of polynomials over the finite field of order q with value set of size at most 2q/d where d is the degree of the polynomial given an upper bound on d on the order of q^{1/4}. We survey the key steps of their proof and explain how, via elementary techniques, the bound can be improved to a square root bound. We then suggest some methods of determining which polynomials have a value set of size at most 3q/d, using several other results of Gomez-Calderon.

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 and Edward Lee: Keeping it Real(izable)

Mentor: Bar Roytman

Equivariant homotopy theory has families of generalizations of commutative monoids with G-actions, where G is a finite group. These generalizations arise from G-operads and differ in which symmetric multiplication operations they admit. A G-transfer system is a combinatorial object that encodes these generalizations. Complex linear isometries operads can be used to turn a collection of irreducible G-representations into a transfer system. However, we are interested in the inverse direction. For which G-transfer systems does such a collection, a "G-universe," exist? Which transfer systems are "realizable"? When G is cyclic, this question reduces to an incidence geometry problem in the subgroup lattice of G. We will outline our strategy for realizing large families of transfer systems, which involves creatively omitting cosets from the complete set of representations.

Yotam Chopra, Charlie Jiang, and Emily Ma: Enumerating Weak Transfer Systems for Cyclic Groups of Prime Power Order

Mentor: Bar Roytman

Blumberg and Hill introduced N-infinity operads as equivariant analogues of E-infinity operads that encode axioms for commutative and associate operations up to homotopy coherence in the setting of spaces equipped with finite group actions. Recently, Stewart formulated a generalization of N-infinity operads where there need not be a binary multiplication operation fixed by the group action. We introduce weak transfer systems, combinatorial objects that function as a decategorified version of these weak N-infinity operads. For cyclic groups of prime power order, we characterize weak transfer systems as collections of generating sets and reduce them to a synthetic form that is free of redundancy. From this alternative characterization we obtain a closed form formula that enumerates all such weak transfer systems.

Mo Chen and Wesal Hammo: Mathematical Modeling of CD8 T Cell Responses to Metabolic Therapy in Cancer

Mentor: Quiyana Murphy

Glycolysis, the metabolic pathway by which glucose is broken down to release energy, is profoundly upregulated in many cancers, with tumor cells adopting aerobic glycolysis (the Warburg effect) to sustain rapid proliferation while generating a nutrient‑deprived, immunosuppressive microenvironment. This heightened glycolytic demand restricts the metabolic capacity of cytotoxic CD8$^+$ T cells, driving functional exhaustion and enabling tumor progression. Recent work suggests that activating pyruvate kinase M2 (PKM2) can restore T‑cell metabolic fitness, enhance effector function, and promote durable immune memory, positioning PKM2 as a promising target for metabolic immunotherapy. Motivated by these findings, we develop a mathematical framework to investigate the impact of PKM2 activation in metastatic melanoma based on published data from a syngeneic, highly metastatic murine melanoma study. Using published B16‑F10 melanoma growth data, we identify the von Bertalanffy model as the most biologically realistic tumor growth law. We then construct a system of ordinary differential equations describing tumor dynamics, CD8$^+$ T‑cell responses, IFN‑$\gamma$–mediated immune interactions and the pharmacological activation of PKM2. The model was then extended to incorporate CD8$^+$ T cell memory subsets. This framework provides a tool for exploring how metabolic interventions may enhance immediate tumor control while supporting long‑term anti‑tumor immunity.

Shivansh Sharma: Hypergraph Geometry for Multi-Way Chromatin Contacts: Tuning the Inter-Chromosomal Signal in Pore-C Data

Mentor: Anthony Bloch

Pore-C sequencing captures multi-way chromatin contacts, which are single reads in which several genomic loci are physically co-located, making a hypergraph, rather than a pairwise graph, the natural representation of three-dimensional genome organization. We study mouse embryonic stem cell (mESC) Pore-C data as a hypergraph whose nodes are genomic loci and whose hyperedges are individual reads, and apply two hypergraph-geometric measures: a LogExp hypergraph centrality and a Ricci–Forman poset curvature, to extract a structurally meaningful "core" of contacts. From this core we build a population pair graph: a chromosome-by-chromosome map of inter-chromosomal co-occurrence that summarizes large-scale nuclear architecture. A central difficulty is that intra- and inter-chromosomal contacts enter this representation in very different proportions, and the inter-chromosomal signal, which is the part most informative about chromosome territories, is both scarce and unevenly distributed. We show that the curvature-based partition separates an inter-rich edge class from an intra-dominated one, and that mixing these classes in controlled proportions lets us tune the intra-to-inter contact ratio to a target while preserving the structure of the pair graph. We discuss what this reveals, and its limits, for recovering inter-chromosomal organization.

Maya Mustata and Zhiwu Wang: Higher-Order Singularities of Tensor Varieties

Mentor: Alessandro Danelon

Given a vector space V, one can construct its tensor powers, symmetric and exterior products, etc. all of which behave as representations of its automorphism group GL(V). One can then construct families of algebraic varieties that are uniformly defined inside these representations. These objects are called Vec-varieties. Many infinite-dimensional varieties arise as limits of such families, and the induced action of GL allows us to use techniques from representation theory to understand their properties. In fact, it has been shown that many properties of classical algebraic varieties also apply to Vec-varieties. In particular, a recent paper of Chiu, Danelon, and Draisma showed that, in sufficiently high dimension, taking the singular locus of each member of such a family does indeed define a closed Vec-subvariety. In our project, we are investigating whether finer invariants of singularities, such as order, jet-lifting, and embedding codimension, are also compatible with this structure. In this talk, we give an introduction to the theory of Vec-varieties, explain the interesting structures they provide, and give our conjectures on the stable behavior of their higher-order singular loci.

Grant Hacking: D-modules and Rigid Analytic Bernstein-Sato Theory

Mentor: Arun Soor

Classical D-module theory provides an algebraic way to study systems of differential equations. A key phenomenon is that localizations such as C[x,f^{-1}] become finitely generated over the ring of differential operators. In this talk, I will explain how this picture changes in p-adic rigid analytic geometry, where the natural functions on complements are analytic rather than algebraic and one must use completed rings of differential operators. Finite generation is then replaced with a condition known as coadmissibility. I will discuss how Bernstein–Sato polynomials and the arithmetic type of their roots control coadmissibility in a theorem of Bitoun and Bode. Lastly, I will discuss the direction of the research project, where we are exploring whether a similar picture holds for the complement of Z_p in the p-adic unit disc.

Benjamin Singer: Derived Analytic Geometry

Mentor: Arun Soor

Generally, the landscape of geometry sorts itself into 3 bins: differential geometry, algebraic geometry, and analytic geometry. Although analytic geometry is closely connected to the first two by various analytification procedures, it fails to inherit foundational properties that make the differential and algebraic theories so powerful. The main pain point is that analytic geometry requires working with classes of topological rings and modules that have poor categorical properties. Resolving this problem has become significant recently, as advancing important problems in arithmetic geometry (particularly, some aspects of the Langlands program) has been shown to rely on a robust theory of analytic geometry. In this talk, I’ll discuss some of the issues with preexisting theories of analytic geometry and explain how we can overcome them by fusing classical functional analysis with derived and homotopical ideas.

Zachary DeVivo: Bounds on Brehm's Extension Theorem

Mentor: Ian Tobasco

It is known that given a two sets of points {a_1... a_n}, {b_1... b_n} in R^2, with the property |b_i - b_j| <= |a_i - a_j| for all i, j, there exists a linear piecewise distance preserving map h such that h(a_i) = b_i for all i. This h can be generated via Brehm's extension theorem. In this paper, we show that for particular objects, Brehm's extension theorem gives a crease set of size proportional to n.

Joseph Gascho: The Mathematics Behind Origami Springs

Mentor: Ian Tobasco

Mathematical analysis of origami models has become more prevalent in the area of applied math, with findings being implemented in many engineering developments, e.g., the James Webb telescope. Jeff Beynon's Spring into Action is an origami model that behaves similarly to a spring. This model has the unique property of simultaneous extension/compression and rotation. We have found explicit formulas for the extension/compression motion of the spring. We have also conducted experiments to find the spring constant and to see the behavior of springs as they get very long.

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