2025 U-M REU Seminars

David Cheng and Beinuo (Carl) Guo: Ergodic Average on Line Segments

Mentor: Reynold Fregoli

The Pointwise Ergodic Theorem, proved by Birkhoff in the 1930s, asserts that for any integrable function on a probability space with a measure-preserving transformation, time averages converge almost everywhere to space averages. This fundamental result has since inspired a rich theory examining how ergodic averages behave under more general actions. Jones (1993) provided a sharp characterization for convergence of spherical ergodic averages in R^d-actions, showing that pointwise convergence holds for functions in L^p provided p>d/(d-1), and demonstrated that this range of p is optimal. More recently, Bandi, Kleinbock, and Fregoli extended the theory to C^1 submanifolds of arbitrary curvature under the assumption of exponentially mixing group actions and continuity of the averaging function. Their results reveal the delicate interplay between geometric features of the averaging set, regularity of the function, and quantitative mixing properties of the system. In this talk, we will share our attempts on whether these conditions are indeed necessary by playing around R^2 and Z^2 ergodic actions on line segments.

David Pidugu: Energy Aware Machine Learning

Mentor: Martin Strauss

Machine Learning has become the staple of many new innovations of the 21st century. However, as models grow, so does the massive energy consumption for both the training of these models and inference (Using the trained models to make predictions). Thus, a more sustainable way of exploring new innovations in Machine Learning is through the eyes of Energy Aware algorithms, concerning both algorithmic techniques and hardware implementation. Specifically I look into a very energy inefficient method of Machine Learning called Distributed Learning, specifically Decentralized Federated Learning, and approach designing algorithms for this type of learning in an Energy-Aware methodology.

Aidan Coletta: Model equivalence and abstract subspaces

Mentor: Jeff Dunworth

In this talk, we introduce fundamental models of computational neuroscience and their relation to common machine learning algorithms, and establish the central idea of model equivalence: the idea that two fundamentally different mathematical models can produce near identical behavior. Giving a brief introduction to various machine learning principles, we show how the canonical ReLU activation function can be recovered from biological data. We subsequently show how computational models are attained from this biological data, ultimately producing the same result. However, given that many models are capable of producing the same behavior, we seek a general method for reducing said models and eliminating unnecessary (or insensitive) or computationally inefficient parameters. Fundamentally, the idea is to use abstract subspaces to identify relevant directions in parameter space indicating likely relationships between parameters.

Grant Koh and Devayani Pradhan: Let Bigons be Bigons: Van Kampen Diagrams and Word Reduction

Mentor: Alex Wright

In this talk, we introduce Van Kampen diagrams as a way to analyze finitely presented groups. One essential problem in the study of such groups is the word problem, which asks if there is an algorithm to determine the equivalence of any two words. While such an algorithm may not exist in general, it does if words can be reduced to the identity without increasing word length. Following Wise (2012), we will demonstrate how the geometry of Van Kampen diagrams can prove that such a reduction is possible, giving a proof in the case of Right-Angled Artin Groups. We will then discuss the generalization of such techniques in a more complicated setting, the mapping class group of the five-times punctured sphere.

Jiayuan Zhu: A Computational Agent-Based Model for Cancer–Immune Dynamics

Mentor: Nizhum Rahman

We develop a computational agent-based model to study cancer–immune dynamics in a spatially structured environment. In this framework, individual tumor and immune cells are modeled as discrete agents that move, divide, die, and interact locally over time. Immune cells can recognize and kill nearby tumor cells. The model also includes periodic immune cell injections to simulate immunotherapy. By incorporating spatial heterogeneity and local interactions, the model allows exploration of how treatment timing and immune influx influence tumor progression. This model enables exploration of tumor–immune interactions and the potential effects of immune cell–based therapies.

Tianyong Yao: External light schedules can induce night-time sleep disruptions in a Two-Process Model for toddler sleep

Mentor: Victoria Booth

As young children grow, their need for sleep changes from needing a daytime nap every day to skipping the nap on some days to eventually growing out of the need for a daytime nap. Adding to the high variability in sleep behavior during this time are effects of external factors, including light exposure that can significantly affect the circadian rhythm and its influence on sleep timing. Using the Homeostatic-Circadian-Light (HCL) mathematical model for sleep timing based on the interaction of the circadian rhythm and homeostatic sleep pressure, we fit the model to data for sleep behavior and sleep homeostasis in 2 year olds and analyze the effects of external light schedules on sleep patterns. We find that low daytime light levels can induce nighttime sleep disruptions, while excessively extended bright-light exposure also negatively impacts sleep stability. Our results suggest that consistent daily routines are essential; irregular schedules, particularly during weekends, markedly worsen toddlers' consolidation of nighttime sleep. Specifically, weekend shifts in morning wake-up times and evening lighting result in nighttime sleep disruptions. However, our findings offer practical guidelines for parents to manage weekend activities without greatly affecting sleep stability. This study highlights how external light, daily rhythms, and parenting routines interact to shape toddlers' sleep health, providing a useful reference for improving sleep management practices.

Evan Holly: Asymptotic behavior of the intermediate order statistics of large mean-field systems and statistical estimations

Mentor: Nikolaos Kolliopoulos

We study the asymptotic behavior of the intermediate order statistics of a system of real-valued diffusive particles with mean-field interaction in the drift. We show that the intermediate order statistics k(N), 2k(N), and 4k(N) of the system of interacting particles and the system of independent McKean-Vlasov particles have the same asymptotic behavior under normalization and under reasonable assumptions. This  asymptotic behavior enables us to show weak consistency and asymptotic normality of the Pickands estimator with interacting particles. This allows us to estimate the extreme value index when we only observe the system of interacting particles.

Aarya Kumar and Andrew Tung: Behavior of Arthur Packets Under the Theta Correspondence

Mentor: Alex Hazeltine

Our talk concerns the behavior of representations of linear algebraic groups over local fields under the theta correspondence: specifically, the correspondence between representations of symplectic and even orthogonal groups. In the Langlands program, such representations may be sorted according to certain arithmetic data into finite collections known as Arthur packets. Adams conjectured that such Arthur packets are preserved under the theta correspondence, though this is not generally true. The Adams conjecture holds for sufficiently high dimensional even orthogonal groups; we study the extent of its failure by classifying the behavior of the theta correspondence at its first instance of nonvanishing: i.e., the lowest dimension such that the correspondence is nonzero. We utilize previously established techniques of computations, which allow us to view representations as “extended multi-segments.” Our classification begins with the special case of tempered representations and extends to other relevant cases.

Tianyong Yao: Mathematical modeling of dopamine rhythms and timing of dopamine reuptake inhibitors

Mentor: Ruby Kim

Dopamine (DA) plays a vital role in mood, alertness, and behavior, with dysregulation linked to disorders such as Parkinson's disease, ADHD, depression, and addiction. In this study, we develop and analyze a reduced mathematical model of dopamine synthesis, release, and reuptake to investigate how daily rhythms influence dopamine dynamics and the efficacy of dopamine reuptake inhibitors (DRIs). We simplify a detailed mathematical model of dopamine synthesis, release, and reuptake and demonstrate that our reduced system maintains key dynamical features including homeostatic regulation via autoreceptors. Our model captures core autoregulatory mechanisms and reveals that DRIs can exert substantial time-of-day effects, significantly altering the amplitude and variability of extracellular dopamine concentrations, even when daily averages remain unchanged. These fluctuations depend sensitively on the timing of DRI administration relative to circadian variations in enzyme activity. We further extend the model to incorporate feedback from local dopaminergic tone, which generates ultradian oscillations in the model independent of circadian regulation. Administration of DRIs lengthens the ultradian periodicity. Our findings provide strong evidence that intrinsic fluctuations in DA should be considered in the clinical use of DRIs, offering a mechanistic framework for improving chronotherapeutic strategies targeting dopaminergic dysfunction.

Dani Catala and Miriam Vollmayr-Lee: Singular Curves in 2-Jet Space

Mentor: Alejandro Bravo-Doddoli

A geodesic is the shortest path between two points, and understanding when such a path is globally minimizing is a central question in differential geometry. We call globally minimizing geodesics, metric lines. In this talk, we investigate geodesics arising from algebraic curves defined by degree-two polynomial vectors. We classify curves with singularities and use this classification to study metric lines in the 2-jet space. Central to our work is the period map, whose components are integrals over these curves and which encode the asymptotic behavior of geodesics. We show the period map is one-to-one by applying Hadamard’s Global Inverse Function Theorem, focusing on the most challenging aspect: to verify that its Jacobian is non-degenerate.

Gavin Bell, Colin Feeney, and Ethan Harr: The Realizability Problem for Saturated Transfer Systems

Mentor: Bar Roytman

In this talk we provide an introduction to transfer systems as a mathematical object, specifically how one can construct a saturated G-transfer system from a collection of irreducible representations of the group G. What we aim to tackle is the inverse problem: Which saturated G-transfer systems can be constructed in this way? Thanks to previous works, it is already known that all saturated C_{2m + 1}-transfer systems can be realized by a G-Universe via a linear isometries operad. Generalizing this result to cyclic groups of even order has thus far eluded mathematicians. The initial proofs of odd case results required the use of some very heavy mathematical machinery, introduced by McBrough, but these devices have fallen short in the even case.  By revisiting the odd case, and finding a new method of realization, we hope to find methods less dependent on the order of the group which may grant new hope towards classifying the realizability of saturated C_{2m}-transfer systems.

Tristen Brisky: Gamma Factors for Triple Tensor Product Representations over Finite Fields

Mentor: Elad Zelingher

The representation theory of GL_n(k), where k is a local field, is crucial to number-theoretic questions, particularly concerning the local Langlands conjectures. In this work, we focus on finite-field analogs of the local Langlands theory for n = 2. While the classification of irreducible representations of GL_2(F), where F is a finite field, is known, questions remain about how certain operations like the tensor product behave under the local Langlands correspondence. To explore this, we attach local factors encoding various properties of the representation. Specifically, we investigate a construction of the gamma factor corresponding to the triple tensor product of three irreducible representations of GL_2(F). Our work continues Elber and Lheem's research on gamma factors for GL_2(F) \times GL_2(F) \times GL_2(F) by developing explicit expressions for the gamma factor in terms of Bessel functions and aiming to use this to prove a multiplicative property.

Xinning Ma and Yuanjue (Persy) Zhu: Classification of geodesic(visual) boundary of horospheres in metric spaces of non-positive curvature

Mentor: Ralf Spatzier

CAT(0) spaces—complete geodesic metric spaces of non-positive curvature in the sense of Alexandrov—extend the concept of non-positive curvature beyond the Riemannian setting and exhibit many elegant geometric and topological properties. One such feature is the well-behaved structure of their visual boundaries, defined via asymptotic equivalence classes of geodesic rays. In CAT(0) spaces, the visual boundary coincides with the ideal boundary arising from Gromov's compactification, providing a natural framework for studying geometry at infinity.

This talk focuses on classifying the visual boundaries of horospheres, which arise as level sets of Busemann functions. We begin by introducing the visual boundary via asymptotic classes of geodesic rays. Then, we will examine the visual boundaries and horospheres of examples including trees, hyperbolic spaces, as well as the product spaces. Our goal is to classify these boundaries and discuss how such classifications contribute to understanding the visual boundaries of symmetric spaces—of which hyperbolic spaces are examples in rank 1.

Nathan Wang: Median filters and geodesic active contours

Mentor: Selim Esedoglu

The median filter is a standard tool from image processing that is very effective at removing certain types (e.g. salt and pepper) noise from images. Surprisingly, it also turns out to be intimately connected to curve shortening flow: The steepest descent dynamics for the length of a closed curve in the plane. In fact, the median filter turns out to be a discrete in time approximation to what is known as the “level set formulation” of curve shortening flow that preserves a very important qualitative feature of the flow known as the comparison principle.

We ask if there is an analogue of the median filter for steepest descent on weighted length of curves. This weighted version of the curve evolution arises in image segmentation models, which will be demonstrated. It also satisfies a comparison principle. Our work can also be understood as looking for a numerical approximation to this flow that preserves its comparison principle.

Mateo Palomares: An Alternative Model for Card Shuffling

Mentor: Mark Conger

The Gilbert-Shannon-Reeds model for the riffle shuffle has been extensively analyzed since its introduction in 1955. Sharp bounds on the mixing time of the model and methods to calculate exact variation distance to uniform have been proven. In this talk, we explore a modified model of shuffling that accounts for differences in the skill of a shuffler by introducing a parameter that describes the neatness of a shuffle. We ask which value for this parameter results in the fastest mixing rate. Then, we discuss techniques that are used to analyze the mixing rate, and interesting patterns that emerge in our findings.

Bo Gao, Ari Krishna, Sargam Mondal, and Chunye Yang: Phantom Categories for Hirzebruch Surfaces

Mentor: Shengxuan Liu

In algebraic geometry, within the context of derived categories, Bondal and Kuznetzov conjectured that there is no nontrivial smooth projective variety X over the complex numbers that admits a phantom, which is an admissible subcategory with certain vanishing numerical invariants. In 2023, Krah demonstrated the first example of a rational surface with a phantom, which was a blowup of the projective plane. We will display our work towards showing that the Hirzebruch surface F_3 admits a phantom using Kuznetsov's techniques, which conjecturally extends to all Hirzebruch surfaces. This will provide more evidence of the non-uniqueness of semiorthogonal decompositions, the crucial challenge to using derived categories as a tool to study the rationality of algebraic varieties.