Fall 2024

Given a discrete group of symmetries acting on a negatively-curved space X, there is a class of measures (“Patterson–Sullivan measures”) on the boundary of X which reflects the data of that group action. The objective of this somewhat exploratory project will be to compute and visualize some Patterson–Sullivan measures.

We will concentrate on actions of free groups, surface groups, or Coxeter groups on the hyperbolic plane or on convex domains in the real projective plane. In this case, the boundary of X, where the Patterson–Sullivan measures live, is homeomorphic to a circle, and there are existing Python packages that can be used.

Faculty Mentor: Feng Zhu

Graduate Student: Ron Yang

Undergraduate Team Members: Wil Cram, Judy Li, Ruiqi Li, Kayley Seow

This project is about distribution of vectors coming from translation surfaces which are shapes that locally look like R^2, outside of finitely many cone points. In previous semesters, we have done lots of experimental work to generate conjectures, and now we are in a position to prove some theorems and generate more conjectures! We are looking for 2-3 undergraduate students who are proficient in python to improve the current code, and finish the project. The advanced mathematical background can be “blackboxed” so that one can only focus on computing, linear algebra, and multivariable calculus.

Faculty Mentor: Caglar Uyanik

Graduate Student: Gabriela Brown

Undergraduate Team Members: Michael Beers, Tanisha Raaj, Xingke Sun

My collaborators and I recently figured out an algorithm for the calculation of the inverse of the Artin-Mazur zeta function for 1d rational maps on locally compact non Archimedean fields. We are interested in implementing it as well as optimizing it to cut down the time complexity.

Faculty Mentor: Chenxi Wu

Graduate Student: Eiki Nourizuki

Undergraduate Team Members: Pritam Kayal, Elijah Schwab, Yuwen Yang, Minghao Yin

This project is exploratory in nature, its main goal is to understand and demonstrate the unique properties of hyperbolic geometry through the medium of crochet. Hyperbolic geometry, unlike Euclidean geometry, is defined by its constant negative curvature, leading to interesting shapes akin to lettuce leaves. As we crochet these models other avenues of investigation will be explored depending on the interest of the team members.

Faculty Mentor: Grace Work

Graduate Student: Allison Byars

Undergraduate Team Members: Emi Bell, Tyson Franke, Alyson Geisler, Jackson Grabowski

The project explores a periodicity of Kayles game on graphs.

We will start with computational aspects of classical Kayles game. Kayles is a simple impartial game played with rows of pins. Each player, on his or her turn, may remove either any one pin, or two adjacent pins. This game is periodic, which can be shown only after computations, which we will provide.

On the second part we will extend our tools to the game on some graphs’ families. If we think of the classical rows of pins as path graphs, then we can have game on stars, caterpillars, general trees, etc.

Faculty Mentor: Mikhail Ivanov

Graduate Student: Ian Seong

Undergraduate Team Members: John Fitzpatrick, Jason Kolbly, Shutong Ni, Chenxi Xia

The goal of this project is to develop a framework for network-level regression, two make sense of approximating an unknown network X by a “linear combination” of a set of known networks for assessing structural similarity and application in graph classification problems. During summer REU 2024, we have developed a method for network regression by combining k-path subgraph sampling and supervised matrix factorization. We would like to continue developing this method and benchmark its performance on graph classification problems with popular graph neural network methods.

Faculty Mentor: Hanbaek Lyu

Undergraduate Team Members: David Jiang, Phoebe Kuang, Yi Wei

The aim of the project is to find some numerical evidence for the Kakeya maximal function conjecture in 3 dimensions. This conjecture roughly states that a collection of tubes in 3-space of equal radii and separated directions is mostly disjoint. It is folklore that if the placement of each tube is random (uniform), then the conjecture is true, so we will first aim to verify this numerically (and possibly theoretically). Then, we will try to find a better way to search parameter space for counterexamples to test whether 3/2 is the correct exponent. We may also try to develop some of these approaches in 2 dimensions first, where the conjecture is known.

Faculty Mentor: Terence Harris

Graduate Student: Kaiyi Huang

Undergraduate Team Members: Tianle Chen, Daniel Alexander David, Vivek Saravanan, Marissa Stolt

This project explores innovative approaches to solving partial differential equations (PDEs) using neural network-based methods. PDEs are fundamental in describing various phenomena in physics, engineering, and other fields, but traditional analytical and numerical solutions can be complex and computationally intensive. Neural networks, with their ability to approximate complex functions, offer a promising alternative for solving PDEs more efficiently.

The primary goal of this project is to develop and implement neural network models to solve different types of PDEs, such as heat equations, wave equations, and fluid dynamics equations. Students will learn how to design, train, and evaluate these models, comparing their performance with traditional methods. By the end of the program, participants will gain hands-on experience in applying machine learning techniques to classical mathematical problems, preparing them for advanced research and practical applications in computational science.

Faculty Mentor: Yukun Yue

Graduate Student: Martin Guerra

Undergraduate Team Members: Lucas Allen, Sophia Cohen, Ethan Hanhold, Alexis Liu, Zhiqian Xu

A translation surface is a collection of polygons in the planes with sides identified by translation to form a surface. Translation surfaces are geometric objects that inherit the notion of a straight line from the plane. When straight lines close up, they form curves that exist in families called cylinders. Some translation surfaces, for instance flat tori, have the property that every cylinder completely covers the surface. Indeed, Maryam Mirzakhani and Alex Wright showed that flat tori are the only examples of translation surfaces with this property. However, curiously, Giovanni Forni, Carlos Matheus, Jean-Christophe Yoccoz, Paul Apisa, and Alex Wright (in various combinations) showed that the situation is fundamentally different for half-translation surfaces, i.e. polygons in the plane with sides identified by translation or translation composed with rotation by 180 degrees. These authors produced infinitely many half-translation surfaces with the property that every cylinder completely covers the surface (call them one-cylinder surfaces). Nonetheless, their examples were again special in that they were all cyclic covers of the sphere branched over four points. This project will investigate a conjecture of Apisa and Wright which says that every one-cylinder half-translation surface is a cyclic cover of the sphere branched over four points.

Faculty Mentor: Paul Apisa

Graduate Student: Gabriela Brown

Undergraduate Team Member: Malak Abdalla

The moduli space of translation surfaces admits a GL(2, R) action. Work of Eskin, Mirzakhani, and Mohammadi shows that given a translation surface, its GL(2, R) orbit has a closure which is defined by linear equations. A major open problem is understanding these orbit closures. In low genus, classifying these orbit closures often reduces to understanding simpler ones, called cylinder rigid orbit closures. These have been classified in genus one. But their classification in genus two is wide open. This project hopes to make progress on this problem.

Faculty Mentor: Paul Apisa

Graduate Student: Ruocheng Yang

Undergraduate Team Member: Pramana Saldin