Fall 2024

We are interested in numerically computing a type of measure that arises in hyperbolic spaces known as a Patterson-Sullivan measure. Given a group acting on hyperbolic space and the corresponding orbit of some basepoint, certain points on the boundary become “accumulation points” of this orbit. The Patterson-Sullivan measure quantifies the proportion of the boundary that is made up of these accumulation points.

Some groups under consideration were free matrix groups and Coxeter groups. We examined two different methods of finding the measure, at first taking a direct combinatorial approach and afterwards using an algorithm which utilizes the eigenvalues of a “transition matrix”.

Faculty Mentor: Feng Zhu

Graduate Student: Ron Yang

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

The goal for this project was to find a new gap distribution for square-tiled surfaces that differed from Hall’s distribution. We developed code in Python and Mathematica to codify a process that found winning saddle connections, produced Poincare sections and yielded an overall gap distribution. Focusing on non-visibility tori, we believe we have found a subset of shapes with a new desired gap distribution that is multi-modal and presents multiple new points of non-differentiability not present in Hall’s distribution.

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

Hyperbolic geometry is a type of geometry in which the surface has constant negative curvature. It is occasionally seen in nature, such as in kale leaves, and is an example of when the rules made for Euclidean geometry are not followed. As a team, we crocheted shapes with this characteristic. In our final model, we demonstrated how rules such as the parallel postulate, angle sums, and circles look different on a hyperbolic surface. Additionally, we studied two models for hyperbolic geometry, the upper half-plane and Poincare disk model, and why these are good models for drawing hyperbolic shapes. Once we understood the foundation, we decided to look into how hyperbolic geometry can be used to model Farey fractions and continued fraction approximations in addition to a tessellation of a hyperbolic shape. A tessellation was also stitched into the final shape as a visualization of tiling hyperbolic space.

Faculty Mentor: Grace Work

Graduate Student: Allison Byars

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

The Kayles Game is a combinatorial two-player game where players alternately knock down one or two adjacent pins from a sequence of bowling pins. The objective is to make it impossible for the opponent to make a legal move. The game can be thought of as a graph made up of paths, where each player removes either (i) an edge, or (ii) a vertex of positive degree and all the edges attached to it. We used a certain function called the Grundy function to describe the nature of the games. We developed a Python program to compute the Grundy values of arbitrary graphs. Using this program, we were able to search for patterns in various families of graphs, such as paths, complete graphs, complete bipartite graphs, and variations of a path.

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

We searched for 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, with 1 for each direction, is mostly disjoint. We worked out a proof that if the tubes are placed randomly, then the conjecture is true. We then tried to search the parameter space using methods other than random sampling, to find counterexamples or evidence for the conjecture. A greedy algorithm approach found the well-known ball example; where all tubes pass through a given point. This example shows that the value of p in the Kakeya maximal inequality is at best 3/2. Our reinforcement learning approach was able to find the “SL2 hairbrush example’’, which was known but only found (by hand) in 2023 (see picture). This example shows that value of p in the Kakeya maximal inequality is at best 3/2, even in the SL2 case (prior to this example it was believed to be p=2 in the SL2 case).

Faculty Mentor: Terence Harris

Graduate Student: Kaiyi Huang

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

In this project we looked to design a neural network that could assist in solving partial differential equations (PDEs). We focused on the Vlasov-Poisson equation which is used in plasma physics. The classical solvers of Vlasov-Poisson use the Semi-Lagrangian Method which involves interpolating a function using cubic splines which is the computational bottleneck of the process. We implemented a DeepONet model architecture which could compute faster than spline interpolation and had accuracy comparable to the Semi-Lagrangian Method. In the future, we would like to extend our networks to interpolate higher dimensional functions which has the potential to greatly improve the efficiency of numerically solving PDEs that use the Semi-Lagrangian Method like the Vlasov-Poisson equation.

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

This project focuses on studying “rank 1.5” affine invariant submanifolds, a concept equivalent to cylinder rigid rank 1 rel 1. These submanifolds are closely tied to the action of GL(2, R) on the moduli space of translation surfaces. Our project specifically examines genus 2 translation surfaces, classifying rank 1.5 affine invariant submanifolds in strata of the form H(2, 0^k) and H(1, 1, 0^k).

Using tools from flat geometry and supported by Gröbner basis computations, we have established the following results:

For k < 7, all rank 1.5 affine invariant submanifolds in H(2, 0^k) are trivial.

For k < 3, only one exceptional case arises in H(1, 1, 0^k).

Illustrative examples of these findings are shown in the accompanying figures.

One key motivation for studying these objects is their role in understanding higher genus cases, and the rank 1.5 affine invariant submanifolds frequently appear as degenerations in the GL(2, R) orbit closures of higher genus translation surfaces. These findings provide valuable insights for classifying orbit closures in higher genus, and we plan to explore this topic further. We hope that this work will contribute to a deeper understanding of orbit closure classifications in the future.

Faculty Mentor: Paul Apisa

Graduate Student: Ruocheng Yang

Undergraduate Team Member: Pramana Saldin