Project descriptions and team members for Spring 2024 can be found below. The Spring 2024 MXM Open House will take place on Tuesday, April 30, 2024 from 2-5pm in Van Vleck 911.

This is an accordion element with a series of buttons that open and close related content panels.

### Frequency cascades for the continuum Calogero–Moser equation

The continuum Calogero–Moser equation is a partial differential equation that arises in fluid dynamics. One main application of this system is to describe a hydrodynamic limit of a particle gas interacting pairwise through an inverse square potential.

In 2022, a family of exact solutions to this equation were discovered that exhibit a frequency cascade. Such behavior is a key characteristic of turbulence; this is a topic in fluid dynamics that has been heavily studied over the past half century, but can be quite complicated to describe in general. The discovery of this explicit example of a frequency cascade poses an exciting opportunity to understand more about turbulence.

Nevertheless, there is still much to be understood about this phenomenon: What do these exact solutions look like? Does similar behavior occur for other initial conditions? Is this the most rapid frequency cascade that can occur, or could there be solutions that blow up in finite time? This project seeks to investigate these questions by numerically modeling solutions using MATLAB.

**Project Mentor: **Thierry Laurens

**Graduate Student Mentor: **Xiang Li

**Undergraduate Team Members:** Megan Case, Morgan Kosch, Nancy Liu, Mridini Thippisetty

### Hyperbolic Geometry Through Crochet

What happens when you break the “rules” of geometry? What if the interior angles of a triangle did not have to add up to 180 deg? Breaking the rules of Euclidean geometry leads to new geometries, including hyperbolic geometry. We will explore this geometry through the medium of crochet with the goal of producing physical objects to illustrate these ideas and concepts and enhance understanding. We will be using Daina Taimina’s book *Crocheting Adventures with Hyperbolic Planes: Tactile Mathematics, Art and Craft for all to Explore* for inspiration.

**Project Mentor: **Grace Work

**Graduate Student Mentor:** Yandi Wu

**Undergraduate Team Members: **Isabelle Bubnick, Hetvi Pethad, Sophia Reiner, Maritza Santiago-Martinez

### Geometry and geology

This will be an interdisciplinary project to examine the geometry of nannofossils, the fossilized remains of single-celled marine phytoplankton. Due to their rapid evolution over time, the fossils of these nannoplankton provide an essential tool in determining the ages of rocks and can also provide paleoenvironmental data to help understand climate change.

Each species of nannoplankton can be identified by the arrangement and geometry of the calcareous plates (coccoliths) the single-celled organism builds around itself. For this project, we would learn about these nannoplankton while studying the geometry of their forms, looking for geometric patterns in how species evolved over time. One goal would be to understand if geometry/symmetry can help give us information about why certain evolutionary branches might have ended in extinction.**Project Mentor: **Nate Fisher

**Graduate Student Mentor: **Becky Eastham

**Undergraduate Team Members: **Hajar Bendada, Haley Czerniejewski, Haakon Gulbrandsen, Leo Shen, Katerina Stuopis

### The Ups and Downs of Convection

Convection refers to the spontaneous motion of a fluid in the presence of a temperature gradient and gravity. Heat transfer by convection is familiar in weather patterns by hot air rising and cold air sinking, and more generally, convection plays an important role in stars and planetary atmospheres.

Natural convection has a research history of 100 years, and yet, there are few known exact solutions to the partial differential equations that determine the associated winds and temperature fields. Some steady solutions have been computed numerically for fluid between two flat boundaries, heated from below and cooled from above, in a configuration driven by boundary forcing.

In this MXM project, students will find solutions characterizing radiatively driven convection in planetary atmospheres and stars, that is, convection forced by internal heating rather than by boundary heating. Such solutions will be found using the freely available software Dedalus (https://dedalus-project.org/) and interpreted using mathematical and statistical analyses. The new solutions will help to elucidate the distinguishing features of updrafts and downdrafts in planetary and solar convection.

**Project Mentor: **Leslie Smith

**Graduate Student Mentor: **Varun Gudibanda

**Undergraduate Team Members: **Tianyang Huang, Haoting Ling, Haoxuan Mu, Alex Williamson Junk

### Limit distributions of random matrices

The project will study scaling limits of eigenvalues of large random matrices. The goal is to set up efficient simulations for various limit distributions. Some of these limit distributions are described via continuous time stochastic processes (stochastic differential equations), or via partial differential equations. The ultimate goal is to numerically verify (or disprove) conjectures about certain random matrix limits.

**Project Mentor: **Benedek Valko

**Graduate Student Mentor:** Yahui Qu

**Undergraduate Team Members: **Yinong Huang, Nitit Jongsawatsataporn, Elijah Schwab, Yuqi Tu

### Explaining Congruences in Character Tables

Associated to a finite group G is its character table, a square array of complex numbers which describes how G can map into groups of matrices. Suppose G is a finite group whose character table is integer-valued. If you look at the character table mod 2, some of the columns become the same because of the Freshman’s Dream identity (x+y)^2 = x^2 + y^2 mod 2. It is a theorem that this is the “only reason” why columns would be the same mod 2. However, it is less clear what happens modulo 4. We have a necessary criterion arising from the identity (x+y)^4 = (x^2 + y^2)^2 mod 4, but not all coincidences of columns mod 4 follow from this identity. For some groups, however, such as the symmetric group, this identity is enough to explain all congruences!

The project will involve:

- Learning necessary background in the representation theory of finite groups;
- Writing and executing software to search for congruences modulo prime powers in character tables of finite groups;
- Develop (and possibly prove) conjectures about behavior of congruences modulo prime powers in finite groups.

**Project Mentor: **Joshua Mundinger

**Graduate Student Mentor:** Ian Seong

**Undergraduate Team Members:** Pramod Anandarao, Ethan Ji, Sanan Sharifzade, Suyu Wang

### Mahler measures and Galois groups of reciprocal polynomials

This project concerns Galois theory and algebraic number theory. Understanding all the words in the following description is not a pre-requisite for participating.

Let f(x) be an irreducible, monic polynomial with integer coefficients. The Mahler measure of the polynomial is a real number M(f)>=1, defined as the absolute value of the product of those roots of f(x) which lie outside the unit circle in the complex plane. A classic theorem of Kronecker states (under the assumptions above) that M(f) = 1 if and only if f(x) is a cyclotomic polynomial. A famous still-open conjecture of D. H. Lehmer from 1933 is that there exists a universal constant c>1 such that if f is not cyclotomic, then M(f) >= c. It is widely conjectured that c can in fact be taken to be the single root outside the unit circle of the famous polynomial appearing in Lehmer’s original paper:

P(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1; this root is approximately 1.176.

C.J. Smyth in 1971 proved the Lehmer conjecture for all polynomials which are not reciprocal. The polynomial f(x) is reciprocal if its coefficients read the same forward as backward (as in the polynomial P above), which is equivalent to saying that alpha is a root of f if and only if 1/alpha is. Therefore, only reciprocal polynomials are of interest in studying the Lehmer conjecture. This condition of being reciprocal restricts the Galois group to be strictly smaller than the full symmetric group. The degree of the f(x) must be an even integer 2n, and the Galois group will be a subgroup of a certain permutation group of order 2^n * n!, which is a wreath product of a group of order 2 and S_n. On the other hand, an important result of Amoroso and David from 2006 shows the Lehmer conjecture holds if the Galois group is very small (say bounded by any polynomial in n). So we have existing upper and lower bounds on the Galois groups of polynomials which might be counterexamples to Lehmer’s conjecture. We would like to explore the possibility of improving either of these bounds.

The first goals of this project are:

- To learn important background material.
- Write code (probably using sage or magma) to compute the Mahler measure and Galois groups of many polynomials.
- Formulate some refined conjectures based on the data.

If we make a lot of progress on these goals, it may be possible to spend some time and effort trying to prove new results along these lines.

**Project Mentor: **Bobby Grizzard

**Graduate Student Mentor: **John Yin

**Team Members:** Hernan Aldabe, Erkin Delic, Matthew Norton, Sammy Ross

### Machine learning for algorithms in pure math

This is a continuation of a fall project, adding new students because some students are graduating or studying abroad. We are investigating the question of whether neural nets can learn interesting algorithms in pure math. In the fall, we worked on the question of how well neural nets could learn to find short representations of elements in SL_2(Z), SL_3(Z), and the Heisenberg group; in the spring, we will continue with the latter two investigations and potentially branch out to further questions.

**Project Mentor: **Jordan Ellenberg

**Graduate Student Mentor: **Karan Srivastava

**Undergraduate Team Members: **Sophia Cohen, Noah Jillson, Pramana Saldin, Jimmy Vineyard, Alex Yun

### Dynamics of p-adic polynomial maps

This is a continuation of the project in fall 2023. The goal is to investigate if it is possible to estimate entropy of p-adic polynomial maps via finite approximation.

**Project Mentor: **Chenxi Wu

**Graduate Student Mentor: **Ivan Aidun

**Undergraduate Team Members:** Andrew (AJ) Butler, Beining Mu, Kevin Williams, Allison Yusim

### Non-planarity in SL(2,Z)-orbits of origamis

This is a continuation of the project in Fall 2023.

**Project Mentor: **Luke Jeffreys

**Graduate Student Mentor:** Ron Yang

**Undergraduate Team Members:** Phoebe Kuang, Shu-Han Liao, Zhenliang Liu

### Numerical Approaches for Gradient Flow Problems

This is a continuation of the project in Fall 2023.

**Project Mentor: **Yukun Yue

**Graduate Student Mentor: **Sanchita Chakraborty

**Undergraduate Team Members:** Adam Distler, Alexis Liu, John Marek