Project descriptions and team members for Spring 2024 can be found below.

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## Frequency cascades for the continuum Calogero–Moser equation

We studied the Continuum Calogero-Moser (CCM) equation this semester to learn about the turbulence behavior in fluid dynamics. CCM is a partial derivative equation, which is a continuous version of Calogero-Moser (CM) equation where CM represents particle systems. Currently, we saw a connection between the CCM solutions associated with the 1-solition and 2-solitions of the CM particle system. In particular, we explored what the solitons look like and the interaction between each other as time passes. Generalization to n-soliton system requires understanding the connections between the CM and CCM, where the particle system solution could reveal solutions to the CCM.

**Project Mentor: **Thierry Laurens

**Graduate Student Mentor: **Xiang Li

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

## Hyperbolic Geometry Through Crochet

In this project we aimed to understand and conceptualize the rules of hyperbolic geometry. In hyperbolic geometry, the parallel postulate of Euclid is replaced with a different axiom, leading to spaces with constant negative curvature. Unlike Euclidean space, hyperbolic space exhibits a negative curvature, and locally looks like a saddle. Throughout completing this project, we explored three methods to physically model hyperbolic geometry. Our initial focus of creating crochet representations led to other creative outlets, including tiling and manipulating Earth projections. An overarching goal of conceptualizing hyperbolic models was the objective throughout these distinct approaches.

The pictures below are all representations of our applications to understand hyperbolic models and their unique boundaries. The map projection is based on area-preserving maps between the Poincaré disk and Euclidean space and helps visualize rapid increases of areas as one travels further from the origin in hyperbolic space. Tiling is a great for visualizing the fact that the total angle of a triangle in hyperbolic space is less than π radians. Finally, our crochet models help visualize the exponential increase of distances in hyperbolic space as one travels further from the origin.

**Project Mentor: **Grace Work

**Graduate Student Mentor:** Yandi Wu

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

## Metric Geometry of Stars

The main goal of our project is to understand stars of polygonal metrics in **R**². Stars characterize a notion of closeness between points on a horofunction boundary. For the polygonal metrics we work with, our goal is to show that stars recognize the geometry of the polygon we used to define the metric. We prove this relationship for the taxi-cab metric in **R**². Following this, our goal is to generalize these results. In doing so, we are able to recover results pertaining to the geometric structure of horofunction boundaries. We also give partial results on how stars reflect this geometric structure.

**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.

We compared the hot and cold temperature plumes and statistics associated with heat transfer for convection driven by boundary heating and convection driven by a heat source in the interior of the fluid. Methods included linear stability analysis, scaling analysis based on asymptotic thinking, and numerical computation using the Dedalus code https://dedalus-project.org/.

Results show that boundary dominated convection exhibits symmetry of the updrafts and downdrafts (top panel in the figure), while internally driven convection has up-down asymmetry determined by the form of the internal heating together with the boundary conditions (bottom panel of the figure). Furthermore, internally driven convection is associated with larger heat transfer across the fluid layer, as predicted by heuristic arguments.

**Project Mentor: **Leslie Smith

**Graduate Student Mentor: **Varun Gudibanda

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

## Limit distributions of random matrices

The project explored various ways to simulate the Tracy-Widom-beta distribution, which describes the limit law of the largest eigenvalue of a wide family of random matrix models. The used methods included Monte-Carlo simulation using random matrices, stochastic differential equations, and also numerical methods using partial differential equation techniques. The group also investigated whether the Tracy-Widom distribution can be recovered as the scaling limit of the largest eigenvalue of a certain block-tridiagonal random matrix model.

**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

Over this semester, we have primarily been working with signed permutation matrices, which are permutation matrices with entries ±1. Denote by B_n the group of n × n signed permutation matrices. Before we generated the character table of B_n, we first tried to generate signed permutation matrices. A significant amount of time went toward making the matrix generation algorithms more efficient with respect to run time. After we determined these efficient algorithms, the matrices were made into a Matrix Group using Sage Math’s libraries, allowing us to take the character table of the matrix group and thus B_n. To be precise, we found a way to generate the character table of B_n up to n = 9 in reasonable time. Another large task we worked on was finding which columns in the congruent module p^r , where p is a prime. This proved easier than the former task

**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

Computing the index of subgroups is a generally nontrivial task. In this project, we sought to determine the index of certain subgroups of SL_2(**Z**) using generalizable machine learning algorithms. We accomplished this task by training various machine learning models to predict the optimal subgroup generator to apply to reach identity in as few transformations as possible, given an arbitrary element of the group, then analyzing which coset representatives they traveled through on their paths back to identity.

**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

Our group explored the entropy of polynomials over finite rings and fields. Calculating entropy directly can be very difficult, so our goal was to look into other possible methods to do so. During our first semester of this project, we looked into the behavior of the leading eigenvalues of matrices with specific conditions as well as integer sequences relating to the matrices. We wrote a program that then allowed us to find and prove some trends that we observed. This semester, we focused on computing the Markov Decomposition of p-adic polynomials, which gives us a way to look at their behavior and calculate their topological entropy. The following images depict the eigenvalues of matrices generated from the Markov Decomposition of *f*(*x*) = *x*(*x*−1)² in in **Z_**3, **Z_**5,and **Z_**7, respectively. We expect that behavior of these eigenvalues will give us a good approximation of the topological entropy of *f* .

**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

A square torus can be obtained from the unit square by identifying the top and bottom sides to form a cylinder and identifying the left and right sides of the cylinder to form the torus. Generalising the construction of the square torus, an origami is a potentially higher genus surface obtained by identifying the sides (tops to bottoms and lefts to rights) of a collection of unit squares. There is a natural action of the group SL(2, **Z**) on (primitive) origamis and each SL(2, **Z**)-orbit can be turned into a finite 4-valent graph. The goal of this project was to investigate the (non-)planarity of these graphs in genus three in the hope of proving that the graphs are non-planar once the origamis are built from enough squares. This work is related to a conjecture of McMullen that such graphs in genus two, where non-planarity has recently been established [2], form a family of expander graphs.

The project used SageMath [3] and surface_dynamics [1], two python-based mathematical software systems, to investigate the structures of these graphs for certain genus three origamis. Using the data from our computational investigations, we have established a conjecture describing a family of K3,3 minors contained in an infinite family of orbit graphs. Over the summer, we intend to prove this conjecture and, in doing so, establish that all of the orbit graphs in this infinite family are non-planar.

[1] Delecroix, V. et al., surface dynamics – SageMath package, Version 0.4.1, https://doi.org/10.5281/zenodo.3237923

[2] Jeffreys, L. and Matheus, C. (2023), Non-planarity of SL(2, Z)-orbits of origamis in H(2). Bull. London Math. Soc., 55: 2258–2269.

[3] SageMath, the Sage Mathematics Software System (Version 10.2), The Sage Developers, 2023, https://www.sagemath.org.

**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

Here we further dive into simulations and analysis of our SAV scheme that was developed for the Chan-Hiliard-Navier-Stokes System using the Scalar Auxiliary Variable Method. We present the beginnings of simulations of a slightly modified decoupled scheme, and also lay the requirements for the convergence analysis.

**Project Mentor: **Yukun Yue

**Graduate Student Mentor: **Sanchita Chakraborty

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