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## Statistics of saddle connections on square-tiled surfaces

This is a continuation of the “Fine scale statistics of straight lines on flat surfaces” project from Spring 2023. In this project, we examined gap distributions of square-tiled surfaces by looking at an equivalent problem using Poincaré sections. We found “winning saddle connections” at certain points in our Poincaré sections and with these winners, we were able to produce some graphs that differed from Hall’s distribution. The goal for this project in the future is to figure out how to produce Poincaré sections and graphs for all cusps of an STS and determine what characteristics of a surface result in a different gap distribution.

**Project Mentor: **Caglar Uyanik

**Graduate Student: **Anna Pham

**Undergraduate Team Members: **Michael Beers, Hannah Sell, Willow Shi, Benjamin Stroeher, Yixuan Wu

## Geometry of Diestel-Leader graphs

The goal of this project is to understand the interaction between geometry and probability on Diestel-Leader graphs. These are infinite graphs that arise in group theory (they are Cayley graphs of “lamplighter groups”) but have also been well studied from a probabilistic point of view.

The ultimate goal will be to try to understand the geometry of Voronoi cells in these graphs. Voronoi cells have been well studied in R^n where they have many real-world applications but Diestel-Leader graphs are exotic objects primarily of interest to pure mathematicians. Therefore, the motivation behind this project is purely mathematical. We will spend time visualizing Diestel-Leader graphs, understanding the geometry of shortest distance paths, considering the shape of balls and finally simulating Voronoi cells. To get a glimpse of the objects we’ll be looking at google Diestel-Leader graph or Voronoi cells and check out the images.

**Project Mentor: **Tullia Dymarz

**Graduate Student:** Becky Eastham

**Undergraduate Team Members:** Hajar Bendada, Tianyang Huang, Yicen Yang, Caroline Zhang

## Slow k-NIM

We wish to generalize a strategy for the three-dimensional slow k-nim games. Every game starts with a certain number of heaps and stones. During the game, the players take turns removing a specified amount of stones from one or more heaps. Each game has a set of rules that will indicate how many stones the players get to remove each turn. The player that removes the last stone will win the game. The P-position indicates that the previous person has a winning strategy, whereas the N-position indicates that the next person has a winning strategy. For example, the position in which there are no stones left in either heap is always a P-position. We started by understanding one and two-dimensional games, then moved to the three-dimensional game. The dimension of the model corresponds to the number of heaps used in the game.

**Project Mentor: **Mikhail Ivanov

**Graduate Student:** Ian Seong

**Undergraduate Team Members:** Irene Bian, Morgan Kosch, Ayushi Saigal, Jessica So, Kylin Sun

## Entropy of polynomials on finite rings and fields

For this project, we 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. Our group 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. The following image depicts a graph of thousands of eigenvalues, which allowed us to find an upper bound

for the leading eigenvalues based on the conditions of the matrix we are looking at.

**Project Mentor: **Chenxi Wu

**Graduate Student: **Ivan Aidun

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

## Numerical Approaches for Gradient Flow Problems

We present an overview and a numerical scheme for the Cahn-

Hilliard-Navier-Stokes system by using both the Scalar Aux-

iliary Variable method along with a projection method. We

began with a first-order scheme with plans to move to a

second-order scheme along with obtaining simulation results

in the future.

**Project Mentor: **Yukun Yue

**Graduate Student: **Sanchita Chakraborty

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

## 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 are in the process of developing a conjecture on the existence of certain subgraphs (called minors) that would give rise to non-planarity of the orbit graphs.

[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:** Ron Yang

**Undergraduate Team Members:** David Jiang, Shu-Han Liao, Zhenliang Liu

## Machine learning for example generation in pure mathematics

There’s a huge amount of current interest in the application of machine learning techniques to pure mathematics. In this MXM, we will experiment with using ML methods to generate interesting mathematical *examples*. Two notable recent papers we’ll use as models are the paper from DeepMind about finding decompositions of tensors related to algorithms for matrix multiplication:

https://www.deepmind.com/blog/discovering-novel-algorithms-with-alphatensor

and Adam Zsolt Wagner’s work on finding counterexamples in graph theory using reinforcement learning.

https://arxiv.org/abs/2104.14516

What kind of mathematical examples we decide to look for will depend to some extent on the mathematical interests and experience of the participants, but one strong possibility is that we’ll build on the DeepMind work in multilinear algebra and look for low-rank decompositions of other natural tensors arising in algebra and combinatorics.

**Project Mentor: **Jordan Ellenberg

**Graduate Student Mentor: **Karan Srivastava

**Undergraduate Team Members:** Sophia Cohen, Donald Conway, Jimmy Vineyard, Alex Yun

## Tangled Labelings on Posets

In this project, we studied the promotion algorithm, which can be viewed as a sorting algorithm for partially ordered sets, or posets. Posets can be visualized as Hasse diagrams, where vertices that are “above” other vertices are > those vertices in the partial order.

A labeling of an n-element poset is an assignment of the numbers 1,2,…n to the elements in the poset, or equivalently to each vertex on the Hasse diagram. A “sorted” or “natural” labeling is one such that smaller labels (numbers) are below (in the Hasse diagram) larger labels.

A single iteration of the promotion algorithm “sorts” at least one label. Hence, for a n-element poset, it takes at most n-1 applications of the promotion algorithm to reach a natural labeling, since it is impossible to just have one “unsorted” label.

We studied tangled labelings, which require the most iterations to become natural. The image shows a tangled labeling (top left) on an 8-element poset, with 7 applications of promotion yielding a natural labeling (bottom right).

**Project Mentor:** Paul Terwilliger

**Graduate Student Mentor: **Owen Goff

**Undergraduate Team Members: **Lucas Allen, Zachary George, Han Liu, Lejun Xu

## Aperiodic monotiles

Our project originated from an investigation into the recently discovered aperiodic monotile known as the “hat.” As our scope expanded, we ventured into the realm of Heesch numbers, which represent the count of times a shape can be encircled by complete layers of identical copies of itself. Subsequently, we extended our research to delve into the Heesch numbers within the context of polykites. Our project culminated in the development of visualization software designed to provide insights into the relationship between Heesch numbers and polykites. As a result, we were able to identify non-trivial Heesch numbers for certain polykites.

**Project Mentor:** Albert Ai

**Graduate Student:** Robert Argus

**Undergraduate Team Members: **Gulinazi Julati, Sameer Narendran, Lucas Ortengren, Nathan Sullivan

## Numerical Exploration of Stochastic Compartment Models

Our project focuses on compartment dynamics, which consist of a network of compartments and i.i.d. chemical reaction networks inside each compartment. In this project, we utilized a modified version of Gillespie’s algorithm to simulate and make conjectures about the behavior of this compartment model. At the end of our project, we try to use the different mathematical representations for the general models to develop new methods for parametric sensitivity analysis.

The graphics below are our simulation results for the compartment model with parameters chosen as they fall in an area where the long-term behavior of the Markov chain is unknown.

**Project Mentor:** David Anderson

**Graduate Student Mentor: **Aidan Howells

**Undergraduate Team Members:** Carina Guo, Olivia Guo, Leo Shen, Yikai Zhang

## Quantitative uniqueness results in complex analysis

Last semester, we developed a quantitative version of Schwarz lemma. This semester, we worked on more generalizations, where we consider estimating how “close” a function is to some more complicated functions than the identity map, with information of values and derivatives of the function at points other than the origin. To do so, we need to come up with some notion of “distance” on jets. We have done this for 1-jets, and the “distance” is invariant under Mobius transformation. Currently, we are working on proving the conjecture in the image.

**Project Mentor:** Andrew Zimmer

**Graduate Student: **Aleksander Skenderi

**Undergraduate Team Members:** Yixuan Hu, Fanchen Meng