Important dates for Fall 2022 MXM participants are listed below:

**Kickoff Meeting**: Tuesday, Sept 12, 2022, 5:00pm. Van Vleck 911**Midsemester Meeting:**Tuesday, Nov 1, 2022, 5:00pm. Van Vleck 911**Open House:**Tuesday, Dec 13, 2022, 2:00-5:00pm. Van Vleck 911

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## Equilibrium distribution of points on metric graphs

Any connected metric graph can be seen as a resistor network, hence one can define the concept of “effective resistance” and the question is to investigate the n points on a given graph that maximizes the sum of pairwise effective resistance. In last semester’s MXM project we have implemented algorithms for finding those points. We will experiment with the code from that project to investigate the following questions:

(1) How would the points vary when we change the edge length?

(2) Is the optimal sum of effective resistance convex or concave?

(3) Are there any good algebraic or geometric interpretation of these points?

**Faculty Mentor: **Chenxi Wu

**Graduate Student: **Andrew Krenz

**Undergraduate Team Members:** Shashank Bangalore, Yijie Shi, Charles Wang

## Numerical study of coupled tilings

The study of random tilings contains a rich interplay between probability, combinatorics, and physics. A important example is that of counting the number of ways to tile a region of a checkerboard with dominos that cover two adjacent squares. Of particular interest is the Aztec diamond, a diamond shaped region cut out of the checkerboard. The number of tilings of the Aztec diamond was computed by N. Elkies, G. Kuperberg, M. Larsen, and J. Propp in 1992. It was later shown that large Aztec diamonds exhibited the limit shape phenomenon: with high probability a tiling chosen uniformly at random from all possible tilings would be `close’ to a limiting tiling. This limiting tiling has distinct geometric features. Further work studied the fluctuations around this limiting shape. These are believed to be universal.

Recently, S. Corteel, A. Gitlin, and D. Keating described a way to introduce an interaction between a pair of domino tilings. Numerical studies displayed interesting limit shape behavior in the coupled tilings. While some results were proven for certain limits of the interaction strength, little is known in general.

The goal of this project is to introduce undergraduates to the world of random tilings. They will learn about domino tilings of the Aztec diamond, limit shapes, and universal fluctuations. They will observe the tilings through numerical simulations. After spending some time on the numerical study of single tilings of the Aztec Diamond, we will move on to coupled tilings. Possible work includes numerically studying fluctuations with the coupling introduced by CGK or studying limit shape formation under different choices of coupling.

**Faculty Mentor: **David Keating

**Graduate Student:** Xiang Li

**Undergraduate Team Members:** Braeden Bertz, Harsha Kenchareddy, Stephen Wei, Ying Zheng

## Describing a hyperbolic surface: from lengths and twists to matrices

This project serves as an introduction to the geometry of the hyperbolic plane, as well as the link between this geometry and the algebra of 2-by-2 matrices.

Most closed surfaces admit hyperbolic metrics. In fact, they often admit lots of hyperbolic metrics, and we can describe each of these metrics geometrically in terms of a finite set of numbers. Half of these numbers are lengths of simple closed curves on the surface, and half of these numbers are “twist” parameters which (loosely speaking) describe how the metric behaves as one crosses one of these curves.

A different way of encoding one of these metrics is to consider the hyperbolic surface as the quotient of the hyperbolic plane by a discrete group of isometries. This gives us a subgroup of the isometry group of the hyperbolic plane (which can be thought of as a group of 2-by-2 matrices) associated to the hyperbolic metric on the surface. One can now ask: given the length and twist parameters for a given hyperbolic metric, can we compute the subgroup of hyperbolic isometries we get?

**Faculty Mentor:** Feng Zhu

**Graduate Student:** Aleksander Skenderi

**Undergraduate Team Members:** Kai Huang, Tim Huang, Billy Li, Jack Westbrook

## Causality based data-driven model identification and dynamical interpolation

In this project, we propose to use a recently developed model identification tool to learn the underlying dynamics of certain observational data from the incomplete observations. Causality based data-driven technique will be utilized to facilitate the model identification and parameter estimation that automatically includes the physical information. The data will either be the sea surface temperature or the atmospheric wind, where the missing observations are due to the intermittent cloud cover that blurs the satellite images. Then the identified stochastic model will be utilized to do a dynamical interpolation that recovers the missing observations. The result will provide a complete data set that can be utilized to study the physical and statistical properties.

The student(s) will be given a chance to learn several applied mathematical tools and earn hand-on coding experience.

**Faculty Mentor: **Nan Chen

**Graduate Student: **Yinling Zhang

**Undergraduate Team Members:** Noah Blum, Jack Maloney, Hanzhang Mao, Zihong Xu

## Computational Geometry: Carpenter's Rule Problem (and more!)

The Carpenter’s Rule problem asks if any polygon or any arc composed of line segments can be straightened to a convex polygon or line, respectively. During this process, the polygon should not be allowed to cross itself, and the lengths of edges must stay fixed. This problem was solved by Connelly, Demaine, and Rote in 2003 by turning this question into an optimization problem very similar to linear programming.

Our goal in this project is to find numerical evidence of a universal “speed” limit, which, under suitable assumptions and normalizations, depends only on something called the chord-arc constant of the curve, and further study the relationship between these two quantities. Along the way, we will learn some interesting theory and background, try to find good animations of this process, and explore other natural questions that arise.

Time permitting, we will explore other areas of computational geometry related to the Carpenter’s rule problem, including hinged dissections, origami folding, efficient methods for calculating the chord-arc constant for a curve, and anything else we find interesting!

**Faculty Mentor: **Jack Burkart

**Undergraduate Team Members:** Caroline Cheng, George Ekman, Harry Hu, Carrie Lu

## Identifiable linear compartmental models

In biology, pharmacokinetics, and various other fields, linear compartmental models are used to represent the transfer of substances between compartments in a system, e.g. the interchange of a drug between organs in the body or the progression of a toxin through an environment. To fully understand the system, one should know the rates at which the substance moves between different sites, but it is often difficult to measure these rates in practice. Therefore, it is important to know if the rate parameters are recoverable only from input and output data. If this is the case, we call the model *identifiable*.

The question of which models are identifiable is yet unanswered, even for models based on linear differential equations. In this case, the model can be represented by a directed graph, as pictured below. This graphical representation adds algebraic and graph theoretic techniques to the tools available to answer the question of model identifiability. The aim of this project is to familiarize students with linear compartmental models, their applications and their characteristics, and to experiment with computing the identifiability of such models, with an aim toward predicting identifiability directly from the graph structure.

**Faculty Mentor:** Aleksandra Sobieska

**Graduate Student:** John Cobb

**Undergraduate Team Members: **Nuha Dolby, Nicole Miller, Seungyeon Oh, Uma Parhar

## Random symmetries of hyperbolic spaces

Hyperbolic plane is an example of a non-Euclidean geometry, and it plays an important role in various branches of mathematics. A Gromov hyperbolic space is a generalization of both the hyperbolic plane and a tree (as in graph theory), and has far reaching implications for a space. This project will explore the following question: Given a subgroup of the isometry group of a hyperbolic space, what do two-generated random subgroups look like?

This project is a continuation of the project “Random Integer Matrices” from Spring 2022. Students who have a background in group theory, metric spaces, and complex analysis are encouraged to apply.

**Faculty Mentor: **Caglar Uyanik

**Graduate Student: **Yandi Wu

**Undergraduate Team Members:** Erkin Delic, Yinong Huang, Holden Swindell, Kasey White

## Averages of polynomials of entries of a random orthogonal matrix

This is a continuation of a similar project from the Spring 2022 semester. The goal is to find formulas for mixed moments of entries of uniformly chosen orthogonal matrices, both experimentally and with rigorous proofs. The team in the spring found some promising results for certain special cases, and I hope that the team in the fall can build on that.

**Faculty Mentor:** Benedek Valko

**Graduate Student:** Jingyun Jia

**Undergraduate Team Members: **David Broga, Jiashu Qu, Ran Tao, Yirun Wang

**Undergraduate Student Requirements: **This is an advanced project suitable for students who have taken multiple upper-level mathematics courses. Students should have taken probability (Math 309, 431 or 531), linear algebra, and at least one proof based class would be important. Math 521/522 and 632 would be useful as well.

## Numerical exploration of singularities in curve shortening

This project continues the theoretical exploration of the topology of singularities in Curve Shortening that was the subject of an MXM project in spring 2022 (https://mxm.math.wisc.edu/past-semesters/sp22/) . The students working in that project indentified a number of possible singularity models. The goal in the current project is to use numerical simulation to

1) exhibit the singularities that were previously suggested, in particular the “n-loop” solutions

2) determine their asymptotic shape (blow-up rate for curvature, which figure appears after rescaling the curves bounding box to a unit square? for the two loop curve it is know the limit shape is a bowtie–what about the 3-loop or higher?)

Unlike the previous MXM project on Curve shortening this project will involve a lot of numerical experimentation, so this is for students who have some familiarity with python/scipy or matlab and who enjoy programming.

**Faculty Mentor:** Sigurd Angenent

**Graduate Student:** Devanshi Merchant

**Undergraduate Team Members:** Ellie DeCleene, Paige Ellingson, Ziheng Feng, Yamin Zhou

## An SIR model of Adderall use disorder in college students

The ordinary differential equations (ODE) models known by the term SIR are commonly used to model infectious disease epidemics, but have also been used to model phenomena as diverse as the spread of ideas on social media and the opioid epidemic. We propose their first application to Adderall substance use among college students. Adderall is reputed to provide increased focus and concentration when used as a study aid by neurotypical students, despite a lack of scientific evidence to support any claim that it increases focus or concentration outside the intended use community of individuals with ADHD/ADD.

Our project has three goals. First we will develop and analyze an SIR-type model. This includes a review of the scientific literature concerning Adderall use disorder among college students, developing a mechanistic ODE model incorporating the current science, and standard equilibrium and stability analysis. Second we will use Python and/or Matlab to fit parameters for our model to data on Adderall use among college students. We will interpret the resulting best-fit parameter values in terms of the biological and sociological meanings of those parameters which we determined during model development, and determine whether the values we have identified align or do not align with current theoretical understanding of Adderall use disorder. Third, we will perform optimal control analysis to determine which interventions in which combinations would result in the biggest gains to our most desired outcomes with given resource limitations (cost, staffing, etc.).

**Faculty Mentor: **Skylar Grey

**Graduate Student: **Sanchita Chakraborty

**Undergraduate Team Members:** Sara Murley, Lilah Tascheter, Eline van Ophem, Chang Yuan

## Shedding light on unexplained behavior of phylogeny estimation methods (continued)

The estimation of evolutionary trees, or “phylogenies”, from genomic data has many important biomedical applications, e.g., elucidating the origins of the COVID-19 pandemic or identifying new variants of concern of the SARS-CoV-2 virus.

Despite extensive work in this area over many decades, much remains to be understood about the theoretical properties of standard estimation methods such as maximum likelihood — even the simplest case of four tips, or “quartet trees”, where the problem reduces to the constrained optimization of a low-degree polynomial.

In this (continued) project, we will explore unresolved mathematical conjectures on the intriguing behavior of maximum likelihood on quartet trees in various asymptotic regimes through a combination of analytical calculations, numerical computations and stochastic simulations. The ultimate goal is to provide insights into a formal proof of these conjectures.

**Faculty Mentor: **Sebastien Roch

**Graduate Student: **Max Bacharach

**Undergraduate Team Members:** Achutha Balaji, Yuheng Cai, Laura Huang, Megan Lolling, Mengwei Sun

## Geometry of horospheres in graphs

The goal of this project is to demonstrate the use of algorithms to solve geometric problems in Cayley Graphs. Students will learn about and implement word reduction algorithms in various groups and use them to draw pictures of spheres, and balls. The ultimate goal is to implement an algorithm to draw pieces of horospheres in hyperbolic Right-Angled Coxeter Groups.

This project will be relatively theory-light and application-heavy. It will start with the problem of pathfinding in particular highly-symmetric graphs. From there, we will develop and implement general algorithms in classes of graphs to calculate distances between points. Students will need some coding experience and familiarity with basic graph theory, but not Abstract Algebra. The students will have the option to delve into the group theoretic background if they so choose, especially if they have had Math 541.

**Faculty Mentor: **Tullia Dymarz

**Graduate Student:** Daniel Levitin

**Undergraduate Team Members:** Noah Jillson, Katerina Stuopis, Erica Wang, Kaicheng Xue

## How do they synchronize?: Interpretable feature learning for networked coupled oscillators

**Faculty Mentor: **Hanbaek Yu

**Undergraduate Team Members: **Binhao Che, Agam Goyal, Bella Qu, Bryan Xu