The current projects for Spring 2022 can be found below.

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## Sports Analytics

Students will use basic techniques in machine learning to predict the outcome of college football games. Students will first build a simple “Elo-type” model and then a more complicated model using linear regression. The models will be built using the Python programming language, but no previous experience in programming is needed (if you are willing to learn!).

The goals of this project are to teach students some basic tools in machine learning and how to use the standard machine learning packages in Python.

**Faculty Mentor:** Andrew Zimmer

**Difficulty:** Beginner

**Prerequisites:** Students who have taken Math 234, some programming experience (in any language) might be helpful, but not necessary.

## Geometry of Random Integer Matrices

Groups are abstract mathematical objects that aide in the construction of many foundational subfields of physics, chemistry, and other sciences; perhaps most importantly, they are a central object of interest throughout numerous fields of mathematics, so acquiring knowledge about the structure and behavior of different groups can be very meaningful.

This project will explore subgroups of SL(2, Z)–the group of integer matrices of determinant 1 by studying their effect on various geometric objects, such as the Euclidean plane, the hyperbolic plane and the circle. In particular, we want to explore the following question: Given two random elements of SL(2,Z) what is the probability that they generate a free group? Along the way students will learn about group actions, ping-pong lemma, and hyperbolic geometry.

**Faculty Mentor: **Caglar Uyanik

**Difficulty:** Beginner

**Prerequisites:** Students should have taken Math 234 and have seen some linear algebra and the basics of group theory. Experience in programming using Python, Sage, and/or Mathematica would also be beneficial.

## Averages of Polynomials of Entries of Uniformly Chosen Orthogonal Matrices

For a fixed $n$ there is a way to choose an $n\times n$ orthogonal matrix uniformly at random. The goal of the project is to study and explore expectations of polynomials of the entries of this matrix for various values of $n$. More precisely, if $X$ is an $n\times$ uniformly chosen orthogonal matrix then we would be interested in the expectation of expressions of the form $X_{a_1,b_1}^{k_1}\cdots X_{a_m,b_m}^{k_m}$ for some choice of parameters $a_1,b_1,k_1,\dots, a_m, b_m, k_m$.

The outcomes of the project could include:

– Producing a Mathematica code that computes the expectation for a given input via simulation.

– Producing a Mathematica code that computes the expectation for a given input exactly.

– Finding patterns for certain special special forms of input parameters, and possibly proving theorems about these patterns.

**Faculty Mentor:** Benedek Valko

**Difficulty: **Intermediate

**Prerequisites: **Students should have taken Intro to Probability (309/431/531) and Linear Algebra (320/340/341/375). Experience in proofs (341/375/421/461) would be useful and Intro to Stochastic processes (632) would be a bonus. Some experience with coding would be useful. Students will also be working with Mathematica so experience using it would be beneficial, but a quick bootcamp can also be given.

## Sampling Random Lozenge Tilings

You will design a program, which produces perfect samples of random lozenge tilings of polygonal domains. For that you will learn the Markov chain Monte Carlo and Coupling from Past methods and implement them for the random lozenge tilings. The ultimate result will be a software which produces pictures like at https://people.math.wisc.edu/~vadicgor/research.html (but for more complicated domains, while this page has only simulations for hexagons)

Eventually, we want the simulations to guide and complement theoretical mathematical results about such random tilings. For general information and motivations for the research on random tilings you can browse through my book: https://people.math.wisc.edu/~vadicgor/Random_tilings.pdf

*Warning*: this is an advanced mathematical book and it takes efforts to understand the content in details. You do NOT have to understand the book in order to be successful in this project.

**Faculty Mentor:** Vadim Gorin

**Difficulty:** Intermediate

**Prerequisites:** Students should be familiar with Markov chains. Experience in programming is also expected as the ultimate goal of the project is to create software to produce visualizations and simulations.d

## Topology of Curve Shortening Singularities

Curve Shortening (CS) deforms a plane curve by its curvature. If the initial curve has self intersections then its evolution by CS can become singular, i.e. small loops can pinch off. It is unknown what kind of configurations of loops can collapse at the same time in a singular solution. There is one simple example of a pretzel-like curve in which two loops with three self intersections vanish at the same time. Its construction uses a topological argument, and the proposal is to find more examples of this kind by considering more complicated initial curves. Another goal would be to formulate reasonable conjectures for what all possible singular solutions look like.

The arguments should involve topology or real analysis at the undergraduate level. No programming will be involved, but the project should require lots of paper and (colored) pencil (erasers too).

For background see Wikipedia https://en.wikipedia.org/wiki/Curve-shortening_flow, and definitely play with the animation at https://a.carapetis.com/csf/.

**Faculty Mentor: **Sigurd Angenent

**Difficulty: **Advanced

**Prerequisites: **Students who have taken multiple upper-level mathematics courses, including Math 521 and Math 551

## Dual Graphs of Homogeneous Ideals in Polynomial Rings

The dual graph of a homogeneous height-unmixed ideal in a polynomial ring S over a field K is an interesting and powerful combinatorial invariant. Using it, one can determine that if I = (x,y) \cap (z,w) in S = K[x,y,z,w], then its dual graph G(I) is disconnected. Hartshorne’s connectedness theorem says that if I is Cohen-Macaulay, in addition, that is, S/I is a Cohen-Macaulay ring, then its dual graph G(I) is connected.

In this project, we will determine whether the converse of Hartshorne’s theorem is false! The project mentor expects that it is.

We will review the prerequisites on homogeneity in polynomial rings over a field K, symbolic powers of radical ideals, and what Stanley-Reisner ideals are. Then the proof should be doable to piece together.

**Faculty Mentor:** Robert Walker

**Difficulty:** Advanced

**Prerequisites:** Students who have taken multiple upper-level mathematics courses including Math 541, 542 and 340. Students should also have experience using Python.

## Shedding light on unexplained behavior of phylogeny estimation methods

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 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

**Difficulty:** Advanced

**Prerequisites:** Students who have taken multiple upper-level mathematics courses including Math MATH 535 or 632 strongly preferred. Students should also have experience using Python.

## Vectors with Smallest Slope

A translation surface is a 3d object obtained by gluing parallel sides of a polygon, for example gluing parallel sides of a square results in a torus (or doughnut). On each translation surface we can define a set of vectors corresponding to straight line trajectories that start at one vertex of the planar polygon and end at another, potentially passing through parallel sides and wrapping around (as in the arcade game Pac-Man). One question we can ask about this set of vectors is “How randomly are they distributed?” A measure of this randomness is the distribution of the differences between consecutive slopes. This problem has been answered in some specific cases, like the square and the octagon, and steps have been made towards answering the question in more general settings. Part of the process to answer this question requires applying specific matrix transformations to the set of vectors to determine which has the smallest positive slope and horizontal component less than or equal to 1. The goal of this project will be to implement an algorithm to determine this vector of smallest slope for a specific class of translation surfaces.

**Faculty Mentor:** Grace Work

**Difficulty:** Beginner

**Prerequisites:** Students should have taken Math 234 and have seen some linear algebra and the basics of group theory. Familiarity with programming and implementing algorithms, slight preference for Python.

## Equilibrium distribution of points on metric graphs

A problem I am working on with my collaborators is to find the equilibrium non negative measures on a metric graph, I am interested in the discrete version of it. Let G be a finite connected metric graph, n be a natural number. One can turn the graph into a resistor network, then between any two points one can measure the effective resistance. What is the optimal configuration of n points such that:

1) The sum of all pairwise distances is maximized?

2) The sum of all pairwise “effective resistance” is maximized?

**Faculty Mentor:** Chenxi Wu

**Difficulty:** Beginner

**Prerequisites:** Students should have taken Math 234 and have seen some linear algebra. Familiarity with programming and implementing algorithms in any language.

## Random Walks on Groups

A random walk is a type of Markov chain, a random process, which gives us a way to explore a space in a random way. A simple example is to start at 0 on the number line, and then with equal probability, either take a step to the left (to -1) or to the right (to +1). Now we repeat the process and take a step either to the left or to the right, possibly returning back to where we started or getting even farther away from 0. We can then repeat this process taking a large number of (or infinitely many) random steps.

There are many questions we could ask about this random walk. For example, after taking a large number of steps, where is the random walker most likely to be? Or how long does it typically take a random walker to reach 1,000,000 on the number line?

The plan for this project is to start by simulating random walks in simple spaces, such as the number line and 2D and 3D Euclidean space. As we consider more complicated spaces, we can ask more interesting questions. The goal is to then simulate random walks in the 3-dimensional Heisenberg group in order to better understand the long-term behavior of random walks. Time permitting, we could also simulate and explore random walks in other groups or spaces of interest.

**Faculty Mentor:** Nate Fisher

**Difficulty:** Intermediate

**Prerequisites:** Students should have taken Math 234. Experience with basic group theory and basic probability would be useful. Familiarity with programming would be helpful. We have some code written in Mathematica, which could be adapted to Python or Sage.