Spring 2023

Consider a smooth map from an interval to itself that has finitely many critical points, one can decompose the interval into subintervals at those points and study the dynamics (i.e. behavior of points under compositions of this map) by looking at which subintervals the critical values may land on under the map and its compositions. This is called the “Milnor-Thurston kneading theory”. There are some attempts at extending this to maps on trees but I think their conclusions can be improved.

Faculty Mentor: Chenxi Wu

Graduate Student: Robert Argus

Undergraduate Team Members: Beining Mu, Anvit Thekkatte, Zihan Zhao

This is a continuation of the project from Fall 2022.

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

Undergraduate Team Members: Lucas Allen, Braeden Bertz, Harsha Kenchareddy, Stephen Wei

The goal of this project is to take uniqueness results in complex analysis and prove quantitative versions. A uniqueness result usually has the form of “there is a unique object with a certain property P” while a quantitative version has the form “an object with properties close to P is close to the unique object.” A challenge is often discovering exactly what “close” should mean.

The project will involve complex analysis, (hyperbolic) geometry, and a little bit of algebra (polynomials and groups). However, we will only use a little bit from each area. So a student who has a strong background in epsilon/delta type arguments and who is willing to learn some new mathematics could be very successful on this project.

Faculty Mentor: Andrew Zimmer

Graduate Students: Aleksander Skenderi & Vicky Wen

Undergraduate Team Members: George Ekman, Mason Holcombe, Yixuan Hu, Fanchen Meng

For this MXM Project at UW Madison in Spring 2023 we will study polynomial systems arising in biology in the context of the Wnt signaling pathway. Motivated by a nearly decade old conjecture to understand these solution sets, we will leverage tools from computational algebra, numerical homotopy continuation, and more. It is known that each of these polynomial systems has nine complex solutions. But, it is an open problem to determine the maximum number of real solutions. A new result would be to find a system with at least four real solutions, and to use this information to make new conclusions about the biological process.

Our main reference will be the article “Algebraic Systems Biology: A Case Study for the Wnt Pathway” by Elizabeth Gross, Heather A. Harrington, Zvi Rosen, and Bernd Sturmfels.

Learning outcomes include

1. formulating an applied problem in terms of algebra
2. using mathematical software to gain insights through computation, and
3. comparing exact algebraic methods to heuristics derived from other areas

Faculty Mentor: Jose Israel Rodriguez

Graduate Student: Aviva Englander

Undergraduate Team Members: Noah Blum, Moulik Mehta, Yutaro Yokoyama

An essential question in analyzing human learning is determining whether humans are model-free or model-based learners. Model-based learners develop a model of their environment to map decisions to outcomes, while model-free learners do not. The 2-stage Markov task purports to distinguish between the two types of learners. However, strong criticisms of the task have since emerged.

Our goal in this project is to develop a task that can actually distinguish between the two types in simulation using methods from reinforcement learning and probability. Along the way, we will review the relevant literature to understand current algorithms for model-free and model-based learning. We will then simulate both types of learners to test out task designs.

Expected Outcomes:

• Implement Markov Decision Processes to define tasks that tests human learning
• Identify, compare, and implement model-free and model-based learning algorithms
• Understand the considerations that arise in simulation experiment and task design

Faculty Mentor: Amy Cochran

Graduate Student: Haley Colgate Kottler

Undergraduate Team Members: Peizhe Li, Jack Maloney, Seungyeon Oh, Jimmy Vineyard

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. In particular, we will continue work of Bortner et al. and the Fall 2022 MXM group on identifiability of catenary models with at least two leaks or output sites.

Faculty Mentor: Aleksandra Sobieska

Graduate Student: John Cobb

Undergraduate Team Members: Nuha Dolby, Nicole Miller, Jack Oakes

A flat surface has a simple description in terms of polygons in the plane where pairs of edges are identified via Euclidean translations. For example, gluing the opposite sides of a square gives a flat torus, and gluing the opposite sides of a regular octagon gives a genus two surface which is locally flat except at a single cone point of total angle 6π. This project will focus on understanding the geometry of a special class of length minimizing paths called “saddle connections” on flat surfaces. Saddle connections can be thought of as building blocks for all distance minimizing paths, and play an important role in the geometry of the surface. The question we would like to address is “how random are saddle connections distributed considered as subsets of the plane?” Motivated by a classical problem in number theory, we will work on to compute distribution of gaps between saddle connections on an important family of examples called square tiled surfaces and seek for exotic number theoretic behavior.

Faculty Mentor: Caglar Uyanik & Grace Work

Graduate Student: Joel Schargordsky

Undergraduate Team Members: Michael Beers, Mia Bhushan, Kwasi Debrah-Pinamang, Jessie You

In Fall 2022 we designed an ODE model of stimulant misuse by college students using the SIR structure as our starting point. We also wrote a survey designed to help collect data to fit this model. In Spring 2023 we will be administering this survey and using the data collected together with MATLAB program(s) we will write to fit our model to data. Then we will interpret the parameter fit that our simulations have provided and see what conclusions might be drawn.

If time allows, we will also perform optimal control analysis of the system. Optimal control is a technique used to determine how best to deploy resources in order to achieve desired outcomes. In this case desired outcomes could include increasing the number of students who recover from stimulant misuse and decreasing the number of students who begin stimulant misuse. Interventions could include educating student populations about stimulants (as the literature indicates a lack of knowledge) and providing appropriate support for those in recovery.

Faculty Mentor: Skylar Grey

Graduate Student: Sanchita Chakraborty

Undergraduate Team Members: LeYao Huang, Tyler Joseph Jones, Eline van Ophem, Lilah Tascheter

This is the continuation of a project from the Fall. It will focus on the extension of techniques and algorithms for graphing horospheres to more complicated groups.

The goal of this project will be to optimize algorithms for determining graphs of horospheres in some specific three-dimensional hyperbolic groups. A secondary goal will be to determine how best to create pictures of these non-planar graphs.

Faculty Mentor: Tullia Dymarz

Graduate Student: Daniel Levitin

Undergraduate Team Members: Noah Jillson, Pramana Saldin, Katerina Stuopis