The projects from Spring 2023 can be found below.

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## Dynamics on trees and graphs

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

## Numerical study of coupled tilings of the Aztec tiling (cont.)

The Aztec Diamond is constructed by the union of unit squares in the plane whose edges lie on the lines of a square grid and whose centers (x,y) satisfy |x| + |y| < n

We can create a tiling of 2 by 1 non-overlapping rectangles over this shape. At a large rank n, we see two distinct regions: A frozen region where dominoes of the same orientation group together, and a disordered region where the tilings follow no such order. These regions are visually separated by a boundary called the Arctic Curve. The random fluctuations of this boundary are known as well.

We can further look at coupled tilings, which are pairs of tilings that we can bias with some weighting parameter t to show certain interactions. Some theoretical results about these exist at t = 0 and infinity. We implemented an algorithm to generate coupled AD’s and then numerically test the theoretical results. We verified the fluctuations in the border of the frozen region at t = 0, 1, and infinity and obtained novel data on coupled tilings with weighting parameters in between these values.

**Faculty Mentor: **David Keating

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

## Quantitative uniqueness results in complex analysis

In this project, we looked at the Schwarz Lemma, a famous theorem from Complex Analysis that describes holomorphic functions from the disk to the disk. In particular, it has a corollary that the only function that is holomorphic, maps the disk to the disk, maps zero to zero, and has derivative one at zero is the identity. Our goal was to prove a quantitative version of this uniqueness result, where a holomorphic function that maps zero close to zero and has derivative close to one, will be close to the identity. We measure closeness using the Poincare metric on the disk.

**Faculty Mentor:** Andrew Zimmer

**Graduate Students: **Aleksander Skenderi & Vicky Wen

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

## Algebra in systems biology

Our project centered around a polynomial system of equations that describes chemical reactions in the Wnt Pathway. Mutations in Wnt Pathway are associated with cancer and neurodegenerative diseases, which is one of the reasons it is studied. Our goal was to find parameters that give us the maximum number of positive real solutions to this polynomial system of equations. In biology, this would mean rate constants and conservation quantities that allow having the most number of steady-states. To achieve this, we first learned theory involved in Numerical Algebraic Geometry, then wrote several scripts in Macaulay2, which helped us better understand the polynomial system of equations.

The graphic below shows how changing three of our parameters changes the number of positive real solutions. We can see the complex decision boundary between getting one and three positive real solutions, where parameter values with two positive real solutions should exist.

**Faculty Mentor: **Jose Israel Rodriguez

**Graduate Student: **Aviva Englander

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

## Task design to distinguish between model-free and model-based reinforcement learning

Reinforcement learning studies algorithms for how an agent learns to make decisions sequentially over time. The main question for this semester was to distinguish whether human decision-making uses model-based (MB) or model-free (MF) learning. We started with understanding Markov Decision Processes (MDP) which provide a mathematical framework for modeling a sequential decision-making environment where an agent learns. We then explored MB and MF algorithms, but there is no widely accepted method of distinguishing between the two; algorithms are only purported to be one or the other. We designed and tested a new task to analyze the difference between MB and MF algorithms, using R-max as an example MB algorithm and Q-learning as an example MF algorithm. Our task, shown in the figure, can be applied to humans and combines a new framework, local change adaptation (LoCA), with an established psychological task, the two-stage Markov task. Since LoCA is the best measure of an agentâ€™s model-based performance in its ability to extrapolate local changes in the environment into global policy changes, we chose this as our base concept for the task. Phase 1 is the usual two-stage Markov task. In Phase 2 the optimal decision is changed and the agent is restricted to one state so they learn locally. Phase 3 is a return to the usual two-stage Markov task but with the updated parameters from Phase 2 so we can tell if they generalize the local learning in a model-based way. This semester, we worked on simulating MDPs and this task to test how MB and MF algorithms behave.

**Faculty Mentor:** Amy Cochran

**Graduate Student: **Haley Colgate Kottler

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

## Identifiable linear compartmental models (cont.)

This semester, we studied the identifiability of linear compartmental models. Linear compartmental models have broad real-world applications, ranging from describing ecological systems to modeling infectious diseases.

Our group looked into how various operations can affect whether certain parameters in these models are identifiable. Specifically, we continued the work of the Fall 2022 MXM group in studying the behavior of catenary models with two leaks, and how moving these two leaks between different compartments affected the parameters’ identifiability. We made partial progress on proving our “leak sandwich conjecture,” that is, in a catenary model with two leaks, any parameter between and including the leaks is unidentifiable, while all other parameters are identifiable.

**Faculty Mentor:** Aleksandra Sobieska

**Graduate Student: **John Cobb

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

## Fine scale statistics of straight lines on flat surfaces

For this project, our goal was to find a square-tiled surface whose gap distribution differs from Hall’s distribution. We were able to find Hall’s distribution for the square torus, shown below, using Farey fractions and then expanded on this result to create an algorithm to find the gap distributions on any square-tiled surface. Our next goal is to use this algorithm on a specific subset of square-tiled surfaces that we think may result in a different gap distribution.

**Faculty Mentor: **Caglar Uyanik & Grace Work

**Graduate Student: **Pubo Huang

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

## An SIR Model of Stimulant Misuse Among College Students (cont.)

Prescription stimulants such as Adderall are sometimes used as study aids because of their reputation for improving focus in people with ADHD. Current research suggests that prescription stimulants are not effective study aids for those without ADHD, however the medications are highly addictive. We built a mathematical model of the SEIR type to study the spread of stimulant use among college students. Due to the lack of data available on this topic, we collected our own data using an IRB exempted survey methodology. We used the next-generation matrix method to compute the equation for R_0, which we will calculate once we have finished parameter estimation. We performed a sensitivity analysis of our model to determine which parameters were significantly correlated with the ratio of Susceptibles to Susceptibles with prescription. This summer we will work on parameter estimation and data analysis.

**Faculty Mentor: **Skylar Grey

**Graduate Student: **Sanchita Chakraborty

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

## Horospheres in Hyperbolic Groups (cont.)

This project aims to understand and visualize horospheres in hyperbolic right-angled Coxeter groups. Using tools such as finite state machines and fiber products we are able to efficiently generate pieces of the horospheres of right-angled Coxeter Groups. The quick computation allows us to visualize the geometry in a manner not allowed for before. This gives us insight into the boundaries of such groups.

**Faculty Mentor: **Tullia Dymarz

**Graduate Student:** Daniel Levitin

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