The projects from Fall 2022 can be found below.

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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 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 associate a tiling of 2 by 1 non-overlapping rectangles and an associated height function to each tiling. 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 circular boundary. Regarding the height function of large rank AD, several theoretical results about the fluctuations of the average height at a point and the covariance of the height exists in the literature. We implemented the domino shuffling algorithm to generate uniformly at random a rank n AD and then numerically test the theoretical results. We verified that the fluctuations of the height at a point (x, y) in the disordered region of the AD was normally distributed as the rank of the AD approaches infinity.

**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

In this project, our goal is to compute a holonomy representation based on the Fenchel–Nielsen coordinates. Specifically, given a hyperbolic surface, we would like to know which group actions would help us generate such surface. Our strategy is applying the constraints about hyperbolic characteristics to help us determine each action. We managed to solve for various types of pair of pants, and we anticipate similar process of “gluing” the pair of pants in a similar fashion in the future.

**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

Identifying the underlying dynamics from data is of practical importance. In this project, we developed a sparse model identification method based on causation entropy inference. This method incorporates physics via causality into the model identification and is thus different from the traditional LASSO regression. It also facilitates efficient parameter estimation. With the identified model, another practical issue is to estimate the state from partial observations with uncertainty quantification. Thus, an ensemble data assimilation method is applied to achieve this goal. The methods are applied to El Nino-Southern Oscillation and Madden-Julian Oscillation data sets.

**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

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 looked into the effect of adding outputs to longer catenary models, and how this can impact the model’s overall identifiability. We also studied the behavior of models with two leaks, and how moving these two leaks between different compartments affected the parameters’ identifiability.

**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

The project builds on the results of a related project from Spring 2022. The goal was to find joint moments of the entries of a uniformly chosen random nXn random orthogonal matrix, both via rigorous arguments and also numerical simulations. We identified some exactly solvable cases (joint moments of the entries of the 2X2 matrix, joint moments of the entries of the first row of the nXn matrix), and using the invariance properties of the model we managed to prove formulas for a number of special cases for the general nXn matrix. We also provided Mathematica simulations to support our results, and to set up conjectures.

**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

Curve Shortening Flow (CSF) is a process that describes time evolution of smooth curves, where points along the curve move along the inward normal vector, with velocity proportional to the curvature at that point. Employing Python to explore the singularity behavior of 3-loop and 4-loop curves, we determined the parameters and conditions such that all loops vanish at the same time, preserving the singularities, as the loop shrinks toward disappearance. Under these conditions, we observed the loop’s unique limit shape resembling a bow tie. Our findings lead to the resemblance of the central segments to a polynomial dependent on the number of singularities: the 3-loop approaching a second degree polynomial, and the 4-loop approaching a third degree. We hypothesize that for any n-loop, its limit shape can be described by a polynomial of degree n – 1.

**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

Stimulant abuse on college campuses is a generally under-researched topic. Our goal is to represent stimulant use and abuse on college campuses using a mathematical model that looks specifically at how stimulants spread among the population. We designed a system of differential equations and developed the equation for the basic reproductive number (Ro), which indicates the likelihood of the behavior to spread. In phase II we will collect data and fit the data to our model. Using these results, we hope to continue to gain a better understanding of stimulant use/abuse on college campuses and hope to propose policies for prevention.

**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)

A phylogenetic tree is a diagram that depicts the lines of evolutionary descent of different species that share a common ancestor. The goal of phylogeny estimation using maximum-likelihood is to recover the topology of a phylogenetic tree from datasets of genes for some set of species or other taxa. Long branch attraction is a systemic error incurred when distantly related, i.e. long-branched lineages, instead appear to be closely related. In this project, we investigated whether long branch attraction introduces bias to maximum-likelihood estimation in phylogenetic tree reconstruction even for simple evolutionary models and small-scale trees. Specifically, we translated the maximum-likelihood problem into a least-squares problem, with the aim of computing the optimality conditions (i.e. the tree parameters that give the maximum likelihood) of all possible configurations of the unrooted four-species tree with two long branches through a combination of analytical solutions and minimization software. After implementing a Homotopy Continuation approach and a Nonlinear Optimization approach in Julia, we found that the results from the two methods agree in certain cases, but questions remain.

**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

This project aimed to understand horospheres in hyperbolic graphs. To achieve this goal, we needed a relatively fast way to compute and render large pieces of these horospheres. We started with the problem of path-finding in an order-4 pentagonal tiling. This allowed us to compute a shortest path between any two vertices in our hyperbolic graph. We then defined a canonical form for these vertices and created a finite state machine specific to our order-4 Cayley Graph over which canonical vertices are accepted. With these three pieces, we were able to write an algorithm to compute horospheres within our hyperbolic graph. Our next goal is to modify the algorithms used on the order-4 pentagonal tiling to generate horospheres for a Cayley Graph defined as a flag triangulation of torus. In particular the Cayley Graph we are interested in is made of three concentric 7-gons and contains no squares.

**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