# Autumn 2020 Teams

## Random Inelastic Billiards

**Faculty Mentor:** Dr. Jayadev Athreya

**Project Description:** We’ll illustrate the dynamics of sticky particles on the real line, using a mixture of probability theory, dynamical systems, programming, and illustration. Our work is motivated by the beautiful work of Ryan Hynd.

**Team Members**: Joshua Ramirez, Aidan Mager, Jackson Zariski, Yinan Guo, Marcelo Cunningham, Laurel Safranek

**Project Level:** This project is open to all students who have taken Math 126.

**Additional Course Requirements:** Ideally students will have taken Math 324 and some intro physics.

**Programming Requirements:** Preferably know some basic Python or Javascript.

## Mathematics of twistronics

**Faculty Mentor:** Dr. Alexis Drouot

**Project Description:** Twistronic is a new area of solid state physics. It studies how a twist angle between 2D-layers affects electronic properties. It can be quite radical: twisted graphene, for instance, dramatically ranges from an insulator to a superconductor even with a tiny change in the twist angle.

Twistronics is a promising field for the conception of novel quantum devices. It produces beautiful quasi-periodic graphs, whose mathematical analysis is rapidly growing. It relies on linear algebra, periodic and quasi-periodic functions, and numerical analysis. We will investigate theoretically and computationally (a) density of atoms; (b) continuous approximations; and (c) energy distribution.

**Team Members**: Jocelin Liteanu, Kyosuke Saito, David Shiroma

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** 309, 327

**Programming Requirements:** Matlab (not necessary)

## Constructing Quantum Theories

**Faculty Mentor:** Dr. Benjamin Feintzeig

**Project Description:** This project investigates the properties of so-called deformation quantizations that model the relationship between classical and quantum physics. We will analyze various differential equations representing distributions of matter in space. Deformation quantization considers algebras of complex-valued functions on the space of solutions to such differential equations, and defines a non-commutative product capturing the fundamental features of quantum physics. Since the deformed non-commutative product approximates a classical commutative algebra in a precise sense, one can use this structure to investigate the “classical limit” of aspects of quantum physics. We will be especially concerned with analyzing the classical limit of quantities associated with energy and particle content in physics.

**Team Members**: Brendan Seamas Murphy, Yin Fu, Legrand Jones II, Tim Ferrin

**Project Level:** This is an advanced project.

**Additional Course Requirements:** Some background in linear algebra, analysis, and abstract algebra will be helpful for this project. No background in physics is required.

**Programming Requirements:**

## Topological structures hiding in sequences of real numbers

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:** Given a sequence of n distinct, real numbers, there is a nice and fairly canonical way of generating a 4-regular graph (that is a graph where every vertex has exactly 4 neighbors). This graph was recently introduced to measure the randomness of a sequence and is quite simple: you connect points based on the order in which they appear in the sequence and in order of their size as real numbers.

For specific deterministic sequences, the arising graphs seem to be amazingly structured: the sequence x_n = n* sqrt(2) mod 1 seems to result in the torus. The van der Corput sequence (a funny sequence from combinatorics that is essentially counting in binary) leads to graphs that look like a Klein bottle(?).

The goal of this project is to investigate classical sequences from number theory and combinatorics and to see which ones have interesting structures hiding underneath. If some are discovered, it would be nice to try to prove their existence. Last but not least, it could be interesting to investigate other types of Graphs, maybe there are even better ways of finding out what’s hiding underneath. Being comfortable with very light programming is a plus.

**Team Members**: Ruimin Zhang, Dana Korssjoen, Biyao Li

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** 461, 462 (Combinatorial Theory 1/2) would be a big plus; Math 301 couldn’t hurt, Differential Geometry (442/443) might help a little; none are strictly necessary.

**Programming Requirements:** Basic programming abilities in any language.

## Markov chains and Monte Carlo simulations

**Faculty Mentor:** Dr. Soumik Pal

**Project Description:** The newly introduced Math 396 suffers from a lack of textbook. Tim Mesikepp and I are writing lecture notes from 396 as a free textbook for all future students. We need good students to produce figures and run simulations that folks can use for the class. In return, they get to learn Markov chains and Monte Carlo algorithms.

**Team Members**: Celeste Zeng, Jamie Forschmiedt

**Project Level:** Intermediate: students who have taken Math 300

**Additional Course Requirements:** Math/Stat 394 in the last couple of years

**Programming Requirements:** Should be able to produce figures/ diagrams in R/Matlab etc. and also run simulations in R/ Matlab or Python

## The mathematics of ranking: from Perron to PageRank

**Faculty Mentor:** Dr. Sinan Aksoy

**Project Description:** From ordering web pages in response to keyword queries, to measuring gene associations with diseases, to guessing which NBA team is going to win Game 7, the problem of ranking spans multiple domain areas. While approaches to ranking are diverse and multidisciplinary, their mathematical foundations often serve as a unifying theme. This project will explore these foundations and the methods to which they give rise, including Perron-Frobenius theory, random walks on graphs and PageRank, and centrality measures from network science. Depending on student interests, additional topics to explore may include recently proposed analogs of the Perron-Frobenius theorem for tensors and multidimensional arrays, hypergraph ranking, and analytical methods for rank comparison. Students will then put their understanding into practice by (1) collecting a dataset of their choice, (2) implementing their favorite ranking algorithm, and (3) analyzing the results and/or the computational complexity of their approach.

All motivated students are welcome, although prior experience with linear algebra, data science, and coding is helpful.

**Team Members**: Haley Paige Riggs, Chuan Shi, Jiaqi Su

**Project Level:** Intermediate: students who have taken Math 300

**Additional Course Requirements:** linear algebra

**Programming Requirements:** basic familiarity with python or MATLAB

## Geometry of zeros of complex polynomials

**Faculty Mentor:** Dr. Harry Richman

**Project Description:** The goal of this project is to explore patterns between the location of zeros of a polynomial in the complex plane and the zeros of its derivative. More specifically, we will look for ways to generalize a basic observation known as Rolle’s theorem: for a real-valued polynomial p(x), between any two (consecutive) zeros of p(x) there must be a zero of p'(x). The case of complex polynomials is much less well-understood. For complex polynomials, the relation between zeros of p(x) and zeros of p'(x) is equivalent to the following electrostatic problem: given the position of n electrons fixed in the plane, where could I place an additional electron such that it would not be pushed around by the fixed electrons?

We will explore this problem using computational visualization tools, using Mathematica or python. (See here for example.) Using these observations on specific families of polynomials, we will then formulate conjectures about what happens for general complex polynomials and try to prove them.

**Team Members**: Sarwesh Rauniyar, Caiwei Tian, Mark Lamin, Omar Shaikh Omar

**Project Level:** Open: students who have taken Math 126

**Additional Course Requirements:** in addition to calculus, familiarity with differential equations or electromagnetism is preferred

**Programming Requirements:** some experience with programming; willingness to learn Mathematica and/or python

## Modeling transport by nanobubbles

**Faculty Mentor:** Dr. Amanda Howard

**Project Description:** Nanobubbles have recently been shown to have applications to lots of interesting problems, including plant and animal growth, medical imaging, and even manufacturing ice cream. Some of the most exciting applications use nanobubbles to transport small particles, like in targeted drug delivery, food seasoning, or waste treatment. However, attempts to model these systems are very computationally expensive and not feasible on large scales. In this project we will develop a continuum-based numerical method to model the transport of solid particles by nanobubbles.

We will use an existing finite-volume code (https://www.sciencedirect.com/science/article/pii/S0021999120305064) developed for nanobubble interactions with solid surfaces and adapt this code to model the movement of solid particles. Open problems include how to calculate the force transferred to the particles from the fluid and nanobubbles, and an exploration of the surface tension and pressure inside the nanobubble. This is a computational fluid dynamics problem, but participants should expect to learn about the mathematical fundamentals of developing rigorous numerical methods.

While existing code will minimize the amount of programming needed, some programming experience, ideally in Matlab or C++, is preferred. Some experience with physics or fluid dynamics is helpful.

**Team Members**: En-Shuo Liu, Nora Fossenier, Brendan Ho

**Project Level:** Open: students who have taken Math 126

**Additional Course Requirements:**

**Programming Requirements:** Basic programming, ideally in Matlab or C++ is preferable.

## The spectra and Jordan chains of cyclic matrices

**Faculty Mentor:** Dr. Pietro Paparella

**Project Description:** In the field of spectral graph theory, it is a well known fact that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. Recently, McDonald and Paparella (2016) gave necessary conditions on the Jordan blocks and Jordan chains of $h$-cyclic matrices. A matrix is $h$-cyclic if and only if the adjacency matrix of its digraph is $h$-partite.

The purpose of this project is to investigate whether these conditions are also sufficient, which would provide a characterization of these matrices.

**Team Members**: Yinxi Pan, Yining Liu, Sela Navot

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** Math 340 (preferred)

**Programming Requirements:** MATLAB