# Spring 2019

# Spring 2019 WXML Projects

### Number Theory and Noise

#### Faculty mentor: Matthew Conroy

##### Graduate mentor: **Kristine Hampton**

##### Team Members: **Mrigank Arora, Miranda Bugarin, Apurv Goel, Aanya Khaira**

### Constructing Quantum Theories

**Faculty Mentor: **Benjamin Feintzeig

##### Graduate mentor: **Charlie Godfrey**

##### Team Members: **Thomas Browning, **Kahlil Gates**, Jonah Librande, Rory Soiffer**

This project investigates the properties of so-called deformation quantizations that model the relationship between classical and quantum physics. We will especially be concerned with using the mathematics of deformation quantization to guide physics in the construction of quantum theories. We will analyze a simple system: a single particle moving in one dimension, whose physics can be captured by a phase space with the structure of the plane. Our strategy will be to focus on algebraic and analytic methods for defining a non-commutative product on algebras of functions on the plane. We will try to do so without relying too heavily on the particular geometrical structure of the plane so that we can generalize to examples with different phase spaces. We will investigate considerations from physics—especially concerning physical and non-physical states—for constructing the product in various ways. This project will make extensive use of algebra and real analysis.

### Mathematics of Gerrymandering

**Faculty Mentor**: Christopher Hoffman

##### Team Members: **Zack Barnes, Aaron Beomjun Bae, Langley DeWitt, Norton Pengra**

Districting and Gerrymandering have been in the news a lot recently, and the mathematical modeling of how to draw districts is a very hot topic. This project will look at the space of all possible redistrictings of a state to see whether the current plan is an outlier. It will involve probability and computing skills.

### Reseating Chinese restaurant process

**Faculty Mentor: **Noah Forman

##### Graduate mentor: **David Clancy**

##### Team Members: **Marques Chacon, Zohebhusain Siddiqui, Nile Wynar**

The reseating Chinese restaurant process is a randomly changing partition of {1,…,n}. Think of n customers in a restaurant; the tables are the “blocks” of the partition. At each step in the process, a single customer leaves their seat and chooses a new one according to a certain probabilistic rule. This process has applications in the clustering problem in machine learning. We will build simulations of the large n limits of these processes and use them to study properties of the processes.

### Locating the Walsh inequalities in the SNIEP

#### Faculty mentor: Pietro Papparella

##### Graduate mentor: **Haim Grebnev**

##### Team Members: **Jon Kim, Jiawei Wang**

Given a multiset Λ (herein, ‘list’) of complex numbers, the longstanding “nonnegative inverse eigenvalue problem” (NIEP) is to find necessary and sufficient conditions on Λ such that Λ is the spectrum of an entrywise nonnegative matrix A. If such a matrix exists, then Λ is called “realizable” and the matrix A is called a “realizing matrix (for Λ)”. If, in addition, Λ only contains real numbers and the realizing matrix is required to be symmetric, then the aforementioned problem is called the “symmetric nonnegative inverse eigenvalue problem” (SNIEP). The above-mentioned problems have been open since 1949 and are unsolved when n>4. Recently, Johnson and Paparella [MR3452738] studied eigenspaces that yield nonnegative matrices and tacitly introduced conjuctive inequalites that may be studied in relation to the SNIEP. The purpose of this project would be to locate their position in the map of other well-known sufficient conditions in the SNIEP [MR3672965] and thus determining their strength. Ascertaining these results would be of great interest to those working on the NIEP and SNIEP and would be suitable for publication in (e.g.) Linear Algebra and its Applications, Linear and Multilinear Algebra, and The Electronic Journal of Linear Algebra.

### Luggage Belt Modeling

#### Faculty mentor: Moumanti Podder and Jayadev Athreya

##### Graduate mentor:** Connor Ahlbach**

##### Team Members: **Andrew Lim, Qiaoxue Liu, Qiubai Yu**

Have you ever waited a long time for your checked-in bag to arrive to the conveyor belt? We’re going to construct different models of luggage arrivals (and passenger arrivals) to try and understand the waiting times and the general process of luggage arrival. We’ll use basic probability, a little geometry, and combinatorics to model this process.

### Rational normal curves

#### Faculty mentor: Bianca Viray

##### Graduate mentor: **Chi-Yu Cheng**

##### Team Members:** Leslie Feng, Alana Liteanu**

Rational normal curves are the simplest smooth curves in any dimensional space. It is known that these curves can always be mapped to quadric curves in the plane, and this project is focused on understanding explicit ways to do so, particularly over fields that are not necessarily algebraically closed.

### Powers of higher nerves

#### Faculty mentor: Bennet Goeckner

##### Graduate mentor: **Amzi Jeffs**

##### Team Members: **Madeline Brown, Spencer Kraisler, Zhongyan Wang**

*nerve complex*is an object that is formed by considering intersections of these sets. In general, the nerve complex is simpler and easier to work with than the original collection of sets, but it shares many of the same properties as S. The

*higher nerve*is a recent generalization that takes into account more information about the intersections of the sets in S. It contains more accurate information than the original nerve but is still easy to calculate. In this project, we want to iterate this process of taking a higher nerve. That is, we will consider the nerve of a nerve complex (or the nerve of a nerve of a nerve, etc.). What happens with this process? Does it ever terminate? Can it say anything interesting about the original collection of sets? We will start by computing examples for well-known objects and see if we can deduce patterns from our work. If participants are interested in coding, we can implement this process in a mathematical language like Sage.

### Triply periodic polyhedral surfaces

#### Faculty mentor: Dami Lee and Jayadev Athreya

##### Graduate mentors: **Charles Camacho, Gabriel Dorfsman-Hopkins**

##### Team Members: **Aadya Bhatnagar, Benjamin Zielinski**

Triply periodic polyhedral surfaces occur in physics, chemistry, and engineering- we’re going to hunt for new examples and explore the properties of existing ones. See http://www.wxml.math.washington.edu/?p=555 for pictures! This project will build skills in geometry and combinatorics, and we’ll also use some basic Python/Sage programming.