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.

[Final Report]

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.

[Final Report]

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.

[Final Report]

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.

[Final Report]

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.

[Final Report]