WASHINGTON eXPERIMENTAL MATHEMATICS LAB

# Autumn 2019 WXML Projects

### Number Theory and Noise

#### Faculty mentor: Matthew Conroy

The Number Theory and Noise project has been ongoing since Spring 2016. In this project, students investigate the possibilities arising from the representation of integer sequences as sound. By listening to integer sequences, students apply an aspect of their perception that is little-used in mathematical investigations. This leads to numerous questions, involving mathematics, coding, and audiology. Many sound examples can be found via our project page: https://sites.math.washington.edu/~conroy/WXML/integerSequenceNoise/home.htm and our sound library page: https://sites.math.washington.edu/~conroy/sequenceNoise/indexWXMLsoundLibrary.htm

This is a project is open for all students who have taken Math 126. Students will be programming, but are welcome to learn as they go.

### Constructing Quantum Theories

#### Faculty Mentor: Benjamin Feintzeig

This project investigates the properties of so-called deformation quantizations that model the relationship between classical and quantum physics. We will analyze various differ- ential 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.

This is an advanced project. Some background in linear algebra, analysis, and abstract algebra will be helpful for this project. No background in physics is required.

### Matching Graphs

#### Faculty mentor: Bennet Goeckner

A graph is a collection of vertices and edges, where each edge is a pair of vertices. If an edge is the pair a,b, then a and b are called the endpoints of this edge. A matching of a graph is a collection of edges such that no two edges share an endpoint, and a matching complex of a graph is the set of all possible matchings of the graph. A matching complex is an instance of a simplicial complex, which is an object that is ubiquitous in many branches of mathematics.
Matching complexes have been well studied and have many connections to topology and representation theory. For the most part, this study has centered around the following question: “Given a graph, what can we say about its matching complex?” In this project, we will ask the opposite question: “Given a simplicial complex, is it a matching complex? If so, what can we say about the related graph?” Students with an interest in combina- torics, topology, or algebra are encouraged to join, though no background in these subjects is required to begin work on this problem. This project has many possible avenues of investigation, and, depending on student interest, we may write computer code to assist in our inquiries.

This is an intermediate project intended for students who have taken Math 300.

### Triply periodic polyhedral surfaces

#### Faculty mentor: Charles Camacho, Dami Lee and Jayadev Athreya

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.

### The card game SET and finite affine geometry

#### Faculty Mentor: Robert Won

The card game SET is a popular pattern-recognition game that is played with a special deck of eighty-one cards. Each card depicts one, two, or three symbols of different shapes (diamond, oval, squiggle), shadings (solid, striped, open), and colors (green, purple, red). Players try to find “Sets” of three cards for which each attribute is all the same or all different. A deck of SET cards gives a very nice way to visualize a geometric space—the finite affine space of order three and dimension four. Under this identification, the cards are the points of the geometry and the Sets are the lines.

When playing a game of SET, a natural question arises: what is the largest collection of cards containing no Sets? Or, equivalently: what is the largest collection of points in this space which contains no lines? We will study some generalizations of this question in geometric spaces of small dimension. This project will involve learning some finite affine geometry, modular arithmetic, combinatorics, and linear algebra over finite fields. We will also generate data using some computer programming.

This is an intermediate intended for students who have taken Math 300.

### Sum of Random Walks

#### Faculty Mentor: Christopher Hoffman

This is a project in probability. We will consider two random walks X(t) and Y (t) for t between 0 and n. We are interested in trying to predict which value t′ maximizes X(t) + Y (t). If we don’t know either of the walks X(t) or Y (t) then all choices of t are about equally likely. (The maximum is slightly more likely to occur near the endpoints 0 and n than near n/2.) If we know both X(t) and Y (t) for all values of t then there is nothing probabilistic. We can just check all the values and find where the maximum occurs. We will consider the problem of trying to predict where the maximum occurs when we know one of the walks (say X(t)) but not the other (Y (t)). This project will involve some theory but will be mostly computational.
This is an advanced project intended for students who have taken Math 394 and 395.