WASHINGTON eXPERIMENTAL MATHEMATICS LAB

# Spring 2020 Projects

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### Constructing Quantum Theories

#### Team Members:Rory Soiffer, Jonah Librande, Timothy Ferrin and Yin Fu

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

#### Team Members: Legrand Jones and Fran Herr

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 combinatorics, 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.

### Random Inelastic Billiards

#### Team Members: Jackson Zariski, Aidan Mager and Laurel Safranek

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: https://www.ams.org/journals/notices/201902/rnoti-p162.pdf

This project is open to all students who have taken Math 126. Ideally students will have taken Math 324, some intro physics and know some basic Python or Javascript.

### The mathematics of covid-19

#### Team Members: Tiffany Tian, Yuchen Shao, Shengkun Ye, Alyssa Mell and Ting Yin

The goal of this project is to model the dynamics of covid-19 outbreak in King County and Washington State. Over the next several months, there will be a lot of data emerging on how covid-19 has spread throughout the community. We will do our best to make sense of this data and attempt to answer a number of fundamental questions concerning the virus. The specific questions that we address may change depending on the data and the realities of how the virus spreads.

A tentative goal will be to estimate various quantities such as the average latency period, average duration of infection, percent of undocumented infections and the differences in transmission rate between documented and undocumented infections. These values will be estimated using a dynamic metapopulation model and a version of the Kalman filter. We will try to incorporate how social-distancing measures and testing affect these values. Similar studies have recently been completed on how covid-19 spread across China and elsewhere, and one of our first goals will be to understand their methodologies and results.

Some of this project will be learning basic modeling theory but a large component of this project will be dedicated to implementation. If you are interested in a project based entirely on mathematical theory, this is not the project for you. You should expect that a significant percentage of your time is spent on locating data sources, coding and running simulations.

This is a project open to all students who have take math 126. A modeling or statistics course would be useful, but not necessary.
Coding experience will be helpful. We will likely implement the model in python or matlab. It might be helpful to have a member of the group fluent in Mandarin.

### Generalized linear algebra over tensors

#### Team Members: Jon Kim and Jiaao Shen

Multivariate structured data commonly result in representations over $N$-dimensional tensors, two cogent exemplars being probabilistic graphical models and certain classes of relational database. We are interested in generalized linear algebra over such tensors. Specifically, consider an algebraic semiring $S=\left$ where $R$ is typically the reals or $[0,1]$; $\oplus$ is a disjunction-like operator like $+$; and $\otimes$ is a conjunction-like operator like $\times$. These functions support marginalization, projection, conditioning, and others operations over subsets of tensor dimensions. Typical semirings include $S=\left<\R,+,\times\right>$ as noted above, but also $S=\left<[0,1],\max,\min\right>$ for fuzzy operations, amongst others. In the semiring, $\otimes$ distributes over $\oplus$, and so this is like linear algebra generalized first to these new algebras, and also to $N$-dimensional tensors instead of just e.g.\ stochastic matrices. We use the python package xarray which supports $N$-tensors rather nicely, but it’s limited to just using only $+$ to able to aggregate over tensor dimensions. Students will first understand the math, and then extend xarray to support some class of $S$, including at least the examples above. PNNL will provide a specification of the problem, pointers to the literature, and sample data.

This is an advanced project intended for students who have taken multiple upper-level mathematics courses and have familiarity with linear and abstract algebra.