December 7, 2018

# Winter 2019

# Winter 2019 WXML applications are now open:

Project descriptions are below:

## New Projects

### Reseating Chinese restaurant process

**Faculty Mentor: **Dr. Noah Forman

**Project Description: **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.

**Project Level: **This is an **intermediate **project. Students should have taken Math 300, and Math 394 is ideal. Some R programming experience is also a plus.

### Constructing Quantum Theories

**Faculty Mentor: **Prof. 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 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.

**Project Level: **This is an **advanced **project. Math 340, Math 426, and Math 402 will be helpful.

## Returning Projects

Note that returning projects often have fewer slots available as many students return to the project.

### Tactile Patterns in Art and Mathematics

**Faculty Mentors **: Prof. Sara Billey and Prof. Timea Tihanyi

**Project Description: **There is a long history of exploration using mathematics and art from ancient Sumerian, Persian, and Greek civilizations, through the Renaissance, all the way to the present. For over a decade, Profs. Billey and Tihanyi have been bridging the gap between art and math by collaborating on various projects at this intersection. Most recently in the area of Cellular Automata and other forms of discrete mathematical algorithms, which output a pattern of binary data or two and three-dimensional matrices for realization as tactile patterns in 3D printed ceramics. See https://www.sliprabbit.org/projects/ for pictures. The focus of this WXML project is to explore open problems and 2-dimensional visualizations that come from the math of sandpile models on graphs. Sandpile models combine aspects of combinatorics, probability, computer science, and art. We are looking for a team of 2-3 students with a variety of skills and experiences in the above areas who can embrace the essential mathematical concepts quickly, implement algorithms for experimentation, attack the open problems, and can communicate well with a broad audience. Students will be expected to work independently for 8-10 hours per week and meet with the instructors 1-2 times per week. Students will also be asked to visit makerspaces on campus and Prof. Tihanyi’s digital ceramic studio off campus as part of this WXML project.

**Project Level: **This project is an **advanced **project. Students should have taken Math 461, Math 381, or Math 394 or similar courses. Some programming experience with graphics in python, Cocalc, or similar language is desirable.

### Mathematics of Gerrymandering

**Faculty Mentor**: Prof. Christopher Hoffman

**Project Description: **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.

**Project Level: **This project is an **advanced **project. Strong programming skills are required.

### Counting k-tuples in Discrete Sets

**Faculty Mentor: **Prof. Jayadev Athreya

**Project Description: **We will count pairs, triples, and k-tuples of integer vectors in the plane, 3-space, and higher dimensions, using both number theory and programming. We’ll also explore other interesting sets arising from dynamical systems and billiards. This project will involve number theory, geometry, and probability, and will be a great introduction to those subjects.

**Project Level: **This project is an **intermediate **project. Completion of Math 300 is required, and some basic Python programming skills are desirable.