WASHINGTON eXPERIMENTAL MATHEMATICS LAB

# Autumn 2018

### Laplacian Eigenmaps for Dimensionality Reduction

Faculty Mentor: Dr. Spencer Becker-Kahn

Project Description: A black and white picture can be represented as a grid of numbers between 0 and 1 (i.e. an n x n matrix with entries between 0 and 1), in which each number represents the “darkness” of the pixel at the corresponding location (for example, if a picture is completely white at the location (i,j) then the matrix that represents it has a zero in the i,j-th entry). In this way, we might represent the set of all grayscale pictures as a set of, say, 100 x 100 matrices. This set is unweildly; it is 10,000-dimensional! However, an *actual* set of pictures that you might want to analyze in a *real-world* application could well just be pictures of only a few different types of similar-looking objects (e.g. faces), taken from a few different angles. In this way, your data set will be “intrinsically” lower dimensional and yet it will “live” in a high dimensional space. It is then typical to try to find a lower dimensional “representation” of the data (i.e. a finite set of points in a much lower dimensional space that somehow preserves important features of the data). This task is called dimensionality reduction and there are various methods for doing it. In this project, we will explore one such method – introduced in 2002 – which uses eigenfunctions of the discrete Laplacian on a graph that is built out of the data. We will focus first on understanding the mathematics of the algorithm (which uses mostly combinatorics and linear algebra) and then trying to get some idea about the deeper mathematical motivations, which come from the spectral theory of the Laplace–Beltrami operator on a Riemannian manifold. If there is enough interest (and/or programming experience), we can acquire some data, apply the method to real data and optimize the parameters. This is an interesting topic because on the one hand there are deeper connections to pure geometric analysis and on the other hand, this method is a type of “manifold learning” that has recently been a hot topic in data science and machine learning.

Project Level: This project is an advanced project. It is essential that team members have taken Math 308, highly desirable for them to have completed Math 340 and analysis at Math 327-level or above. It would be desirable if they had some experience with combinatorics/discrete optimization e.g. Math 381/409. Some Python experience would be desirable as well.

### Anomalous diffusions and fractional order differential equations

Faculty Mentor: Prof. Zhen-Qing Chen

Undergraduate Team Members: Qiaoxue Liu, Zian Chen

Project Description: Time-fractional diffusion equations have been actively studied in several fields including mathematics, physics, chemistry, engineering, hydrology and even finance and social sciences as they can be used to model the anomalous diffusions exhibiting subdiffusive behavior, due to particle sticking and trapping phenomena. Solutions to time-fractional diffusion equations can be represented probabilistically using inverse of time change. Any mathematical modeling is an approximation. In this project, students will learn time-fractional derivatives and anomalous subdiffusions, then study stability of solutions to time-fractional diffusion equations through numerical methods.

Project Level: This project is an advanced project. Team members should have taken Math/Stat 394/395/396, and some knowledge on differential equations is desirable. Some knowledge of computer simulations (Mathematica/Matlab) is ideal.

### Tactile Patterns in Art and Mathematics

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

Graduate Mentors: Connor Ahlbach and Caroline Babecki

Undergraduate Team Members: Hannah Van Wyk, James Pedersen

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

Undergraduate team members: Weifan Jiang, Norton Pengra, Zachary Barnes, Langley DeWitt

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

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.

### Number Theory and Noise

Faculty Mentor: Dr. Matthew Conroy