Fall 2022
The following are the projects held during the Fall 2022 quarter. Undergraduate students and graduate students are expected to meet for 1 hour per week at a time of their choosing. In between meetings, undergraduate students are also expected to spend 4 hours per week reading and working on their own. Graduate students are expected to take the requisite time preparing for meetings on their own time as well.
Combinatorics and Graph Theory
Mentor: Yirong Yang
Mentee: Steven Li
Description: How many ways can King Arthur and his knights sit at the round table? How many five-card poker hands can be dealt from a single deck? A double deck? How many ways are there to change a dollar? How many people are required at a gathering so that there must exist either three mutual acquaintances or three mutual strangers? In this project, we will be able to find the answers to all of these questions and many more. Not only is combinatorics a beautiful subject to study in its own right, it also has a wide range of applications both in and outside of the field of mathematics. We will be following “Combinatorics and Graph Theory” by Harris et al., an incredibly fun book with great quotes and story-telling. The choice of specific topics will depend on the interest and level of the student.
An Introduction to Arithmetic Dynamics
Mentor: Alexander Galarraga
Mentee: Rohan Pandey and Yushan Huang
Description: We will introduce the necessary math to understand what questions are asked in arithmetic dynamics with the aim of selecting a paper to read some of based on the interests of each individual student. To introduce arithmetic dynamics, we will follow an original text based on a previous WDRP project, which will present its own unique challenges (that is, I probably made a bunch of typos for which I apologize in advance). In the process of revising the text, however, we will have the opportunity to actively engage questions surrounding how to communicate effectively in mathematics. In order to engage the active research occurring in arithmetic dynamics, we will aim to select a paper to read based on the individual interests of the student once we have properly introduced the background material. Although not necessary, arithmetic dynamics has a large computational component, so any student interested in computation will be able to engage the material using Sage, an open source computational software.
Mathematics for Sustainability
Mentor: Haoming Ning
Mentee: Lisa Li and Alex Albors
Description: In this project, we will be reading the thought-provoking book “Mathematics for Sustainability”. The book itself is intended for a very general audience, and aims to build mathematical reasoning in the context of real-world problems around the question of sustainability. Regardless of your background in math, the reading is designed to engage you in a mathematical and quantitative approach to studying and embracing the idea of sustainability.
The Graph Theory Toolbox
Mentor: Natasha Crepeau
Mentee: Freya Salsbury
Description: We will use the textbook “Graph Theory You Need Before Undergrad Research” to explore what a graph is and some important properties, such as connectivity, coloring, and planarity. We will also be building intuition for the types of proofs we see in graph theory, as well tackling more open-ended (but not necessarily unsolved) problems. If there’s time and interest, we can also dig deeper into any of the lectures from the book; for example, diving deeper into theorems around trees (a type of graph). This project does not require a student to already know what a graph is or to have taken a proof-based course.
Applied Category Theory
Mentor: Nelson Niu
Mentee: Harper Hults
Description: If you told a math major you were reading up on “applied category theory,” they’d probably look at you a little funny. After all, category theory has a reputation for being notoriously abstract: a fancy piece of machinery designed to tie lots of complicated math subjects together. But it doesn’t have to be this way! It turns out that the same category-theoretic framework that mathematicians use to study things like “schemes” and “homotopies” crop up in much more concrete, easy-to-visualize settings, such as spreadsheets, networks, and the instructions for baking a pie. If you’ve ever wondered how the language of mathematics can shape the way you think about the real world, then this is the project for you. We will be following the book “Seven Sketches in Compositionality: An Invitation to Applied Category Theory” by Fong and Spivak. Along the way, you’ll catch a few glimpses of set theory, logic, abstract algebra, and more. No prior knowledge or experience necessary—there’s something in this project for everyone!
Mathematical General Relativity
Mentor: Sean Richardson
Mentee: Kenneth Wu
Description: What do mathematicians mean when they say “spacetime”? How does spacetime determine how matter moves? How does matter influence spacetime? The answers lie in Riemannian geometry, which mathematicians use to describe curved surfaces like a sphere! In this project we will aim to understand mathematical general relativity, learning the necessary Riemannian geometry along the way. We will follow the first four chapters of Iva Stavrov’s “Curvature of Space and Time, with an Introduction to Geometric Analysis”. The only prerequisites are comfort with linear algebra and multivariable calculus.
An Invitation to Nonlinear Algebra
Mentor: Andrew Tawfeek
Mentee: Jacob Linden
Description: In linear algebra, we would often limit ourselves to linear expressions: just variables but raised to one single power — and never, ever consider multiplying your variables with eachother. But why did we stop there? And here, we won’t. Once we take the plunge into these “higher degree polynomials,” a fascinating and wildly different world arises: one with, not just lines and planes, but curves and surfaces — where singularities begin to peak in and you have to learn to “blow up” things to deal with them. As we traverse through this abstract world, we’ll explore applications of this “nonlinear algebra” to fields across science and engineering along the way.
Large Networks and Graph Limits
Mentor: Raghavendra Tripathi
Mentee: Terence Wang
Description: Large networks are ubiquitous objects in modern world. There are many interesting and challenging questions pertaining to large networks (or dense graphs). In this project, we aim at reading Part I of the Lovasz’s book “Large networks and graph limits”. The goal of the project is to appreciate the questions one can ask about large graphs and understand the need for advanced tools to answer some of those interesting questions.
Groups and Geometry
Mentor: Jackson Morris
Mentee: Nathan Louie and Ansel Goh
Description: The goal of this project is to study topology. Using Jack Lee’s “Introduction to Topological Manifolds” as our guide, we will study objects such as topological spaces, manifolds, quotient spaces, and surfaces, and study characteristics such as connectedness, compactness, homotopy, and coverings. We will pick up any necessary algebra along the way.