Spring 2023
The following are the projects held during the Spring 2023 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.
Applied Category Theory
Mentor: Nelson Niu
Mentee: Anthony Xing
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!
Prerequisites: None.
Tilings and Patterns
Mentor: Herman Chau
Mentee: Dang Phan
Description: Tiling and patterns occur throughout the arts and sciences, from the mesmerizing works of M.C. Escher to lattice structures in crystals. As another example, consider soccer balls: they are tiled using a mixture of pentagons and hexagons, but is there a reason why? Could we throw some squares into the tiling? Or if you like video games, have you thought about how some games are able to randomly generate the wall tiles in a level? The different tiles used come from a set of tiles called Wang tiles.
Here is a problem you may have heard before: “How many ways are there to tile, using dominoes, a chess board with two opposite corners cut off?”. The answer to this problem will serve as a stepping stone into our study of tilings. In this WDRP project, we will follow the book “Tiling and Patterns” by Grünbaum and Shephard. The book is comprehensive and we will pick and choose topics to study based on the participant’s background and interest.
Prerequisites: None.
Chip-Firing on Chains of Loops
Mentor: Caelan Ritter
Mentee: Gabriela Salaben
Description: Suppose you’re a monarch with too much time on your hands. To alleviate your boredom, you call your vassals to the great hall to have them play the following inane game. Each person is given a (possibly negative) number of french fries. When you point to a person, they must give each of their close friends a fry (by means of a mini-trebuchet especially designed by your engineers); call this action a “chip-firing move”. As you play, in your limitless wisdom, you begin to ponder: if some people start in fry-debt, is there always a sequence of chip-firing moves that ends with everyone out of debt? Consulting with your court mathematicians, you realize that this line of questioning leads to profound results in combinatorics, algebraic geometry, and other fields. Huzzah! You easily solve all of mathematics, the prestige of your kingdom grows, and your people rejoice.
In this project, we will learn about chip-firing on graphs from the ground up, with special emphasis on chains of loops. We will chiefly follow Sam Payne and David Jensen’s lecture notes, found at https://web.ma.utexas.edu/users/sampayne/Math665.html. Prior experience with linear algebra and reading proofs is encouraged but not required.
Prerequisites: None.
Computational Complexity Theory
Mentor: Raghav Tripathi
Mentee: Sam Tacheny
Description: Is it easier to add than to multiply? Complexity theory allows one to make sense of such questions. Computational complexity theory studies the resources required to solve a computational task. The goal of this project would be to get familiar with the terms used in the complexity theory, for example, to be able to understand P vs NP problem. We will follow Arora and Barak’s computational complexity book. We will not get into rigorous proof, but the goal would be get an essence of the results in this area.
Prerequisites: Intro to proofs would be required.
p-adic Numbers
Mentor: Ting Gong
Mentor: Rashad Kabir
Description: Real numbers are complete, meaning that all Cauchy sequences are convergent. This is a fact anyone will encounter when they first approach real analysis. However, for rational numbers, there is another way to make things complete, namely through p-adic norms. What is surprising is that, Ostrowski proved that there are only two nontrivial absolute values on the rational numbers, either the regular absolute value or the p-adic one. We will explore the surprising facts of p-adic numbers by following the book p-adic Numbers: An Introduction, by Fernando Q. Gouvêa.
Prerequisites: Some general facts about basic real analysis and point-set topology, if not known, we can make it up along the way too.
Projective Geometry
Mentor: Haoming Ning
Mentor: Anish Gupta
Description: The study of projective geometry is a rich and beautiful subject, with its origin dating back to 17th century mathematicians exploring the principals of perspective arts. In modern mathematics, it has an important presence in the areas of algebraic and differential geometry. In this project, we will study the foundations of projective geometry following two approaches, a purely synthetic one using Pappus’ Axiom, and one over the real numbers, relying on facts from Euclidean geometry. We will see how these two perspectives interact and supplement each other, including some highlights such as the classical Theorem of Pappus and Desargues’ Theorem. Depending on interests and backgrounds, we will follow this with an introduction of the modern treatment of projective geometry using algebra.
To participate in this project, you should be comfortable with reading and writing proofs and have taken linear algebra. Although not strictly necessary, a first course in abstract algebra can greatly enhance your experience.
Prerequisites: Intro to proofs + linear algebra. First abstract algebra course optional.
Triangulations
Mentor: Yirong Yang
Mentor: Dianna E
Description: Roughly speaking, triangulation just means cutting a space into “triangles.” As a natural way to decompose a region, it has various applications in algebra, computer science, and optimization. We will follow the book “Triangulations” by De Loera, Rambau, and Santos, which gives a comprehensive introduction to triangulations and its applications. The goal is to understand the rich geometric structure of the space of triangulations of a given set of points, and explore its connections to other areas in math depending on the student’s interest and time.
Prerequisites: MATH 318 or a more advanced linear algebra course, and at least one proof-based course.
Analysis, Combinatorics, and Number Theory (Continued)
Mentor: Ryan Bushling
Mentee: Logan Garwood
Description: In this project, we will study problems that unite ideas from a breadth of mathematical subdisciplines. The purpose is (at least) twofold: first, to gain facility using tools from different areas of mathematics in tandem, and second, to gain familiarity with some of the most interesting and important unsolved problems in harmonic analysis. To do this, we will read A View from the Top: Analysis, Combinatorics and Number Theory by Alex Iosevich. It is a light and pleasant read that masterfully speaks to a broad mathematical audience, from those new to mathematical proofs all the way to graduate students.
Prerequisites: MATH 300, MATH 308, MATH 327
Linear Algebra
Mentor: Marty Bishop
Mentee: Makenna Cannahan