Winter 2023
This Winter 2023, the important dates and times for WDRP are as follows:
Saturday, November 26th: Undergraduate student applications open (Link here).
Wednesday, December 28th: Undergraduate student applications due at 5:00 pm.
Friday, January 6th: Latest possible day that we will announce project pairings.
Outside of these times, 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: Mitchell Levy
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.
The Brachistochrone. Fermat and the Bernoullis
Mentor: Begoña García Malaxechebarría
Mentee: Songnan Lin
Description: In this project my main goal is to get to the section of the brachistochrone (curve of fastest descent) in the book Differential Equations with Applications and Historical Notes by George F. Simmons, which very much inspired me when I was an undergraduate. For this we will go over Chapter 1 giving as much background as needed on differential equations and we will also talk about other applications of your interest
Prerequisites: MATH 207.
Combinatorics and Graph Theory
Mentor: Yirong Yang
Mentee: Alicja Misiuda
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 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 are combinatorics and graph theory beautiful subjects to study in their own right, they also have a wide range of applications both in and outside of the field of mathematics. The specific choice of textbook will depend on the interest and level of the student, but some great options include “Combinatorics and Graph Theory,” “Catalan Numbers,” “Graph Theory and Its Applications,” and “Graph Theory with Algorithms and Its Applications.”
Prerequisites: None.
Fields and Number Theory
Mentor: Alex Wang
Mentee: Dhruv Ashok
Description: Why do we do calculus over the real numbers? Why do we use matrices whose entries are real numbers? What properties of the real numbers makes them the correct setting to do all of these things? We explore the mathematical object of fields, their properties, and their numerous applications in algebra, number theory, cryptography, and more. Possible directions include finite fields, p-adic fields and complete fields, number fields, and field extensions.
Prerequisites: Some form of linear algebra (e.g. 208) would be good to have, but can certainly be substituted for an enthusiasm for mathematics.
Projective Geometry For Computer Vision
Mentor: Jessie Loucks-Tavitas
Mentee: Eugene Kim
Description: Even though parallel lines in the world never meet, they appear to do so in many images. This is captured particularly well in Renaissance paintings incorporating a vanishing point. The foundations of computer vision rely on a model of the world that allows for points “at infinity” which can behave as vanishing points in images. We will study and work problems involving lines, planes, and higher-dimensional objects in projective space along with their transformations. Our end goal will be to describe the motivation and utility of projective geometry with regards to computer vision.
Possible sources include:
* Multiple view geometry in computer vision, by Hartley and Zisserman (book)
* An Introduction to Projective Geometry (for computer vision), by Birchfield (expositional notes)
* Introduction to Projective Geometry, by Wylie (book)
* Projective Geometry: A Short Introduction, by Boyer (lecture notes)
Prerequisites: Math 126 (or equiv. vector calculus) and Math 208 (or equiv. linear algebra).
Analysis, Combinatorics, and Number Theory
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
Groups and Geometry
Mentor: Jackson Morris
Mentee: Jack Zhang
Description: Groups are a primary object of study in abstract algebra. The goal of this project is to demystify how “abstract” they actually are! We will be working towards understanding the following sentiment: “Algebraic properties of a group are reflected in the geometry of the spaces on which the group acts.” In understanding this, we will learn about groups, metric spaces, graphs, and other geometric objects, and how to use one to gain information about the other. Our primary text will be “Office Hours with a Geometric Group Theorist”, though we could shift to Sagan’s book on the Symmetric group if interest is there. The only strict requirements are comfort with reading/writing proofs, though some linear algebra would be helpful.
Prerequisites: MATH 208, MATH 300
Hands-on Introduction to Random Matrix Theory
Mentor: Garrett Mulcahy
Mentee: Leyu Zou
Description: This project will be a hands-on introduction to the theory of random matrices following the text “Introduction to Random Matrices” by Livan, Novaes, and Vivo. In a first probability course, the principal object of study is random variables: functions from some sample space to the real line. The objects we will consider in this project are random matrices, which are matrix-valued random variables. Random Matrix Theory (RMT) is an exciting area of mathematics that combines probability and linear algebra. For example, a typical question would be to describe the distribution of eigenvalues for NxN symmetric matrices drawn from some distribution and then to consider the distribution as N increases to infinity (look up the Wigner semicircle law)! Additionally, RMT is a field with many applications in statistics and physics. The text also is accompanied with several examples of code that we will work through and tinker with to help build intuition throughout the quarter.
Depending on student background and interest, we will likely spend some time reviewing linear algebra and probability at the beginning of the term or as needed throughout the quarter. Students should have taken at least one course in linear algebra, probability, and multivariable calculus.
Prerequisites: Linear algebra, probability, and multivariable calculus
Magical Mathematics
Mentor: Raghav Tripathi
Mentee: Daria Akselrod
Description: Well! All mathematics is magical, isn’t it? But in this project we will discover the application of mathematics to `magic’. We will follow the book “Magical Mathematics” by Persi Diaconis and Ronald Graham. The book explores some very interesting and profound mathematics which has been used in creating card games. (Fun Fact: Persi Diaconis was a magician before becoming a mathematician!) So without much further ado, let the magic begin!
Prerequisites: Having taken a course in probability would be useful! (And it would help if you have seen a deck of card already!)
p-adic Numbers
Mentor: Ting Gong
Mentee: Mark Polyakov
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.
Visual Differential Geometry
Mentor: Sean Richardson
Mentee: Alicia Stepin
Description: Tired of endless symbols? If so, “Visual Differential Geometry and Forms” could be for you! This text has more than just lots of pictures; every argument in the book is inherently visual. In working through the book, we will learn the author’s visual and intuitive style of proof inspired by Newton. Some possible objectives depending on your background and interests could be Gauss’s “Remarkable Theorem”, the Gauss-Bonnet Theorem, Einstein’s Field Equations, or anything else in the book. There are many ways to work through the text, but in every case we will gain a completely visual understanding of classic results in differential geometry.
Prerequisites: Multivariable calculus.