Applications
This Winter 2025, the important dates and times for WDRP are as follows:
Applications
Tuesday, December 3, 2024: Undergraduate student applications open (Link here).
Thursday, December 26, 2024: Undergraduate student applications due at 11:59pm.
Friday, January 10, 2025: Latest possible day that we will announce project pairings.
Events
Monday, January 13, 2025: Kick-off Event 5:30pm-6:30pm.
Wednesday, February 12, 2025: Midquarter Event 5pm-6:50pm.
Wednesday, March 12, 2025: Final Presentations 5pm-6:50pm.
Wednesdays, January 29 and February 26: WDRP Seminars 5pm-6pm
The events are mandatory: only apply if you can attend all events during the listed times. In addition, mentees are required to attend two seminars (described here). 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.
Explorer Level: All math levels welcome
The Structure of Mathematical Revolutions
Mentor: Daniel Rostamloo
Description: The past century of development in algebraic geometry and adjacent areas has seen several significant efforts to restructure the foundations of the subject which have yielded remarkable progress toward long-standing problems across mathematics. At the same time, various philosophers have proposed models for the processes which underlie progress in science as a whole. These models are especially well-tailored to the physical sciences, but generally do not address the rigorous and abstract nature of pure mathematics today. The aim of this project is to conduct close readings of these models and various historical texts which lie close to the modern development of algebraic geometry to reconcile this difference. Familiarity with proofs and various mathematical concepts will be developed as necessary. Possible final projects include a written analysis providing an interpretation for the structure of mathematical revolutions.
Prerequisites: N/A
Intro to Proofs: Discrete Math
Mentor: Nathan Cheung
Description: If you’ve been wondering what more upper level math courses are like, or are wondering if you should be a math major, this is the book for you! Written to give undergraduates an introduction to proofs and different proof techniques, this book combines various important math topics such as combinations, graphs, logic and sequences.
Prerequisites: None
Mathematics for Sustainability
Mentor: Haoming Ning
Description: Some of the most important problems in the 21st century center on sustainability questions like “can this (key part of our social or economic system) last?” For example, how long can fossil fuel supplies last? How much greenhouse gas can we emit before climate instability becomes dangerous? For many of these discussions on sustainability, we need some knowledge of mathematics in order to participate in a well-informed way. The aim of this project is to help you gain that mathematical knowledge and the ability to apply it to sustainability questions. We will learn about key mathematical concepts such as measurement, flow, connectivity, change, risk, and decision-making. We will find out how to use these tools to model sustainability on local, regional, and global scales. Whatever conclusions you reach, this project will prepare you to think critically about your own and other people’s arguments and to support them with careful mathematical reasoning.
Prerequisites: None! Less background is preferred.
Applied Category Theory & The Joy of Abstraction
Mentor: Nelson Niu
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, relationship 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. Depending on your preference, we will be following one (or both!) of the books “The Joy of Abstraction: An Exploration of Math, Category Theory, and Life” by Eugenia Cheng and/or “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, as well as applications to science, engineering, and/or social justice. No prior knowledge or experience necessary—there’s something in this project for everyone!
Prerequisites: None
Combinatorics, Graph Theory, and Computer Science
Mentor: Grace O’Brien
Description: Combinatorics is the mathematics of counting. Although it may seem like even small children can count, there are interesting questions that arise when we try to count complex objects. How many different outfits can you make with the clothes in your closet? How many routes are there to travel between your home and school? Related to combinatorics is graph theory, the study of networks. Graph theory has application to many situations from classic travelling salesman problems to Google’s Page Rank algorithm. We will cover the basics of counting and graph theory to prepare ourselves to see graph algorithms or another special topic. Reference: Miklos Bona’s “A Walk Through Combinatorics.” We may also explore connections of combinatorics to computer science and some key mathematical ideas behind current hot topics in CS today such as machine learning.
Only enthusiasm required! We can adjust the content to the level and interests of the student.
Prerequisites: None
Chip-firing on chains of loops
Mentor: Caelan Ritter
Description: Suppose you are 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 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
Beginner Level: May require some Calculus
Poker and Stats
Mentor: Nathan Cheung
Description: Note that this is not an endorsement to gamble. Gambling can become a dangerous and addictive issue in many people’s lives. This book will not help you become a better gambler. In particular, the author clearly states in the preface that “If you are reading this book in the hope that you will learn strategy tips on how to be a better poker player, you are bound to be disappointed.”
Instead, I hope that this book uses a game many people are already interested in to learn about statistics and probability. We will start with covering basic probability concepts and counting problems and hopefully end with talking about discrete and continuous random variables.
Prerequisites: Math 124
Exploring Upper-Level Math
Mentor: Kevin Tully
Description: Are you interested in math but aren’t sure if you’d like to become a math major? Then this project is for you! It’s a low-stakes way to see what pure math is like. Following the book “Transition to Advanced Mathematics” by Diedrichs and Lovett, we’ll learn about how to write proofs, discuss the distinctive aspects of upper-level math, see the vocational possibilities the math major opens up, and explore any other topics that interest you. Some background in calculus would be helpful, but only mathematical curiosity is required.
Prerequisites: Math 124 and 125 would definitely help, and Math 208 couldn’t hurt but isn’t required.
Visualizing Complex Numbers and Functions
Mentor: Leo Zhang
Description: This will be an introduction to complex numbers and functions, with an emphasis on visualization. We will follow the first 2-3 chapters of Tristan Needham’s “Visual Complex Analysis.” Here is a review of the book:
“This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book’s intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book’s use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.”
Prerequisites: Calc 126
Mathematics of Quantum Mechanics
Mentor: Curtiss Lyman
Description: Ideas from quantum mechanics and mathematics have long influenced one another, and many impactful results and theorems have come about due to this exchange of ideas. Unfortunately though, mathematicians and physicists often use different language and notation to describe identical concepts, and this can often make topics in these fields more confusing than they have to be. In this reading project, we will work through the first few chapters of “Quantum Theory for Mathematicians” by Brian C. Hall to both develop a basic understanding of quantum mechanics, and also to bridge the gap between mathematics and physics. This project is designed to be accessible to anyone with some exposure to both calculus and linear algebra, but if we ever encounter any unfamiliar topics (which will likely happen, and that is okay!), there are a number of supplementary resources beyond the recommended text that we can use to fill in any gaps. If you have an interest in the nature of the universe though, this is the project for you!
Prerequisites: Math 126 & Math 208
Derivatives Markets: Options and Futures
Mentor: Junaid Hasan
Description: Have you wondered what are options and how can contracts have value incorporated in two dimensions: the price and the time. We will delve into this world with a more rigorous bent, and hope to unwind some mysteries. First few weeks we will be reading through [1] and/or [2] and the last couple of weeks maybe do some Monte Carlo simulations.
Remember this is a reading group and not investment advice.
[1]: John C Hull, Options Futures and Other Derivatives
[2]: Sheldon Natenberg, Option Volaitility and Pricing
Prerequisites: Math 208, Math 207
Intermediate Level: May require Math 300 (proofs) and possibly other 300-level courses
Geometrical Explorations
Mentor: Bryan Lu
Description: Ever notice that maps of the world always look a little strange? For instance, land near the north/south poles is almost always drawn super large, and straight lines on a map are almost never the shortest path between those point, unlike in a plane. In this project, we will try to explain these phenomena and continue the student’s study of curves and surfaces, now focusing on the intrinsic geometry of these shapes in three-dimensional space. We will start with thinking about curves and figure out how to measure how “twisty” or “curvy” a curve is, and progress to analyzing surfaces and what “straight lines” look like on them. We will follow Andrew Pressley’s Elementary Differential Geometry and emphasize being able to visualize and compute with examples.
Prerequisites: MATH 126 and MATH 208. Analysis experience (MATH 327) encouraged but not necessary.
The Combinatorics and Geometry of Kaleidoscopes
Mentor: Michael Zeng
Description: Have you ever found the scenes inside a kaleidoscope mesmerizing? If so, you’ve got yourself in great company, because great mathematicians think so too. The word ‘kaleidoscope’ is termed from Greek roots meaning “observation of beautiful forms.” Mathematicians have observed that the beautiful symmetries that appear in a kaleidoscope are captured by something called reflection groups, and these groups appear in many areas of mathematics: algebra, geometry, combinatorics, harmonic analysis, etc., leading to many bizarre coincidences. In this project, we will explore these many different guises of reflection groups. This project can either serve as an illustrated introduction to group theory, or to either of the listed areas depending on interest. Recommended references are Mirrors and Reflections – The Geometry of Finite Reflection Groups (Borovik & Borovik) and Combinatorics of Coxeter Groups (Björner & Brenti).
Prerequisites: Basic linear algebra and 3D calculus are required. Experience with proofs will be highly beneficial.
Ideals, Varieties, and Algorithms
Mentor: Andrew Tawfeek
Description: In this project we venture into a growing classic book in mathematics dedicated to peering into the geometry behind the notions of “groups, rings, and fields” and using this geometry to tackle and understand real-world problems. We’ll not only learn neat and deep mathematics, but explore using computational tools, and investigate various applications of mathematics used everyday.
Prerequisites: Prerequesties are linear algebra and a proof-oriented course.
Introduction to Lie Groups
Mentor: Varun Shah
Description: Can we describe the “symmetries of a sphere”? Every rotation of the sphere can be described by a three-dimensional orthogonal matrix, and these matrices together form the three-dimensional orthogonal group — a Lie group!. Group theory provides a mathematical framework to study symmetries of discrete objects like a deck of cards as well as infinite geometric objects like high-dimensional spaces and spheres. The symmetries of geometric objects are often seen to vary smoothly and this is captured by the idea of a Lie group. Lie groups and Lie theory form a foundational part of modern mathematics with a very interesting history and finds applications to areas such as representation theory, mathematical physics, differential equations and algebraic geometry. In this project we will study various concrete Lie groups (such as the group of symmetries of a sphere), and develop tools to study how their geometric and algebraic structures interact. We will begin by reading Naive Lie Theory by John Stillwell, and if time permits, take detours to see how Lie theory connects to other interesting mathematics described above.
The only prerequisites are a good handle on calculus and linear algebra; we will learn all the required group theory and topology along the way!
Prerequisites: Math 300
Solving polynomial equations and more
Mentor: Ting Gong
Description: In an effort of finding formulas for solving polynomial equations, people developed many techniques in algebra and geometry. In this project we study Galois theory which is developed to tackle the quintics (and many variants and applications depending on the interest). The variants include the basic Galois theory, the geometric version or ramification theory in number theory.
Prerequisites: Math 300 or equivalently Math 134 is required. Math 402 is recommended but not required.
Mathematical Finance – How Math Ru(i)ns Wall Street
Mentor: Linhang Huang
Description: In this project, we will learn some basic concepts of mathematical finance and how to think about stocks and other financial products mathematically. We will also talk about how the misuse of the math models could lead to devastating impacts in real life. The book “Arbitrage Theory in Continuous Time” by Tomas Björk will be used to learn the basic mathematical ideas and we will also read some new reports on quants’ role in the 2008 Financial Meltdown. This project is meant to be a gentle introduction to the field so no prior knowledge of finance or measure theory is needed. Some experience in numerical simulation can be helpful.
Prerequisites: MATH 394 or MATH 392
Investigating Number Theory
Mentor: Bryan Lu
Description: What numbers are the sum of two perfect squares? How can people hide their darkest secrets in plain sight? How many primes are there below 17^17? This and other questions are in the purview of number theory, a rich area of study with tons of questions that are easy to ask and sometimes very difficult to answer! In this project, we will attempt to answer some of these questions and learn a lot about the structure underlying the integers that you know and love. This project is very flexible and can accommodate a wide range of mathematical backgrounds. Number theory is a very accessible subject in which one can develop a lot of theory from a few basics, but it can also lead to deep insights about the nature of various algebraic structures. A lot of emphasis will be placed on being able to compute with examples, either by hand or with a computer. Depending on the interests of the student, this project can go in a lot of different directions — as an endpoint, we might explore cryptography, different number systems, reciprocity laws, or any other topic the student finds interesting!
Prerequisites: MATH 300. Any other algebraic experience (MATH 402 especially) is helpful but not necessary.
Analytic Number Theory
Mentor: Zawad Chowdhury
Description: Prime numbers distribute amongst the natural numbers in a random manner, making it impossible to give a formula for all primes. But if you look at the proportion of numbers less than some fixed number x which are prime, that proportion seems to be about x/log x. And as you take bigger and bigger values of x, this proportion matches x/log x better and better! This result, called the Prime Number Theorem, fascinated mathematicians for hundreds of years. It led to the development of complex analysis and was the reason Riemann defined his now famous Zeta function. But nowadays, we have a proof that is accessible to anyone familiar with elementary calculus (though it is intricate and will take some effort to read)!
This question, and others like it (such as the Twin Prime conjecture or the Riemann Hypothesis, if you want to start with the hardest questions) are the study of Analytic Number Theory. In this DRP Project, we’ll try to take a first step into the field using Tom Apostol’s “Introduction to Analytic Number Theory”. Even if you have no prior experience with Number Theory, you’ll end the course with some knowledge about the ideas of fourier analysis, asymptotics, and perhaps an elementary proof of the Prime Number Theorem! Depending on prior background and interest, we can take some detours into other topics such as finite groups and their characters, quadratic reciprocity or the Riemann zeta function.
Prerequisites: MATH 300 (experience with proofs). MATH 301 helpful but not required
Advanced Level: Students who have taken multiple upper-level 400 level mathematics courses
Algebraic Curves, an introduction to algebraic geometry
Mentor: Ting Gong
Description: Algebraic curves are geometric objects of dimension 1 defined by polynomials. Because of its low dimension, it owns very special properties. And it is a good setting for people to get hands on tangible examples in the infamously abstract subject of algebraic geometry. The goal of this project is to get through the famous theorems in algebraic geometry such as Bezout theorem and the Riemann Roch theorem. We are going to use the book Algebraic Curves by Fulton.
Prerequisites: Math 402
Exploring algebraic number theory
Mentor: Bryan Boehnke
Description: The prime number 13 can be written as a sum of two square integers (13 = 2^2 + 3^2) while the prime number 19 cannot. What else can we say about the prime numbers that can be written as a sum of two squares? Are there infinitely many? If so, can we classify them?
While the statements of these questions only involve the integers at first glance, we can find answers by utilizing our knowledge of abstract algebra to study certain larger rings containing Z. In this project, we will begin by studying these more general “rings of integers” and can see how such settings where numbers like 5 and 13 are no longer “prime” can be useful for solving problems.
Depending on your background and interests, further topics could include factorizations of ideals in rings of integers, lattices and the geometry of numbers, applications of continued fractions, or connections with algebraic geometry.
Prerequisites: Math 402
Graphs, Optimization, and Transportation Networks
Mentor: Cameron Wright
Description: If you’re anything like me, you’ve probably used (and appreciated) Seattle’s public transit network. You may have even noticed that, at its core, a transportation network can be modeled as a graph in the combinatorial sense. With this observation, a little knowledge of graph theory and optimization opens a whole toolkit of mathematical insights about transit networks and how to optimize them. In this project, we will start from this model and will learn about how to optimally design, run, and grow a transit network into a useful and efficient system. We will begin with a review of the necessary graph theory and optimization concepts before diving into modern approaches to transportation modeling. Prerequisites are minimal, but students will benefit from a degree of mathematical maturity.
Prerequisites: MATH 208, MATH 407/8/9 (optional), MATH 461 (optional), MATH 381 (optional)
Spectral Sequences
Mentor: Jackson Morris
Description: Spectral sequences are a nifty tool for computing derived functors such as homology and cohomology. If you are comfortable with homological algebra, then they are very efficient at spreading out a computation into more manageable pieces coming from chain complexes: it is then up to the user to assemble these pieces together and get a desired answer. We will peruse spectral sequences from an example based point of view, sourcing from algebra and topology.
Prerequisites: Undergrad algebra sequence. Some homological algebra/topology would be nice
Introduction to Algebraic Geometry
Mentor: Haoming Ning
Description: The field of algebraic geometry occupies a central place in modern mathematics, with profound connections to multiple fields such as number theory, topology, and complex analysis. It has the reputation of being incredibly abstract, but in return, it allows us to prove far-reaching conclusions. For example, Wiles’ proof of the longstanding conjecture known as Fermat’s Last Theorem uses techniques in algebraic geometry, and demonstrates the power of this subject.
In this project, we will begin building the foundations of algebraic geometry by learning the language of algebraic varieties, with an emphasis on geometric intuitions. We will follow the excellent introductory textbook “Algebraic Geometry” by Robin Hartshorne, and this will be supplemented by other texts depending on your background. If you are an advanced undergraduate looking to learn more graduate math or even apply to graduate school in math, this will be a very useful and enriching project that’s perfect for you!
Prerequisites: Required: modern algebra (MATH 402-404 or equivalent). Preferred but not strictly necessary, in the order of importance: commutative algebra (parts of MATH 504-506), point-set topology (MATH 441), complex analysis (MATH 427).