Applications
This Winter 2026, the important dates and times for WDRP are as follows:
Applications
Tuesday, December 2: Undergraduate student applications open. (Link Here)
Monday, December 22: Undergraduate student applications due at 11:59pm.
Friday, January 2: Latest possible day that we will announce project pairings.
Events
Monday, January 5: Kick-off Event 5pm-6:30pm.
Monday, February 9: Midquarter Event 5pm-6:50pm.
Monday, March 9: Final Presentations 5pm-6:50pm.
Mondays, January 26 and February 23: 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
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.
Explorations in Combinatorics
Mentor: Mont Cordero
Description: Have you heard of combinatorics? It’s not a word many outside of math know, but it is one of the fields of math that requires some of the least background to get started! So this can be an opportunity for you to dip your toes in math beyond calculus. Combinatorics deals with problems of counting and uses the patterns and structure of the things you’re interested in to understand them better. We will learn some of the basics of combinatorics from scratch and then pick a more advanced topic to explore (such as graphs, permutations, generating functions, or magic squares).
Prerequisites: None
Mentor: Jonas Golm
Description: Do you enjoy that split second when a problem finally clicks and you think, “Ohhh… that’s beautiful” despite having been hard stuck for a week? This project aims to chase that feeling on purpose. Join me for a problem-solving adventure built around Peter Winkler’s legendary puzzle collection compiled in his book “Mathematical Puzzles.” Each week choose a collection of puzzles that speak to you and try to solve them. No tedious definitions, no endless theorem lists – just clean, clever ideas hiding inside stories and riddles, leading you to build serious reasoning and proof skills. No prior math background needed! What you will need is an iron will, as Winkler warns, “the existence of an easy solution does not mean that a solution is easy to find.”
Prerequisites: None
Introduction to Graph Theory
Mentor: Jenny Zhan
Description: Can you travel through a town without retracing your steps? How many colors do you need to color a map so that no neighboring regions share the same color? How big does a class need to be to guarantee that at least three people are mutual friends or mutual strangers? The answers to these questions all lie in graph theory. In this project, we will follow the book “Pearls in Graph Theory: A Comprehensive Introduction” by Nora Hartsfield and Gerhard Ringel to explore the basics of graphs. Topics may include coloring, circuits, spanning trees, and more, with the specific focus depending on the student’s interest and level.
Prerequisites: None
Beginner Level: May require some Calculus
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.
Discrete Models for Physical Systems
Mentor: Charlie Magland
Description: There are many physical phenomena which are determined by forces. For example, a hot mug of coffee in a cold room will cool down proportional to their difference in temperature. If we can understand the forces at play, we can model how any system will progress. As these systems become more complicated, finding an exact equation to represent them becomes more challenging. Additionally, once some chaos and small errors are introduced exact solutions may differ from the physical realizations. We will use discrete models to explore how slight changes impact outcome, and look at applications to population growth, spread of disease, heat flow, and more.
Prerequisites: MATH 125, 207 useful but not required
Mathematical Cryptography
Mentor: Ting Gong
Description: Cryptography is a subject making information safe from eavesdropping. With the development of online technologies, data safety turns out to be more important. In this project, we are going to explore cryptography from a mathematical point of view, both classical and postquantum. The eventual goal is to understand current trends in cryptographical research and understand what role mathematics is going to play.
Prerequisites: Math 208 Linear algebra. But of course more won’t hurt
Mentor: Andrew Aguilar
Description: Polyhedra have long been a part of human history; appearing in art, engineering, medicine, chemistry, architecture and especially mathematics. Mathematicians have studied polyhedra for thousands of years which means much of the interesting modern mathematics is rooted in their study. In this project we will explore the geometric properties of polyhedra using some less ancient ideas than you probably saw in a high school geometry class. Our main topics of interest will be constructions and existence, symmetry groups, and Euler characteristic but there’s plenty more to see when it comes to polyhedra. For reference we will use selected sections from the book “Polyhedra” by Peter Cromwell. Given the geometric nature of the topic we will spend a lot of time looking at pictures and studying figures. So this project would be great for anyone interested in seeing some beautiful math while playing with shapes!
Prerequisites: Math 124 and 125
Intermediate Level: May require Math 300 (proofs) and possibly other 300-level courses
Numerical Stochastic Differential Equations
Mentor: Linhang Huang
Description: Stochastic differential equations (SDEs) are powerful tools for modeling random processes and have found wide applications in physics, biology and finance. In this project, we want to look into how to approximately solve those random equations numerically, and compare the strengths and limitations of different numerical methods. If time permits, we will also discuss Monte-Carlo sampling for SDE processes and its practical application.
Prerequisites: Some exposure to probability and experience in NumPy (Python) will be helpful
Computer-Assisted Proofs
Mentor: Bryan Lu
Description: Nowadays, mathematicians are increasingly using computers and proof assistants such as Lean to verify the correctness of their proofs. In this project, we will see what it takes to actually “convince” a computer that a proof is correct and produce some proofs of our own that can be machine-checked. It turns out that the relevant technology that makes this work is a fairly old idea and appears as a feature in every major programming language! We will talk about types, play with proof assistants, and learn some (programming) language theory and constructive mathematics. Depending on the student’s background, time-permitting, we might look at higher-order logics, verified software, or different type theories.
Prerequisites: MATH 300 (or some experience with proofs)
Mentor: Isaiah Siegl
Description: A graph consists of a set of vertices and a set edges connecting pairs of vertices. The degree sequence of a graph records the number of neighbors of each vertex. These definitions suggest several natural questions. When is a sequence of nonnegative integers the degree sequence of a graph? When do two graphs have the same degree sequence? What properties of a graph can we determine by looking at its degree sequence? How many different degree sequences can appear when looking at subgraphs of a given graph? Some of these questions have elegant answers while others remain mysterious.
Prerequisites: MATH 300
Mentor: Tony Zeng
Description: Visualizations and animations can be excellent tools in communicating mathematics. This project aims to (1) explore some of the lesser-known features of Desmos that allow it to be an extremely powerful, self-contained platform in which a math presentation can be designed and given, (2) allow students explore and read about a topic (to be determined, open to discussion) whose presentation would benefit from some kind of visualization or animation, and (3) create and give a presentation about said topic using only Desmos.
Prerequisites: Math 12X or equivalent. At least one of Math 200, 208, or 300 is preferred, but not strictly necessary. Some coding proficiency is recommended.
Mentor: Jay Reiter
Description: Since its invention, category theory has been an essential tool in modern math for phrasing new questions and unlocking new theorems. The language of categories provides elegant and concise ways of understanding topics in countless areas of math from algebra to geometry to topology. But category theory has found many applications outside of math, too, especially in computer science. Functional programming languages like Haskell provide a way to write code which can be mathematically proven to be error-free, and the theoretical underpinning of these languages is category theory. Via this application to computer science, we’ll learn about essential concepts in category theory like functors, the Yoneda lemma, adjunctions, monads, and more!
Prerequisites: Math 300; some basic programming experience would be good, too (e.g. in Java, C++, etc.)
Representations and Lie Theory
Mentor: Wolfgang Allred
Description:Welcome to the wacky world of Lie Groups! What’s a group? It’s a mathematical object that encodes symmetry. What’s a Lie group? It’s a group that also has the structure of a smooth geometric space, called a manifold. They were invented by Sophus Lie to solve problems involving symmetric systems of ODEs, but Lie theory has grown far beyond its humble beginnings and can now be found in many disparate areas of math, physics, and engineering.
Prerequisites: MATH 300 and MATH 208
GAME, SET,… SURREALISM.
Mentor: Bryan Lu
Description:In the beginning, there was naught but the empty set. On the first day, we invented the number “zero,” consisting of an empty set on the left and an empty set on the right. Thenceforth, we continued to build numbers consisting of sets of previously-built numbers, and in the infinite days since, out came all real numbers, infinite ordinals, infinitesimals, and… two-player games??? We will see how surreal these numbers get, learn how to use them to analyze combinatorial games, and open our hearts to the universality of Nim. (Sources include John Conway’s Winning Ways for Your Mathematical Plays and On Numbers and Games, among others.)
Prerequisites: MATH 300 (or some experience with proofs and knowledge of how induction works)
Smooth Groups and Linear Algebra
Mentor: Andrew Aguilar
Description: You might remember from your linear algebra class there are a couple of ways to think about matrices. You can think of them algebraically as encoding information from systems of equations, or geometrically as linear transformations. As it turns out matrices have much more complex and interesting algebraic and geometric properties. The collection of invertible matrices form a group, which means they describe symmetries of objects. Moreover, if you give them a topological structure they become manifolds in a very nice way.
In this project we will study matrix lie groups, which are collections of matrices which are both groups and manifolds. To guide us we will use “Matrix Groups for Undergraduates” by Kristopher Tapp, but depending on your background there are other resources we can include. This project would be great for anyone looking for an intro to different areas of math and how they interact to form a vibrant theory!
Prerequisites: MATH 300 and MATH 208, MATH 441 would be helpful but not necessary
Advanced Level: Students who have taken multiple upper-level 400 level mathematics courses
Introduction to Arithmetic Geometry
Mentor: Ting Gong
Description: Arithmetic geometry studies the solutions of polynomial equations over a field that is not necessarily algebraically closed, for example, the field of rational numbers and the finite field of p elements, and their finite extensions. In this project, we introduce the interesting properties this may give rise to, such as Hensel’s lemma and Mordell’s theorem. We are going to use the set of notes Bjorn Poonen produced for his course.
Prerequisites: Math 402, Math 300
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: 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).
Mentor: Juan Jose Villamarin
Description: This project explores the deep geometric ideas underlying the Hawking–Penrose singularity theorems, which explain why singularities such as the Big Bang and black holes are inevitable under broad physical conditions. Starting from Riemannian geometry—metrics, geodesics, curvature, and the behavior of geodesic congruences—we will introduce the structure of Lorentzian manifolds and the role of curvature in focusing timelike and null geodesics. With this foundation, the student will study simplified versions of the Raychaudhuri equation, energy conditions, and geodesic incompleteness, gaining insight into one of the most celebrated achievements of mathematical relativity. This project is ideal for students eager to connect advanced geometry with modern physics.
Prerequisites: Math 441 Math 442
Mentor: Zihong Lin
Description: We’ll follow Fulton’s classic Introduction to Toric Varieties to explore a beautiful playground where lattices, cones, and polytopes completely control geometric objects called (algebraic) varieties. Along the way you’ll see how to use simple combinatorial data to build and compute a rich collection of varieties. The reading culminates in the orbit–cone correspondence—the key principle explaining how a toric variety can be stratified in accordance with the faces of its polytope. Time permitting, we will selectively study more advanced topics such as divisors and line bundles on toric varieties. This is a great opportunity for motivated undergraduates to get hands-on with modern algebraic geometry through concrete, visual, and highly structured examples.
Prerequisites: A solid understanding of abstract algebra, in particular basic ring theory (MATH 402 or equivalent), is very essential. Previous exposure to algebraic varieties is desired but can be learned along the way.
