Fall 2024
This Fall 2024, the important dates and times for WDRP are as follows:
Applications
Monday, August 12: Undergraduate student applications open (link here).
Friday, September 13: Undergraduate student applications due at 11:59pm.
Wednesday, September 25: Latest possible day that we will announce project pairings.
Events
Monday, September 30: Kick-off Event 5:30pm-6:30pm.
Wednesday, October 30: Midquarter Event 5pm-6:50pm.
Wednesday, December 4: Final Presentations 5pm-6:50pm.
Wednesdays, October 16, October 23, November 6, November 13: WDRP Seminars 5pm-6pm
The first three events are mandatory: only apply if you can attend all events during the listed times. In addition, mentees are expected to attend two of four seminars (described here), although more is recommended. 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
Primes and Number Theory
Mentor: Alex Wang
Description: Prime numbers are one of the most interesting topics in mathematics: at first glance, they appear sporadic and mysterious, but patterns arise as we explore their properties. We’ll investigate how they form the building blocks for arithmetic through the branch of math known as number theory. Possible topics include the Sun Tzu remainder theorem, quadratic reciprocity, and arithmetic of finite fields.
Prerequisites: None!
The Mathematics of Voting
Mentor: Grace O’Brien
Description: At first glance, it is easy to think that math and politics have no overlap. However, when we think of math as the study of logic and reasoning instead of just numbers, we see that there are many ways to connect math and political science. This project will focus mainly on voting systems. We will define properties of “fair” elections and analyze which different methods of choosing winners (simple majority, instant run-off, etc.) are fair in which ways. If we have time, we may also take a look at different methods of apportioning seats in the legislature (choosing how many seats each state should get). No particular knowledge of math or the US government is necessary, just an interest in learning about proofs in the context of voting!
Prerequisites: None! We can adapt based on the student’s background and interests
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 all!) of the books “The Joy of Abstraction: An Exploration of Math, Category Theory, and Life” by Eugenia Cheng; “Seven Sketches in Compositionality: An Invitation to Applied Category Theory” by Fong and Spivak; and “Relational Thinking: From Abstractions to Applications” by Aguinaldo, Fong, Dancstep, and Srinivasan. Along the way, you’ll catch a few glimpses of set theory, logic, abstract algebra, and more, as well as applications to science, engineering, programming, and/or social justice. No prior knowledge or experience necessary—there’s something in this project for everyone!
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.
Mathematical Paradoxes
Mentor: Mallory Dolorfino
Description: Mathematical paradoxes are abundant and really fun to learn about. Some famous ones include the Banach Tarski paradox, Russell’s paradox, the two envelopes problem, and Bell’s spaceship paradox (if you’re interested, look them up!). Some paradoxes are, in a sense, resolvable, but some require a lot more work to think about. For instance, “resolving” the Banach Tarski paradox would require rejecting the axiom of choice. Thus, thinking and learning about these paradoxes is a gateway to thinking about logic, proofs, axiomatizations, and higher levels of math. For this reason, this project would be good for both students who are considering getting more into math, or for those who have already taken a lot of courses. There are many books about mathematical paradoxes, and topics and level of difficulty will be catered to the mentee.
Prerequisites: None
Pythagorean Triples
Mentor: Bryan Boehnke
Description: The Pythagorean theorem famously relates the side lengths of a right triangle via the equation a^2 + b^2 = c^2. If we have a triangle whose three side lengths a, b, and c are all whole numbers, then we call (a, b, c) a Pythagorean triple. Examples include the triples (3, 4, 5) and (5, 12, 13).
Given this definition, we might wonder: how many Pythagorean triples are there? This project will start by considering this question, and we will find that while this question is simple to state, its answer can reveal connections between number theory, algebra, and geometry that you might not immediately expect.
From here, we can continue to explore related connections between these topics. There are several directions this project can take, and we can tailor the project to your mathematical background and interests. In particular, depending on your background, further topics could include modular arithmetic, p-adic numbers, quaternion algebras, projective geometry, Brauer groups, or quadratic reciprocity.
Prerequisites: None.
Combinatorial Games
Mentor: Caelan Ritter
Description: We will discover how to analyze (and win!) sequential games with perfect information using the power of mathematics. Along the way, we will learn about the surreal numbers. Depending on your interest and background, we may focus on “Winning Ways for Your Mathematical Plays”, by Berlekamp, Conway, and Guy, or “Mathematical Go: Chilling Gets the Last Point”, by Berlekamp and Wolfe. Suggestions for alternative texts are also welcome.
Prerequisites: None
Knots
Mentor: Luz Grisales
Description: We will read through “The Knot Book”. Here is a description of the book:
Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics.
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.
Geometry and Surfaces
Mentor: Paige Helms
Description: In our project, we’ll be exploring what a ‘surface’ is, and understanding special surfaces that are made by shapes. For example, if you take a square, tape the top to the bottom and make a long tube. Then stretch the two edge circles so hey meet, and now you have a donut! What else can we make from polygons, and what do we know about these so called ‘polygonal surfaces’?
Prerequisites: Calc 124 would be helpful, but every level is welcome!
Intermediate Level: May require Math 300 (proofs) and possibly other 300-level courses
Knot Project
Mentor: Andrew Tawfeek
Description: Do knots frustrate you? Are you overwhelmed with a desire to untangle things? Is the lack-of-a-knot your favorite type of knot? Then you must either be type-2A DNA topoisomerase or interested in learning about knot theory together in this project.
Prerequisites: Experience with 300-level mathematics recommended.
Elliptic Curves and its Applications
Mentor: Ting Gong
Description: Elliptic curves have been heavily used in modern mathematics and computer science. It is a special type of curve with an algebraic structure. In mathematics, it is closed related to algebraic geometry and number theory – even the proof of Fermat’s Last Theorem heavily studies this geometric object! Elliptic curves are also used in cryptography. In recent years, people have been studying abelian variety (a generalization of elliptic curves) cryptography and had a huge breakthrough in the past year. In this project, we are going to explore the structure of elliptic curves. Depending on the interest and knowledge of students, we can start with Rational Points on Elliptic Curves by Silverman and Tate, or The Arithmetic of Elliptic Curves by Silverman. If time and interest allows (depending on the student), we can hop into abelian varieties and the moduli of elliptic curves, or abelian variety cryptography.
Prerequisites: Math 300, other courses are encouraged, such as algebra and topology, but they are not required. We can learn as we go.
An Introduction to the Dollar Game
Mentor: Natasha Crepeau
Description: The dollar game on a graph is a simple set up – if every vertex has some number of dollars, then is it possible to move dollars around the graph in a specified way so that no vertices are in debt? This seemingly simple game has deep connections to many areas of mathematics, such as algebra, combinatorics, and algebraic geometry. In this project, we’ll talk about and define the dollar game, then highlight the surprising connections to other fields of mathematics, diving deeper into any fields of interest.
Prerequisites: Math 208, Math 403 helpful but not required
Representation Theory
Mentor: Jackson Morris
Description: Linear algebra rules! What if we made it even more interesting? In linear algebra, we study Euclidean spaces and linear transformations between them. In representation theory, we study Euclidean spaces with a type of symmetry on them (like rotating about the plane), and linear transformations respecting these symmetries. This reading course will dive into the representation theory of the symmetric group. If you liked linear algebra and want a hands on approach to some cool math, this project is for you.
Prerequisites: Math 208
Galois Groups and Fundamental Groups
Mentor: Mallory Dolorfino
Description: Two theories, one in algebra and one in topology, are remarkably similar. These are the theories of Galois extensions in algebra and Galois covers in topology. In this project, we will explore these similarities and their consequences, loosely following the book “Galois Groups and Fundamental Groups” by Tamas Szamuely (available through the UW library). Depending on the mentee’s background, we could start with the basics of both theories, or with the surprising consequences of the theories’ relationship. In any case, we will see how these two seemingly unrelated ideas interact to prove super interesting results in both topology and algebra. This project would be especially suitable for someone who wants to explore the relationship between algebra and geometry/topology.
Prerequisites: Math300 (required) Intro algebra and topology classes would help, but are not necessary
Representation Theory of Finite Groups
Mentor: Andrew Aguilar
Description: Many groups naturally appear as symmetries of objects, take for example the dihedral groups as symmetries of regular n-gons. A natural way to write these symmetries explicitly is to use matrices which act on the n-gon sitting in the plane. Representation theory of finite groups studies these “representations” of groups as matrices. This correspondence may seem trivial at first glance but actually leads us to new perspectives. For example, we can take complicated actions of groups and decompose them into simpler pieces. In fact it gives us a way of classifying the simplest kinds of actions in a nice compact package called characters. There is much versatility in representation theory, appearing in chemistry and physics; and other branches of math, like number theory, differential geometry, topology and more.
Representation theory has evolved a lot in the last 100 years so there are perspectives suitable for any background, whether you’ve never heard of a group or if you know what categories are there is something plenty for anyone to learn!
Prerequisites: Linear algebra (math 208) , group theory (math 403) would be helpful but is not necessary
Matrix Lie Groups
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 between spaces. 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: Linear algebra (math 208), group theory (math 403) would be helpful but is not necessary
Algebra, Geometry, and Computer Visualization
Mentor: Joe Rogge
Description: This project will be about the geometry of varieties, which are the solution sets to polynomial equations such as x^2 + y^2 = 1 or y = x^2, the familiar unit circle and parabola. A focus will be placed on explicit computations and visualizations, so we will primarily focus on curves and surfaces, a beautiful and classical area of study that has fascinated mankind since time immemorial and continues to be an active area of study. Our main text will be “Ideals, Varieties, and Algorithms”, the celebrated book of Cox, Little, and O’Shea, supplemented by other sources as needed and informed by student interest. We will also make use of the computer algebra system Macaulay 2, but no prior knowledge is necessary. A first course in algebra would be useful but is by no means necessary.
Prerequisites: MATH 300
Algebra and Categories
Mentor: Justin Bloom
Description: Approaching abstract algebra (groups, rings, fields, modules) from the point of view of universal properties and functors. Connecting abstract algebra back to its roots (pun intended) in solving systems of equations.
Prerequisites: Math 208 and 300 minimum. Math 402 optional but helpful.
Advanced Level: Students who have taken multiple upper-level 400 level mathematics courses
Galois Cohomology
Mentor: Ting Gong
Description: Cohomologies are geometric invariants of a space. If we smash, twist, extend a space without changing the intrinsic properties, cohomology of this space is going to remain the same. Hence, in a way, they are the “unchanging” things of a “changing” object. This concept can be used in algebras as well. We can define the Galois Cohomology to study the invariants of algebraic geometric objects, such as elliptic curves. There are many resources for this project, some of which are: William Stein’s Notes on Galois Cohomology, the book “Central Simple Algebras and Galois Cohomology” by Gille and Szamuelly, Serre’s classical books on “Local Fields”, “Cohomologie Galoisienne”, and “Local class field theory”. We are going to use some of these to study Galois Cohomology and Brauer groups, a very often studied object in modern mathematics.
Prerequisites: Math 402-404, algebra sequence, namely, groups, Galois theory, and Math 441 topology.
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).
Abelian Varieties
Mentor: Soham Ghosh
Description: Algebraic geometry is the study of solutions of polynomial equations and their geometry. One of the most important classes of geometric objects studied are projective varieties. Of special importance among these are those which have a compatible group structure. These are known as Abelian varieties. Abelian varieties play an important role in the proof of Falting’s theorem, one of whose consequences is finiteness of integral solutions to x^n+y^n=z^n for n at least 4- a major progress towards Fermat’s last theorem. In this project we will study the rich properties enjoyed by Abelian Varieties due to their group structure. Along the way we will learn any necessary Algebraic geometry.
Prerequisites: Undergraduate and/or graduate algebra