Spring 2024
This Spring 2024, the important dates and times for WDRP are as follows:
Events
Monday, April 1st: Kick-off Event 5:30pm-6:30pm.
Wednesday, May 1st: Midquarter Event 5pm-6:50pm.
Wednesday, May 29th: Final Presentations 5pm-6:50pm.
The events are mandatory: only apply if you can attend all events during the listed times. 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 as well.
A Mathematical Perspective through History
Mentor: Ting Gong
Mentee: Rowan Surkan
Description: In this project, we will explore history of mathematics by working on problems that concerned the mathematicians at the generation, and we will see the interaction of mathematics with many other areas of study: engineering, physics, arts, and history itself. This is going to focus on the maths, as well as the lives and perspectives of historical mathematicians. A good reference will be Stillwell’s book: Mathematics and its History.
Prerequisites: No prerequisites, but the student should be enthusiastic about mathematics
Singularities
Mentor: Andrew Tawfeek
Mentee: Ben Bioren
Description: You know everything about them and see them everywhere: points where chaos comes to a halt. Or, were you to flip the switch on time, points that birth chaos themselves. From your kitchen sink, to the shadow that reduces your 3-dimensional being to a 2-dimensional facsimile. How much do we know about singularities? How much can we say about singularities, those strange points where life comes to a halt? What does it even mean to be “singular”?
Prerequisites: Nothing!
Combinatorics and Graph Theory
Mentor: Grace O’Brien
Mentee: Ava Michler
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 algroithm. 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.”
Only enthusiasm required! We can adjust the content to the level and interests of the student.
Prerequisites: None
Fibonacci Numbers through Counting
Mentor: Dan Guyer
Mentee: Justin Zhou
Description: The Fibonacci numbers are ubiquitous in mathematics. While we can define them via a simple recurrence, they also appear very naturally when counting various objects. For example, the number of ways to tile an 2 x n floor with dominoes is precisely the (n+1)’st Fibonacci number. We will move onto showing how to reconstruct famous Fibonacci identities by counting the same object in two different ways. My hope is that by the end of the project my student(s) can acquire a new view and appreciation of the Fibonacci numbers. I plan to follow certain chapters of the book “Proofs that Really Count” by Art Benjamin and Jennifer Quinn. While the title of this book has the word “Proofs” in the title, it does not assume familiarity with proof techniques. Each proof, while rigorous, relies solely on directly counting objects. This skill of counting will be developed through the project.
Prerequisites: The only prerequisite is an excitement to cleverly count.
An Introduction to Combinatorics
Mentor: Julie Curtis
Mentee: Eliza Pinckney
Description: Counting is something we learn when we’re very young, and it hardly feels like mathematics. However, deceptively simple things can become very difficult to count! How many different ways are there to arrange your plants in the window? How many different teams for charades could you make with your family? How many different routes are there to walk between classrooms? The answers to these questions lie in the field of Combinatorics, along with many more beautiful ideas. Combinatorics is a broad subject which also has many applications to other fields. We will follow the book “A Walk Through Combinatorics” by Miklós Bóna to gently meander through the basics of counting and potentially graph theory. This course can be as introductory or advanced as the student desires, with many options for topics to focus on, depending on student interest!
Prerequisites: None
Polytopes and Higher Dimensions
Mentor: Josh Hinman
Mentee: Sarthak Mitra
Description: Welcome to the wonderful world of polytopes! Polytopes are solid shapes with flat sides: think cubes, prisms, or pyramids. If you’re into board games, you’ve probably seen other polytopes in the form of eight/ten/twelve/twenty-sided dice.
Since we live in three dimensions, we’re most accustomed to three-dimensional polytopes (AKA polyhedra). But why stop at three? A four-dimensional universe would have its own, four-dimensional polytopes, and each of their sides would be a 3D polyhedron! It’s hard to picture what these polytopes would look like, but with some imagination (and some math), we can study polytopes in four dimensions and even higher.
We’ll use the textbook “Lectures in Geometric Combinatorics” by Rekha Thomas. If you’re interested in geometry, drawing pictures, building models, or thinking in higher dimensions, this is the project for you!
Prerequisites: None
Learning stats with baseball
Mentor: Nathan Cheung
Mentee: Jay Kimerling
Description: Baseball is the most statistically driven sport in the world. Teaching Statistics Using Baseball by Jim Albert uses data from PitchFX and fangraphs to give a nice introduction to basic stats ideas in the context of baseball. If you don’t already, you may learn to appreciate truly how great players like Ricky Henderson, Barry Bonds, and Randy Johnson were (this is of course not an exhaustive list of great players mentioned in the book)! Probably don’t need a ton of math background to read and enjoy this textbook.
Prerequisites: Math 124
Exploring the Complex World
Mentor: Tyson Klingner
Mentee: Brinda Moudgalya
Description: This is an introduction project to the world of complex analysis. It is one of the most classical fields in existence and one of the most useful. Complex analysis concerns the study of complex-valued functions and their properties. The field began in the 18th Century with the work of Euler and Gauss, studying integration and polynomials. The field matured when Riemann proved the vital properties of complex functions and established ties to analytic number theory with the Zeta function. In this reading project, we will be flexible and start learning the foundation of the theory, which could include studying limits, integration along a contour, Cauchy integral formula, Cauchy integral theorem, Residue theorem, and much more. If we discover one exciting avenue in the field, we can read down that direction. This will be a gentle introduction to reading sophisticated, beautiful mathematics!
Prerequisites: Math 125 (Integration)
Tropical Geometry
Mentor: Caelan Ritter
Mentee: Saachi Dhamija
Description: Algebraic geometry is the study of zero sets of polynomials (e.g., the unit circle is the set of points (x,y) in the plane that satisfy the equation x^2 + y^2 – 1 = 0). These sets can get complicated fast; one way to simplify them is to forget some information about the coefficients and only remember the exponents. This is a rough description of a process known as “tropicalization”, which takes a possibly very complicated algebraic set to a much simpler polyhedral complex. The latter can be understood using combinatorial techniques, and studying its properties can tell us something about the original object. These ideas form the heart of the field known as tropical geometry.
We will study tropical curves and how they are obtained from their algebraic counterparts, starting with the gentle introductory paper “A bit of tropical geometry” by Brugallé and Shaw and supplementing with other resources as time permits. This project can easily accommodate students with a wide variety of backgrounds, from those just past pre-calculus to those with some familiarity with algebraic geometry or polyhedral geometry.
Prerequisites: n/a
Computer Assisted Proofs
Mentor: Cordelia Li
Mentee: Pranav Madhukar
Description: Recent developments in machine learning like ChatGPT have drawn lots of attention in many areas. As mathematicians, we may wonder if machines could actually prove theorems for us one day. In this project, we will look into a proof assistant programming language called Lean. In particular, we will start by looking at Lean’s textbook: Mathematics in Lean, follow the examples and code along, and we can go forward with specific topics depending on the participants’ interest.
If you want to know more about this, see Terrence Tao’s talk here: https://www.youtube.com/watch?v=AayZuuDDKP0
Prerequisites: Knowing some basic proof techniques & programming would be great, but it is okay if the mentee has no such experience.
The Mathematics of Ranking
Mentor: Maddy Brown
Mentee: Helinda He
Description: Many things in life lend themselves to ranking, from teams in sports leagues, to Google search results, and even Netflix recommendations. There is a surprising amount of methods available to rank things, most of them involving linear algebra! This project will focus on the book “Who’s #1: The Science of Rating and Ranking” by Langville and Meyer, which explores the most widely used rating and ranking techniques. Those who have taken 208 should be well prepared for this project. Bonus points for interest in sports, since many of the ranking examples in the book involve ranking teams!
Prerequisites: Math 208 (318 would be a plus but not required)
Representation theory
Mentor: Jackson Morris
Mentee: Dylan Rosenlind
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
Mathematics of Quantum Mechanics
Mentor: Curtiss Lyman
Mentees: Amelia Martin, Erik Vank
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: Calculus III and Math 208: Linear Algebra
Representation Theory of Finite Groups
Mentor: Andrew Aguilar
Mentees: Aman Thukral, William Easton
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 understanding of groups. For example, we can take complicated actions of groups and decompose them into simpler pieces. In fact using matrices gives us a way of classifying the simplest kinds of actions in a nice compact package called characters. The versatility of representation theory is huge, appearing in chemistry and physics. Not to mention the appearances in 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. Our main resource will either be online lecture notes or classical books, like Serre’s book, depending on background. Having seen groups would be helpful but is not necessary, although a familiarity with proofs is.
Prerequisites: Math 300 proofs and 208 Linear Algebra.
Introduction to Group Theory
Mentor: Sarafina Ford
Mentee: Kayra Tuncer
Description: What do integers, invertible matrices, and a Rubik’s Cube have in common? They’re all groups! Group theory studies the common structures underlying seemingly unrelated mathematical objects. A solid understanding of group theory is an important foundation for any mathematician. The goal of this project is a low-pressure introduction to groups and their properties, with a focus on how these concepts show up in linear algebra and other concrete settings.
Prerequisites: Math 208, Math 300 preferred
Curve-Shortening Flow
Mentor: Jacob Ogden
Mentee: Woorim Lee
Description: Geometric flows are an important class of partial differential equations which play a central role in geometric analysis. In a geometric flow, a geometric object such as a surface changes over time in a way determined by geometric properties of the object. Famous examples include mean curvature flow and Ricci flow. In this project, we will learn about the simplest example of a geometric flow: the curve-shortening flow, which moves a curve in the plane along its normal vector with speed equal to the curvature. Our goal will be to understand the proof of the famous Gage-Hamilton-Grayson theorem, which says that if we start with any simple, closed curve and evolve it under curve-shortening flow, then the curve eventually collapses to a point, and if we zoom in on the collapsing curve it looks more and more like a circle. No knowledge of partial differential equations or differential geometry is necessary.
Prerequisites: Math 224 and Math 300. Math 327 and Math 207 may be useful but are not required.
Enumerative Geometry, Incidence Geometry, Projective Geometry
Mentor: Michael Zeng
Mentees: Andrew Stephens, Johannes van Vliet
Description: We know that two non-parallel lines in a plane always intersects at exactly one point, but what does it mean if someone says two parallel lines “intersect at infinity?” Why does two conics always meet at 4 points? Why are there always 8 circles that are simultaneously tangent to 3 given ‘generic’ circles in a plane? Why are there 3264 conics that are tangent to 5 given conics?
In this project, we hope to develop a framework to give satisfactory answers to all the above questions. We will investigate the geometry that arises from common roots of polynomials and develop the machinery of projective geometry, with the goal of deriving classical theorems in incidence geometry in a modern framework.
Experience with proofs is highly appreciated!
Prerequisites: Proof-based courses
Introduction to Algebraic Geometry
Mentor: Haoming Ning
Mentee: Runchi Tan
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).
Algorithmic Integration
Mentor: Glenn Sun
Mentee: Samuel Sullivan
Description: Derivatives are easy, but integration is hard. You know this from experience – differentiation is basically just chain rule, whereas integration involves creatively applying integration by parts, recognizing trig integrals, or even techniques like Feynman’s trick. But then how can Mathematica can do it instantaneously? Algorithms for integration are actually very cool and involved a lot of deep mathematics to develop, and surprisingly, the solutions to this analytic problem lie in algebra.
We will follow the book Modern Integration by Brent Baccala, at least in the beginning. (Disclaimer: We’ll be learning together! This is not a topic that I’m deeply familiar with yet, though I work in some adjacent areas.)
Prerequisites: Math 402 would be good. Other algebra courses may be helpful but not required. Programming not expected.