Fall 2021
This Spring 2021, the important dates and times (tentative) for WDRP are as follows:
Wednesday, October 6th, 5:00 pm: Start-of-quarter kick-off event.
Wednesday, November 3rd, 5:00 pm: Mid-quarter event (for undergraduate students only).
Wednesday, December 8th, 5:00 pm: End-of-quarter presentations.
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 prepare for meetings on their own time as well. All meetings will be virtual.
Projects and Participants
Knot Theory
Mentee: Tess Gerrard
Mentor: Alexander Waugh
Description: Intuitively, we define a knot as any loop we can form by tangling an extension cord and connecting the ends. If we do this with two different extension cords, can we (without unplugging the cords) make one knot look exactly like the other? This is a hard question because there is a seemingly infinite number of ways to move the cords, but really no immediately obvious way of saying that we will never be able to make one look exactly like the other. The bulk of the project will explore ways we might indirectly simplify this problem and determine whether two knots are different.
Throughout this project will will use string to construct knots and will follow “The Knot Book” by Collin Adams. The level of rigor will be adjusted based on the mathematical maturity of the student, but a solid understanding of high school algebra is all that is required. If time permits, we will also study some of the applications of knot theory to understand how abstract mathematics applies in real life.
Groups and Graphs 🙂
Mentee: Qichen Xu
Mentor: Paige Helms
Description: The mathematical goal of this project is to learn about what groups and graphs are as objects, and see the ways they can relate to each other. We’ll do this by reading through a little bit of a book called “Office Hours with a Geometric Group Theorist”. There are many things we can explore once we have some idea of what these things are, so the project is open-ended, and very much about enjoying math and having fun while we do it.
Knots!
Mentee: Lily Gibbs
Mentor: Caelan Ritter
Description: Imagine this very real scenario. You have to unknot a badly tangled extension cord, but the two ends have been (irreversibly) joined together by a villain! The clock is ticking, but as you sit there frantically pulling and pushing at ends, trying to make the cord look like a circle, a bleak thought passes through your head: is it even possible?
Thankfully, more than a hundred years of mathematics has gone into trying to answer that very question (among others)! Knot theory is an active field of research with applications for biochemists, physicists, and sailors alike. For this project, we’ll read Colin Adams’ “The Knot Book”, a friendly introduction to knot theory—and mathematical reasoning—that presupposes nothing beyond high school algebra.
Applied Category Theory
Mentee: Seoyoung Cho
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, 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!
Designing Wallpapers
Mentee: Tammy Chau
Mentor: Herman Chau
Description: Have you ever stared at wallpapers and wondered how come they repeat so neatly? It turns out this is because there are only seventeen different “types” of wallpapers and we can classify exactly how they repeat! In this reading program we will look at how these different types of symmetry can be encoded in objects called “groups” and learn about the seventeen wallpaper groups. Depending on your interest and software experience, we will either code a program to generate original wallpaper designs or use existing software to illustrate the different wallpaper symmetries at the end. We will mostly be following the book “Groups and Symmetry” by M. A. Armstrong.
A Tour of Graph Theory
Mentee: Paul Abboud
Mentor: Kevin Liu
Description: Graph theory is a beautiful, visual subject with applications both inside and outside of mathematics. “Vertices” are used to model objects, and “edges” are used to model relationships between those objects. One example is maps, where vertices represent locations and edges represent possible routes. In this situation, one might be interested in how to reach certain locations or ship certain goods either as quickly or as cost-effectively as possible. Another example is networks. If vertices represent people and edges represent interactions, one can ask how closely acquainted certain people are or how quickly we would expect rumors and diseases to spread. Another application with networks is to let vertices represent websites and edges represent links between websites. By combining graph theory with some linear algebra, Google used this idea to rank websites and construct the original version of their famous search engine.
We’ll start with some classical topics in graph theory, such as planarity, colorings, Euler’s formula, and Euler and Hamilton walks. Depending on time and student interest, other topics include optimization on graphs (shortest paths, minimum cost shipping or flows, etc.) or spectral graph theory (i.e. combining tools from graph theory and linear algebra).
A Tour of Graph Theory
Mentee: Anthony Luu
Mentor: Kevin Liu
Description: Graph theory is a beautiful, visual subject with applications both inside and outside of mathematics. “Vertices” are used to model objects, and “edges” are used to model relationships between those objects. One example is maps, where vertices represent locations and edges represent possible routes. In this situation, one might be interested in how to reach certain locations or ship certain goods either as quickly or as cost-effectively as possible. Another example is networks. If vertices represent people and edges represent interactions, one can ask how closely acquainted certain people are or how quickly we would expect rumors and diseases to spread. Another application with networks is to let vertices represent websites and edges represent links between websites. By combining graph theory with some linear algebra, Google used this idea to rank websites and construct the original version of their famous search engine.
We’ll start with some classical topics in graph theory, such as planarity, colorings, Euler’s formula, and Euler and Hamilton walks. Depending on time and student interest, other topics include optimization on graphs (shortest paths, minimum cost shipping or flows, etc.) or spectral graph theory (i.e. combining tools from graph theory and linear algebra).
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Coin flips and elections
Mentee: Conor Fahey
Mentor: David Clancy
Description: Amy and Bob ran against each other to be their senior class president last year. We know that Amy won with115 votes and Bob got 85 votes. Unless we were in the room when the ballots were counted, it’s impossible for us to know if Bob had more votes than Amy after counting just 100 ballots. But, we can say that there is a 85% chance that at some point during the count Bob did have more votes than Amy. Where does this 85% come from and what changes if Amy got 515 votes and Bob got 475 votes (or any other numbers!)?
This is so-called ballot problem and can be proved by counting coin flips. We will explore this using (some of) the classic text “An Introduction to Probability Theory and its Applications” by Feller. We’ll see where it goes from there.
Multi-view Geometry
Mentee: Yuqing (Queenie) Liu
Mentor: Jessie Loucks Tavitas
Description: We will explore a fascinating and relevant application of mathematics (in particular, linear algebra): Computer vision! Multiple-view geometry provides a mathematical model for cameras, with which we can attempt to answer two questions. The first is, “If I have multiple images of the same object, can I reconstruct that object in 3D?” The second reverses the first question: “If I have a 3D object and multiple images, can I determine where the cameras are located?”
These are crucial questions whose answers are directly applied to animation (like Pixar!), 3D modeling (think home interiors, dinosaurs, cities, etc…), self-driving cars (wow! so future!), and much more. We’ll read the first chapter of “Multiple View Geometry” by Hartley and Zisserman and follow our noses after that. Our goal will be to understand the holistic idea of multi-view geometry as well as to study and enjoy the math that goes into it.
Representations and Structures
Mentee: Kaya Wooley
Mentor: Nico Courts
Description:Â The field of representation theory has its foundation in physical chemistry and the study of symmetries that arise in nature. One way to motivate groups (as in the object found in an abstract algebra course) is as the proper structures that encode information about finite sets of invertible transformations on some object or set and representation theory was formalized as a way of attempting to capture all the different ways that groups can act. Modern representation theory has gone much farther, however, and has generalized the work done by the foundational researchers Schur, Frobenius, and Brauer to representations of Lie algebras, (Hopf) algebras, and even categories! Of course the farther you go up the ladder, the more complicated things can get!
As long as a student is motivated to read and think independently, the project can be adapted to students with many different skill levels. Some understanding of groups would be a good foundation for this work, but we can fill in gaps where we need to.
Some ideas for good books include:
Fulton & Harris – Representation Theory
Peter Webb – A Course in Finite Group Representation Theory
Etingof et al. – Introduction to Representation Theory (course notes)
There are many others that can be chosen depending on your experience and interest!
Computational Ill-Posed problems, Calerman estimates and ML application.
Mentee: Mohit Bansal
Mentor: Kirill Golubnichiy
Description:Â This project examines a novel way to predict options prices up to two days in advance, utilizing a new model for the Black-Scholls Equation including new intervals, initial, and boundary conditions for the underlying stock. The Black-Scholls equation, regularized and solved as an ill-posed inverse problem for future data, allows the accurate extrapolation of option price in-formation historical data. This provides high-accuracy results, which can be improved when applying machine learning to include a larger dataset in the calculation.
Random Matrix Theory
Mentee: Mark Lamin
Mentor: Raghavendra Tripathi
Description:Â A random matrix is a matrix whose entries are random. Just like any (deterministic) matrix, we can compute the eigenvalues of a random matrix. For large random matrices the eigenvalues exhibit a universal behavior, that is, the distribution of eigenvalues are independent of how we choose the entries of the matrix. In this project, we will run computer experiments to see verify this. It would be an introduction to a wonderful landscape.