Winter 2022
This Winter 2022, the important dates and times (tentative) for WDRP are as follows:
Wednesday, January 5th, 5:00 pm: Start-of-quarter kick-off event via Zoom.
Wednesday, February 2nd, 5:00 pm: Mid-quarter event (for undergraduate students only).
Wednesday, March 9th, 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.
Projects and Participants
Applied Category Theory
Mentee: Josephine Noone
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!
Tropical Geometry
Mentee: Carla Braunlich
Mentor: Caelan Ritter
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 make the polynomial x^2 + y^2 – 1 equal to zero). These sets can get complicated; 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 understanding what properties it has 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, following the very friendly introductory paper “A bit of tropical geometry”, by Brugallé and Shaw, and supplementing with other resources as time and student background permit. This project does not presuppose any familiarity with algebraic geometry or polyhedral geometry. Some familiarity with linear algebra and ring theory is preferred, but not required; we will cover any missing background as the need arises.
Random walks and electrical networks
Mentee: Ansel Goh
Mentor:Peter Gylys-Colwell
Description: In this project we will explore an interesting connection between random walks on a graph and voltages induced by grounding and applying current to nodes on an electrical network.
Computational Ill-posed problems, inverse problems and Neuaral Network application
Mentee: Zheng (James) Cao
Mentor: Kirill Golubnichiy
Description: We do 2 things to offer:1. Computational ill-posed problems with math finance application .2 inverse problems for nonlinear transport equation.
Computational Ill-posed problems, inverse problems and Neuaral Network application
Mentee: Lisa Lyu
Mentor: Kirill Golubnichiy
Description: We do 2 things to offer:1. Computational ill-posed problems with math finance application .2 inverse problems for nonlinear transport equation.
Hitchhiker’s Guide to Combinatorial Topology
Mentee: Owen Jacobs
Mentor: Andrew Tawfeek
Description: In this project we’ll be adventuring through the subject of “Discrete Morse Theory”: one that fascinatingly combines topology — a subject dedicated to the study of things “morphing” from one shape to another, and combinatorics — a subject dedicated to the more puzzle-like aspects of mathematics, where you work with often rigid objects governed by little rules, and you try to understand how the pieces can fit together. In particular, this subject that we’ll explore requires little background behind desire to train your visualization techniques. We’ll study particular shapes called “simplicial complexes,” which are merely a bunch of triangles stuck together (combinatorial part), and then develop some neat tools to understand how we can mess around with the shape/morph it into various things, as well as gain insight into important properties of it as well. The material covered in this subject is important in subjects ranging from mathematics, computer science, to data analysis. We’ll be using the book by Nicholas A. Scoville on the subject.
Hitchhiker’s Guide to Combinatorial Topology
Mentee: Penghuan Xu
Mentor: Andrew Tawfeek
Description: In this project we’ll be adventuring through the subject of “Discrete Morse Theory”: one that fascinatingly combines topology — a subject dedicated to the study of things “morphing” from one shape to another, and combinatorics — a subject dedicated to the more puzzle-like aspects of mathematics, where you work with often rigid objects governed by little rules, and you try to understand how the pieces can fit together. In particular, this subject that we’ll explore requires little background behind desire to train your visualization techniques. We’ll study particular shapes called “simplicial complexes,” which are merely a bunch of triangles stuck together (combinatorial part), and then develop some neat tools to understand how we can mess around with the shape/morph it into various things, as well as gain insight into important properties of it as well. The material covered in this subject is important in subjects ranging from mathematics, computer science, to data analysis. We’ll be using the book by Nicholas A. Scoville on the subject.
A Study of Elliptic Curves
Mentee: Zach Daniel
Mentor: Juan Salinas
Description: We will explore the realm of elliptic curves and their applications: such as finding rational points on elliptic curves and elliptic curve cryptography (ECC). If time (and interest) allows, we will talk about the moduli stack of elliptic curves. We will use Silverman and Tate’s book “Rational Points on Elliptic Curves”, and other articles for the moduli of elliptic curves.
A Study of Elliptic Curves
Mentee: Ryan Seek
Mentor: Juan Salinas
Description: We will explore the realm of elliptic curves and their applications: such as finding rational points on elliptic curves and elliptic curve cryptography (ECC). If time (and interest) allows, we will talk about the moduli stack of elliptic curves. We will use Silverman and Tate’s book “Rational Points on Elliptic Curves”, and other articles for the moduli of elliptic curves.
Topology through Existence Results
Mentee: Lin He
Mentor:Alexander Waugh
Description:Imagine you have a cup of coffee. You begin stirring the coffee for a few seconds and then stop. From your perspective you believe that all the “points of coffee” inside the cup have been moved when you stop stirring. However, while every “point of coffee” may have been moved while you were stirring, a famous theorem in topology says that there is at least one “point of coffee” which returns to its original position. In fact, this implies it is impossible to have every point end up in a different position than where it started. While this is already counterintuitive, what’s even crazier is that we may not know what point is actually ending up where it started. That is, we know a “fixed point” exists, but we do not know how to find it.
This project will be an introduction to topology through such “existence results”, like the example above. After going through some preliminary topology results, we will examine the one and two-dimensional cases of the above example, the so-called “pancake theorem” (and possibly the “Ham Sandwich Theorem”), a form of the fundamental theorem of algebra, etc… We will follow “First Concepts in Topology” by Chinn and Steenrod. An enthusiastic student without experience writing proofs will be able to complete this project, but having taken Math 300 or a proof based course will help.
A Tour of Graph Theory
Mentee: Andrew Jumanca
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.
In this project, we’ll start with some classical topics such as planarity, colorings, Euler’s formula, Euler and Hamiltonian walks/cycles, and famous problems. Depending on time and student interest, other topics include optimization on graphs (e.g., shortest paths, minimum cost shipping, flows) or spectral graph theory (i.e. combining tools from graph theory and linear algebra).
Graphical Designs
Mentee: Chris Lee
Mentor: Catherine Babecki
Description: Graphical designs are a generalization of quadrature rules to graphs: they are small subsets of vertices of a graph which approximate the graph well in terms of numerical integration. We want to find more connections between graphical designs on cube graphs and error correcting codes. We may also explore random graphs.
Mostly Surfaces
Mentee: Hai Lin
Mentor: Josh Southerland
Description: We will use the textbook “Mostly Surfaces” by Richard Schwartz to explore concepts such as the Euler characteristic of a surface, fundamental groups and covering spaces, and also Euclidean, spherical and hyperbolic geometry. If time permits, and depending on the student’s interests and background, we will discuss Riemann surfaces and translation surfaces. The level of experience needed lies somewhere between intermediate and advanced. Most topics are developed from scratch in the book but a student would get the most out of this project if they feel comfortable with real analysis (Math 327-328) and are to an extent familiar with groups (covered in Math 403).
Mostly Surfaces
Mentee: Runchi Tan
Mentor: Josh Southerland
Description: We will use the textbook “Mostly Surfaces” by Richard Schwartz to explore concepts such as the Euler characteristic of a surface, fundamental groups and covering spaces, and also Euclidean, spherical and hyperbolic geometry. If time permits, and depending on the student’s interests and background, we will discuss Riemann surfaces and translation surfaces. The level of experience needed lies somewhere between intermediate and advanced. Most topics are developed from scratch in the book but a student would get the most out of this project if they feel comfortable with real analysis (Math 327-328) and are to an extent familiar with groups (covered in Math 403).