Spring 2021
This Spring 2021, the important dates and times (tentative) for WDRP are as follows:
Wednesday, March 31st, 5:00 pm: Start-of-quarter kick-off event via Zoom.
Wednesday, April 28th, 5:00 pm: Mid-quarter event (for undergraduate students only) via Zoom.
Wednesday, June 2nd, 5:00 pm: End-of-quarter presentations via Zoom.
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
What do math majors do?
Mentee: Aditi Jain
Mentor: Maddy Brown
Description: Picking a major can be hard. Early in your college career, it’s possible that you have an interest in math, but might not be totally sure what a math major is like! This project is an opportunity for a student to try out pure mathematics without having to commit to a whole major just yet. We’ll explore topics similar to what you might see in a course like Math 300, and learn about basic proof writing and logic that mathematicians use. Since I am a former UW math major myself, we will also have the opportunity to discuss specifically what it’s like in the math major at UW! Some calculus experience is recommended, but only an interest in learning about the math major is required.
Set Theory
Mentee: Molly Stark
Mentor: Juan Salinas
Description: How do we define a number, say 1? One approach is to say it is a set with one element. How do you define a set? In this reading project we will answer this question, and in doing so build the foundations of modern mathematics. We will use Halmos’s “Naive Set Theory”.
Chip-Firing on Chains of Loops
Mentee: Jan Buzek
Mentor: Caelan Ritter
Description: A graph is a collection of nodes with edges between nodes. For example, we could define a graph as follows: each node represents a person, and two people have an edge between them if they are close friends. Now assign to each person a number of potato chips. To perform a chip-firing move, we pick a person and they give one of their chips to each of their close friends. Although this might sound like a simple game, it is closely tied to ongoing areas of research in combinatorics, algebraic geometry, and other fields. It also can be used to produce pretty pictures! (Google images of “abelian sandpile model”.) In this project, we will learn about chip-firing from the ground up, focusing especially on chip-firing on a special kind of graph, the chain of loops. No prerequisites; familiarity with mathematical rigor is encouraged but not required.
Exploring Linear Algebra
Mentee: Helen Li
Mentor: Marty Bishop
Description: Have you recently taken a linear algebra course? Are you interested in exploring more advanced techniques and applications? Then this is the project for you! We’ll let our interests determine what exactly we study, but potential topics include singular value decomposition, principal component analysis, and applications to optimization, data science, machine learning, and more!
M.C. Escher and Hyperbolic Tessellations
Mentee: Emma Favier
Mentor: Josh Southerland
Description: In the middle of the twentieth century, M.C. Escher produced a series of prints entitled “Circle Limit”. These are prints of rather bizarre looking tessellations: the tessellations shrink and seemingly vanish along the perimeter of a circle. In this reading project, we will study the mathematical underpinnings of these tessellations. Surprisingly, the ideas behind these tessellations are connected to many topics in modern mathematics. We will study the math underlying these tessellations, and at the end of the quarter, produce computer-generated tessellations.
M.C. Escher and Hyperbolic Tessellations
Mentee: James Cao
Mentor: Josh Southerland
Description: In the middle of the twentieth century, M.C. Escher produced a series of prints entitled “Circle Limit”. These are prints of rather bizarre looking tessellations: the tessellations shrink and seemingly vanish along the perimeter of a circle. In this reading project, we will study the mathematical underpinnings of these tessellations. Surprisingly, the ideas behind these tessellations are connected to many topics in modern mathematics. We will study the math underlying these tessellations, and at the end of the quarter, produce computer-generated tessellations.
Computational Ill-posed/Inverse Problems
Mentee: Michelle Tan
Mentor: Kirill Golubnichiy
Description: My research presents a new empirical, mathematical model (based on machine learning) for generating more accurate multi-day option trading strategy using new intervals, initial conditions, and boundary conditions for the underlying stock.
The Arcsine Law
Mentee: Raymond Guo
Mentor: Anthony Sanchez
Description: Suppose we play a game consisting of successive tosses of a fair coin. Every time it lands on heads you gain one point and every time it lands on tails I gain one point. Intuitively you expect to be ahead about half the time and me to be ahead the other half in a long enough game with a fair coin.
As it turns out, our intuition is incorrect! One of us will be ahead 85% of the time with a probability greater than 1/2. The exact distribution of who is ahead is given by the arcsine law. The main goal of this project is to understand and to prove this theorem using some analysis and probability theory. We will use the wonderful text, “Heads or Tails” by Lesigne.
The Arcsine Law
Mentee: Zongze Li
Mentor: Anthony Sanchez
Description: Suppose we play a game consisting of successive tosses of a fair coin. Every time it lands on heads you gain one point and every time it lands on tails I gain one point. Intuitively you expect to be ahead about half the time and me to be ahead the other half in a long enough game with a fair coin.
As it turns out, our intuition is incorrect! One of us will be ahead 85% of the time with a probability greater than 1/2. The exact distribution of who is ahead is given by the arcsine law. The main goal of this project is to understand and to prove this theorem using some analysis and probability theory. We will use the wonderful text, “Heads or Tails” by Lesigne.
Ergodic Theory
Mentee: Siddharth Kamath
Mentor: Albert Artiles
Description: Have you ever wondered what the future holds? Well, then ergodic theory is for you! Ergodic theory looks at different systems and tries to predict how they will look far into the future. We will mainly follow the book: “An Invitation to Ergodic Theory” by Cesar E. Silva. The course will be mainly focused in understanding examples and deriving the theory from them.