Spring 2022
This Spring 2022, the important dates and times for WDRP are as follows:
Wednesday, March 30th, 5:00 pm: Start-of-quarter kick-off event.
Wednesday, April 27th, 5:00 pm: Mid-quarter event (for undergraduate students only).
Wednesday, June 1st, 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
An Introduction to Arithmetic Dynamics via Examples
Mentee: Aaron Banh
Mentor: Alexander Galarraga
Description: The goal of this project is to provide a friendly introduction to a branch of math known as arithmetic dynamics, which studies the behavior of rational numbers when a polynomial is repeatedly applied. Using Sage, a mathematics software built on Python, we will generate a wealth of examples. Reading material will be given to provide necessary background, however, the lack of good, readable introductions to arithmetic dynamics means this project will be more exploratory in nature, and unfortunately reading materials will be somewhat ad hoc. After an introduction to arithmetic dynamics in Sage, topics will be chosen within arithmetic dynamics according to the interest of the student. The only strict requirement is willingness to learn how to code in Sage, although knowledge of MATH 124 is helpful.
An Introduction to Arithmetic Dynamics via Examples
Mentee: Dang Phan
Mentor: Alexander Galarraga
Description: The goal of this project is to provide a friendly introduction to a branch of math known as arithmetic dynamics, which studies the behavior of rational numbers when a polynomial is repeatedly applied. Using Sage, a mathematics software built on Python, we will generate a wealth of examples. Reading material will be given to provide necessary background, however, the lack of good, readable introductions to arithmetic dynamics means this project will be more exploratory in nature, and unfortunately reading materials will be somewhat ad hoc. After an introduction to arithmetic dynamics in Sage, topics will be chosen within arithmetic dynamics according to the interest of the student. The only strict requirement is willingness to learn how to code in Sage, although knowledge of MATH 124 is helpful.
Applied Category Theory
Mentee: Hisham Bhatti
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!
Combinatorics and Graph Theory
Mentee: Kirupa Gunaseelan
Mentor: Yirong Yang
Description: How many ways can King Arthur and his knights sit at the round table? How many five-card poker hands can be dealt from a single deck? A double deck? How many ways are there to change a dollar? How many people are required at a gathering so that there must exist either three mutual acquaintances or three mutual strangers? In this project, we will be able to find the answers to all of these questions and many more. Not only is combinatorics a beautiful subject to study in its own right, it also has a wide range of applications both in and outside of the field of mathematics. We will be following “Combinatorics and Graph Theory” by Harris et al., an incredibly fun book with great quotes and story-telling. The choice of specific topics will depend on the interest and level of the student.
Hyperbolic Geometry
Mentee: Alejandro Gonzalez
Mentor: Albert Artiles
Description: Let’s explore a world full of wonder and very different from our own. Here the Pythagorean theorem is not true and we have regular heptagons that tesselate the plane. A world that inspired the artist M.C. Escher to draw “Circle Limit III.” This project will be an exploration of the hyperbolic world. We will follow James Anderson’s book titled, “Hyperbolic Geometry” where will build the hyperbolic plane from scratch and study some of its properties.
Hyperbolic Geometry
Mentee: Angela Wei
Mentor: Albert Artiles
Description: Let’s explore a world full of wonder and very different from our own. Here the Pythagorean theorem is not true and we have regular heptagons that tesselate the plane. A world that inspired the artist M.C. Escher to draw “Circle Limit III.” This project will be an exploration of the hyperbolic world. We will follow James Anderson’s book titled, “Hyperbolic Geometry” where will build the hyperbolic plane from scratch and study some of its properties.
Projective Geometry
Mentee: Evana Sorfina Mohd Nazri
Mentor: Juan Salinas
Description: In this project we study projective geometry: first over real numbers, then over arbitrary fields. The goal is to study major theorems in projective geometry first model-theoretically, then with using tools in algebra. We will use Robin Hartshorne’s “Foundations of Projective Geometry” to study projective space and pick up language from group/ring theory, followed by chapter 1 of “Algebraic Geometry” by the same author.
Projective Geometry
Mentee: Lawrence Tan
Mentor: Juan Salinas
Description: In this project we study projective geometry: first over real numbers, then over arbitrary fields. The goal is to study major theorems in projective geometry first model-theoretically, then with using tools in algebra. We will use Robin Hartshorne’s “Foundations of Projective Geometry” to study projective space and pick up language from group/ring theory, followed by chapter 1 of “Algebraic Geometry” by the same author.
How to teach math to a computer?
Mentee: Tommy Dong
Mentor: Vasily Ilin
Description: Computers are powerful but dumb. The prover-assistant called Lean helps mathematicians harness this power by “teaching” math to a computer. Humans write code to formalize mathematical statements in a form that a computer understands, and in return the computer verifies the correctness of the statements and sometimes even assists the human.
For example, we know that a+b = b+a for integers a and b. How do we convey this knowledge to a computer? Level 4 of the Natural Number Game (https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/?world=2&level=4) does exactly that — teaches a computer that addition is commutative.
There is now an enormous library called mathlib (https://github.com/leanprover-community/mathlib) that contains all the math that humans have taught computers so far. If you want to be part of this community or just see what it’s like to teach math to a computer, join me in this project.
Topology and Groupoids
Mentee: Daniel Dou
Mentor: Alexander Waugh
Description: Imagine you have two clumps of clay. You form a ball out of one and a donut out of the other. Can you, without creating/destroying “holes” or “cutting”, deform the ball into a donut shape? You may try this, but after a short while you will suspect that it is impossible. Spoiler: it is, but how do we prove it? This is one of many motivating examples for this project. We will cover the basics of point set topology, which will give us a language to talk mathematically about “balls” and “donuts” and the process of “squishing”. As you might suspect, the “hole” in the donut is what makes it “different” from the ball. The latter part of the project will be focused on developing an algebraic method of counting holes and showing that two spaces cannot be the same if they have a different number of holes. We will be using “Topology and Groupoids” by Ronnie Brown. This project is accessible to anyone with experience writing/reading proofs.
Chip-firing on Graphs
Mentee: April Li
Mentor: Andrew Tawfeek
Description: Say you have a group of 10 people: some know eachother, and some do not. On paper, you can represent each person with a small dot, and draw an edge between two dots A and B if person A and B know eachother. Now say one person has 5 dollars, another has 2 dollars, perhaps another has -3 dollars (so they are *owed* 3 dollars), and so on, assigning some amount of money to each dot in your picture. When is it possible by only sending money from one dot to another if they’re connected, to turn this into a graph where every dot has a positive amount of money? (You can play with this idea on this website!: https://thedollargame.io/) Pursuing this game leads into the deep rabbit-hole of the mathematics of chip-firing that ventures through graph theory, linear algebra, and even touches upon fun wildly abstract areas like “algebraic geometry”. If you decide to join me on this quest, there’ll be plenty of chips to fire around and chips* to be eaten. (* Funding for this has yet to be approved.)
p-adic Numbers
Mentee: Gavin Pettigrew
Mentor: Caelan Ritter
Description: The familiar notion of distance d between two points x and y in a metric space obeys the triangle inequality d(x,y) ≤ d(x,z) + d(z,y) for all points z. But imagine a world where the following, stronger “ultrametric” inequality holds: d(x,y) ≤ max{d(x,z), d(z,y)}. Our running example of such a space will be the p-adic numbers, an extension of the integers where we declare a number to be “small” if it contains a large power of the prime p in its prime factorization.
In this “non-archimedean” world, all triangles are isosceles; every open ball is closed; every point contained in a ball is a center of that ball; and if you take a small enough step towards a point, your distance to that point does not change. This last fact in particular may feel alarming, since it means we can’t do calculus in the usual sense: breaking intervals up into tiny pieces in order to approximate slopes or areas just isn’t possible.
We will read through part of “p-adic Numbers”, by Fernando Gouvêa. This book is aimed at undergraduates, presents a broad overview of the topic (and all the myriad fields of mathematics that it relies on), and emphasizes problem-solving. To quote the preface, “the reader should be familiar with the language of congruences, with the basic theory of fields and rings, and with basic concepts about point-set topology, continuity, and infinite series”—but don’t take these prerequisites too seriously; we’ll cover any background material as needed. The most important prerequisite by far is the desire and willingness to learn fun mathematics!
Computational Ill-posed Problems
Mentee: Benjamin Jiang
Mentor: Kirill Golubnichiy
Description:My research interests are related to Ill-Posed and Inverse problems particularly focused on economic measurements. In 2015, I proposed to myself to work both analytically and numerically on a very fresh and surprising idea: to predict prices of stock options using the famous Black-Scholes equation.
In mathematical finance, the Black–Scholes equation is a parabolic partial differential equation in both time and space that models the price of common financial assets.
Computational Ill-posed Problems
Mentee: Wanchaloem Wunkaew
Mentor: Kirill Golubnichiy
Description:My research interests are related to Ill-Posed and Inverse problems particularly focused on economic measurements. In 2015, I proposed to myself to work both analytically and numerically on a very fresh and surprising idea: to predict prices of stock options using the famous Black-Scholes equation.
In mathematical finance, the Black–Scholes equation is a parabolic partial differential equation in both time and space that models the price of common financial assets.