Winter 2024
The following are the projects held during the Winter 2024 quarter. 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 on their own time as well.
Polyhedra
Mentor: Andrew Aguilar
Mentee: Annie LeGrand
Description: Polyhedra have long been a part of human history; appearing in art, engineering, medicine, chemistry, architecture and especially mathematics. Mathematicians have studied polyhedra for thousands of years which means much of the interesting modern mathematics is rooted in their study. In this project we will explore the geometric properties of polyhedra using some less ancient ideas than you probably saw in a high school geometry class. Our main topics of interest will be constructions and existence, symmetry groups, and Euler characteristic but there’s plenty more to see when it comes to polyhedra. For reference we will use selected sections from the book “Polyhedra” by Peter Cromwell. Given the geometric nature of the topic we will spend a lot of time looking at pictures and studying figures. So this project would be great for anyone interested in seeing some beautiful math while playing with shapes!
Prerequisites: None
An Introduction to Combinatorics
Mentor: Julie Curtis
Mentee: Heming Gao
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
Mathematics for Sustainability
Mentor: Haoming Ning
Mentee: Jayrin Xie
Description: In this project, we will be reading the thought-provoking book “Mathematics for Sustainability”. The book itself is intended for a very general audience, and aims to build mathematical reasoning in the context of real-world problems around the question of sustainability. Regardless of your background in math, the reading is designed to engage you in a mathematical and quantitative approach to studying and embracing the idea of sustainability.
Prerequisites: None! Less background is preferred.
Beginner Level: May require some Calculus
The Mathematics behind Quantum Computing
Mentor: Cordelia Li
Mentee: Varun Vijayababu
Description: Quantum computing has become a big term nowadays that we always hear in the news. Due to special properties of quantum states, quantum computing offers strong computational power. For example, it can find prime number factorizations within polynomial time, which is impossible with classical computers.
Although quantum computing may sound complex, we could actually describe quantum computing theory in our familiar math terms. For example, basic quantum bits can be viewed as special vectors with tensor products, and quantum gates are represented by unitary matrices that can be applied to these vectors. As a new research area, quantum computing is incredibly interdisciplinary, and it involves a wide range of math concepts, including encryption algorithms, combinatorial optimizations, Lie algebra, functional analysis, and more.
In this project, we will first focus on the fundamental math principles underlying the fancy term “quantum computing,” following the guidance of the book “Quantum Computing: From Linear Algebra to Physical Realizations” by Nakahara and Ohmi. Afterwards, depending on the participant’s interests, we can delve into more advanced topics, such as quantum Fourier transform, quantum error correction codes, or some actual implementations of quantum algorithms on real quantum computers.
Prerequisites: The student would need to know basic linear algebra; no physics background required.
Exploring Upper-Level Math
Mentor: Kevin Tully
Mentee: Grace Zhou
Description: Are you interested in math but aren’t sure if you’d like to become a math major? Then this project is for you! It’s a low-stakes way to see what pure math is like. Following the book “Transition to Advanced Mathematics” by Diedrichs and Lovett, we’ll learn about how to write proofs, discuss the distinctive aspects of upper-level math, see the vocational possibilities the math major opens up, and explore any other topics that interest you. Some background in calculus would be helpful, but only mathematical curiosity is required.
Prerequisites: Math 124 and 125 would definitely help, and Math 208 couldn’t hurt but isn’t required.
Intermediate Level: May require Math 300 (proofs) and possibly other 300-level courses
Elliptic Curves and its Applications
Mentor: Ting Gong
Mentee: Rashad Kabir
Description: Elliptic curves have been heavily used in modern mathematics and computer science. It is a special type of curve with an algebraic structure. In mathematics, it is closed related to algebraic geometry and number theory – even the proof of Fermat’s Last Theorem heavily studies this geometric object! Elliptic curves are also used in cryptography. In recent years, people have been studying abelian variety (a generalization of elliptic curves) cryptography and had a huge breakthrough in the past year. In this project, we are going to explore the structure of elliptic curves. Depending on the interest and knowledge of students, we can start with Rational Points on Elliptic Curves by Silverman and Tate, or The Arithmetic of Elliptic Curves by Silverman. If time and interest allows (depending on the student), we can hop into abelian varieties and the moduli of elliptic curves, or abelian variety cryptography.
Prerequisites: Math 300, other courses are encouraged, such as algebra and topology, but they are not required. We can learn as we go.
Lie Groups are Not a Lie. They are Everywhere.
Mentor: Michael Zeng
Mentee: Hisham Bhatti
Description: Our world is filled with symmetries. From the beautiful geometric symmetry exhibited in crystals and snowflakes, to the fundamental symmetries that govern the laws of physics, understanding these patterns is key to deciphering the language of nature and mathematics. In this project, we will embark on a journey to explore the mathematics that governs the symmetries around us.
We start with an introduction/review of groups and manifolds. Then, we will learn the definition of a Lie group and compute explicit examples in their matrix forms. We will also see how these groups interact with other objects through group actions, leading to their representation theory and concepts like homogeneous spaces and orbifolds. We will also explore how the Lie groups interact with their associated Lie algebras.
Background in linear algebra is highly preferred. Suggested texts include Lie Groups, Lie Algebras, and Their Representations by Hall and Lie Groups: A Problem-Oriented Introduction via Matrix Groups by Pollatsek.
Prerequisites: MATH 318 or similar linear algebra courses
Mathletics
Mentor: Maddy Brown
Mentee: Minh Tran
Description: Analytics have become an increasingly large part of how we consume sports games and content. From situational win probabilities to individual player ratings, we are always looking to stats to help us understand what’s happening and to predict what we think will happen next. In this project, we will read from Wayne L. Winston’s “Mathletics” to explore how mathematics, especially linear algebra, probability, and basic statistics, can be used to analyze sports. Depending on student interest, we may also put our Excel or coding skills to work and do some “mathletics” of our own!
Prerequisites: Math 394 (probability) and Math 208 (linear algebra)
Calculus of Variations – Theorems and Examples
Mentor: Kuan-Ting Yeh
Mentee: Julien Johnson
Description: Calculus of Variations explores optimizing quantities involving functions. Unlike traditional calculus, which deals with functions of one or more variables, it deals with functionals – functions of functions. We seek functions that extremize quantities, like minimizing energy in physics or maximizing efficiency in economics. It’s applied in physics, engineering, economics, and biology. We will start from the very beginning and build all the necessary tools along the way with theorems and examples.
Prerequisites: Math 327
Metric Space Embeddings
Mentor: Joe Rogge
Mentee: Joeph Rafael
Description: When can one metric space be embedded into another in a way that preserves all distances? This kind of embedding, called an isometry, has been studied extensively and touches diverse areas of mathematics and computer science. For an important class of metrics known as lp metrics, there are elegant classifications of isometric embeddability into lp spaces. One can also ask, what happens if you’re allowed to slightly change the distances? For the lp metrics, this question was answered by Bourgain’s logarithmic distortion theorem, which has direct applications to approximation algorithms for sparse graph cuts. Depending on student interest, we can explore either of these lines of inquiry or branch off and pursue other questions about metrics, such as hyper cube embeddings or tree metrics. In the course of reading, we will touch on polyhedral geometry, graph theory, and positive semi-definite matrices (among other topics).
Prerequisites: MATH 300 (strongly encouraged), MATH 318 (useful but not necessary)
Introduction to Group Theory
Mentor: Sarafina Ford
Mentee: Sean Kawano
Description: What do integers, invertible matrices, and a rubik’s cube have in common? They’re all groups! Group theory looks at the common structure underlying seemingly unrelated mathematical objects and seeks to understand these objects through studying this structure. 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, focusing on how these concepts show up in linear algebra.
Prerequisites: Math 208; Math 300 preferred but not required
Commutative Algebra with a view toward Algebraic Geometry
Mentor: Brian Nugent
Mentee: Lukshya Ganjoo
Description: If you like algebra and want to see how it is used in other fields or are interested in algebraic geometry then this course would be a good fit for you. We would be reading through the book Undergraduate Commutative Algebra by Miles Reid. This book is an introduction to commutative algebra that focuses on its connections with algebraic geometry.
Prerequisites: Math 402
Curve-Shortening Flow
Mentor: Jacob Ogden
Mentees: Jaedon Rich, Rohan Pandey
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.