Fall 2023 Projects

Below is a list of the DRP projects from Fall 2023, sorted alphabetically by the mentee’s last name. Click on an entry to see the full project description.

Over the course of the semester we plan to study introductory topics in combinatorial topology. We will begin with some familiarization with mathematical proofs. Simultaneously, we go through introductory concepts of topology through the combinatorial method utilizing “A combinatorial introduction to topology” by Michael Henle. This will be complemented by “Topology Through Inquiry” by Su and Starbird.

Final presentation slides

This semester, we will be studying Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton. We will begin by exploring affine algebraic sets and affine varieties and build up to understanding the Zariski Topology and morphisms of varieties. 

Final presentation slides

In order to bridge the gap between number theory and algebra, we’re going to be studying the book “A Classical Introduction to Modern Number Theory” by Kenneth Ireland and Michael Rosen. Our focus is primarily on approaching number theory through the lens of algebraic number theory and arithmetic algebraic geometry. More specifically, we’ll be discussing topics ranging from congruences, the law of quadratic reciprocity, and polynomial equations over finite fields to Zeta functions.

Human Immunodeficiency Virus (HIV) infection and the resulting gradual decline of CD4 cells pose a perplexing question regarding the slow disease progression observed over a decade. My study introduces a dynamic model that relies on partial differential equations to uncover the nuanced interplay between HIV evolution and the immune system during individual infections. The model incorporates key elements such as antigenic variation, enabling HIV to evade immune responses. A pivotal concept emerging from the analysis is the "diversity threshold," representing a critical parameter influencing disease dynamics. Beyond this threshold, the model predicts uncontrolled viral proliferation, elucidating the mechanisms behind disease progression. The model identifies three distinct parameter regions aligning with observed infection patterns, offering quantitative insights into the spectrum of HIV outcomes—from rapid progression to asymptomatic infection and delayed disease development. This study underscores the utility of mathematical modeling in unraveling the intricate dynamics of HIV infection and provides a quantitative framework for understanding disease evolution.

Final presentation slides

Over the course of the semester, we hope to study concepts from statistics and machine learning such as Bayesian statistics and graphical models. We will also learn about ideas from information theory and stochastic optimization, which will lead us to explore inference algorithms and different subsets of machine learning like deep learning and large language models, as well as how topology may be used in machine learning. We plan to follow and reference “Probabilistic Machine Learning: Advanced Topics” by Kevin Patrick Murphy, supplemented by various papers on machine learning with applications to topological data analysis.

Final presentation slides

This project explores the interplay between geometry and general relativity: specifically, our goal is to explain the theoretical foundations and implications of these tools within the context of black holes and curved spacetime. We first establish a mathematical framework to comprehend the Penrose Inequality and explore spacetime curvature induced by massive bodies. To do so, we make use of techniques such as curvature flow, which allow us to study the geometric properties of black hole spacetimes. Finally, we apply these geometric principles to the study of even more complicated spacetimes in general relativity.
We plan for our independent study to cover some introductory topics in the field of representation theory and their applications to physics. We will begin with an introduction to the basics of lie groups and lie algebras, and we will spend the remaining time covering applications to various topics, such as complex geometry and gauge theory. We plan to follow a few different texts, including “Lie Groups, Lie Algebras, and Representations” by Hall and “Representation Theory and Complex Geometry” by Chriss and Ginzburg.

Throughout the semester, we explored foundational concepts and results in probability theory by following Rick Durrett’s Probability: Theory and Examples. We consolidate our insights by delving into the classic infinite monkey theorem, which states that if you give a monkey hitting typewriter keys randomly for an infinite amount of time will almost surely type any finite string. Specifically, we rigorously dissect the mathematical foundations underpinning this whimsical theorem and its implications in probability theory. Additionally, we highlight the practical applications of this thought experiment and its approach to probabilistic reasoning.

Final presentation slides 

In this project we will study the fundamentals of scheme theory. We will go through part of “The Rising Sea” by Ravi Vakil, with brief detours to learn basic concepts from differential geometry and complex geometry.

I will be presenting on topics chosen from Nonstandard Analysis, following Goldbring's UCLA lecture notes. I will first introduce the hyperreals and nonstandard extensions, discuss their properties and examples, and dive deeper into the transfer principle and convenient results which we can gain from nonstandard analysis.

Final presentation slides

You and 3 friends — Alice, Bob, and Charlemagne — enter an escape room. After a difficult hour, you collectively solve the puzzles necessary to be successful. As you exit, Alice proclaims, "I was the reason we solved the puzzles!" Bob protests, "No, it was me!" Charlemagne interjects, "You are both wrong, I am the best!" As the cunning game theorist you are, you aim to resolve this dispute with a some help from Nobel Prize winner Lloyd Shapley. Using a lot of math and a bit more simplifying, you can effectively divide credit among your friends! This talk will only use elementary linear algebra to prove this fascinating result.

Final presentation slides 

Our project aims to delve into the topic of minimal surfaces in differential geometry. After going over the basics, we will research higher dimensional minimal surfaces and what it means to be minimal in different spaces. Finally, we will look at current research in this field and some popular outstanding problems.

Final write-up

We will study the area of tropical geometry- an analogue of algebraic geometry over the semiring of the real numbers plus infinity with the operations of minimum and addition. We will begin by surveying the first few chapters of Introduction to Tropical Geometry by Maclagan and Strumfels. We then hope to study either tropical curves or tropical varieties in more detail using the same book along with some supplementary papers.

Tools from continuous model theory have gotten a lot of hype recently due to their success in proving new results in fields like functional analysis. But what is continuous model theory, and why is it useful? In this talk, I plan to give a primer into ultraproducts and applied model theory, giving a brief overview of the classical applications, then discussing how to switch over to the continuous side!

Final presentation slides

Over the course of the semester, we aim to study the language of modern algebraic geometry. We will begin with a survey of the language of variety, scheme, and sheaf. We will spend the rest of the semester examining examples like projective varieties and algebraic curves. We plan to follow Adreas Gathmann’s course notes “Algebraic Geometry,” supplemented by Ravi Vakil’s “The Rising Sea: Foundations of Algebraic Geometry.”
Throughout the semester we hope to explore applied algebraic topology, specifically cellular sheaves, and its connections to fields like robotics. We will first start with reading Jacob Hansen’s “Laplacians of Cellular Sheaves” and Hansen and Ghrist’s “Opinion Dynamics on Discourse Sheaves” to study the cellular sheaves, cosheaves, sheaf laplacians, and sheaf cohomology through different perspectives such as category theory and opinion dynamics. We will then work on Farber’s “Topology of Robot Motion Planning”, Ghrist’s “Finding Topology in a Factory: Configuration Spaces”, and other papers to examine the applications of cellular sheaf in discrete morse theory, motion planning, configuration spaces, and other topics in topological analysis.

Categories are among the most general and far-reaching constructions in mathematics. In this talk, we will motivate and define what a category is and explore some interesting properties of functors, maps between categories, using concrete examples. No prior knowledge of category theory is assumed.

Final presentation slides

In Machine Learning, we have different approaches, but essentially, they are all built on matrix calculations and gradient descent. Throughout this semester, we learned and discussed the math behind machine learning and optimization, from simple to more sophisticated scenarios. In this presentation, I will introduce how we deal with linearly separable to how we deal with linearly inseparable cases, in which I will cover the fundamental linear regression techniques up to the ideas of the kernel that helps computers to learn under different conditions of the original data.

Over this Fall semester, our plan is to explore general Topology by reading James R. Munkres’s book “Topology – Second Edition”. We will first be familiarizing ourselves with basic concepts in Topology like Topology spaces and Continuous Functions. Then, we will continue to explore interesting Topology theorems and applications throughout the book. We might also look at other interesting applications supplemented by books such as “Topology of Surfaces” by L. Christine Kinsey.

Both the discrete and real Fourier transforms have many applications, drawing from a rich theory. In this write-up, we will discuss a generalization of the Fourier transform onto locally compact abelian groups, introducing techniques in abstract harmonic analysis. We begin by talking about what it means to integrate over a group before constructing the general Haar measure. We will prove Pontryagin duality and finish by discussing some consequences of this result.

Final write-up