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.

Combinatorial Topology

**Mentee:** Talia Becker Calazans

**Mentor:** Maxine Calle

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.

Algebraic Curves

**Mentee:** Molly Bradley

**Mentor:** Oualid Merzouga and Alvaro Pintado

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.

Algebraic Number Theory

**Mentee:** Angela Cai

**Mentor:** Deependra Singh

Modeling the Dynamics of HIV Disease Progression with Partial Differential Equations

**Mentee:** Sudhish Devadiga

**Mentor:** Xinxuan Wang

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.

Probabilistic Machine Learning

**Mentee:** Iris Horng

**Mentor:** Leonardo Ferreira Guilhoto

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.

Studying the Geometric Properties of Complicated Spacetimes with Curvature Flow

**Mentee:** Sophie Kadan

**Mentor:** Hunter Stufflebeam

Lie Groups and Representation Theory

**Mentee:** Jash Kakadia

**Mentor:** Tianyue Liu

Probability Theory and the Infinite Monkey Theorem

**Mentee:** Dylan Marchlinski

**Mentor:** Jae Ho Choi

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.

Algebraic Geometry

**Mentee:** Eric Myzelev

**Mentor:** Marc Muhleisen

Nonstandard Analysis

**Mentee:** Thomas Rainow

**Mentor:** Krishan Canzius

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.

Giving Credit Where Credit is Due: Escape Rooms and Shapley Values

**Mentee:** Elan Roth

**Mentor:** Ryan Brill

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.

Study of Minimal Surfaces

**Mentee:** Arjun Shah

**Mentor:** Jacob Van Hook

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.

Tropical Geometry

**Mentee:** Ethan Soloway

**Mentor:** Avik Chakravarty

An Invitation to Continuous Logic

**Mentee:** Eric Tao

**Mentor:** Jin Wei

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!

Algebraic Geometry

**Mentee:** Chenglu Wang

**Mentor:** Yidi Wang

Cellular Sheaf Theory and Applications in Robotics

**Mentee:** Ling Xu

**Mentor:** Miguel Lopez

Small Examples in Category Theory

**Mentee:** Eric Yu

**Mentor:** Benjamin Keigwin

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.

General Concepts on Machine Learning and Optimization Techniques

**Mentee:** Xinkai Yu

**Mentor:** Shyam Sankaran

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.

Theory and Applications of Topology

**Mentee:** Larry Zhang

**Mentor:** Andres Mejia

Pontryagin Duality of LCA Groups

**Mentee:** Darren Zheng

**Mentor:** Travis Leadbetter

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.