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

Reductions of Elliptic Curves in Finite Fields

**Mentee:** Quincy Alston

**Mentor:** Jianing Yang

Observe that Q is the field of fractions of Z. Consider an elliptic curve E defined over Q. One can transform E into a curve over Z by clearing denominators and using Weierstrass normal form to create an isomorphic elliptic curve over Z. If we endow Z with the Zariski topology then prime ideals of Z can parametrize the Weierstrass form of E together with the reductions of E in finite fields. Certain properties of E can be verified by confirming those properties across the reductions of finite fields. We plan to read on this method and its generalizations for other schemes and rings.

Algebraic Topology

**Mentee:** Angela Cai

**Mentor:** Marielle Ong

Formal Methods of Verification in the Context of Type Theory

**Mentee:** Kaan Erdogmus

**Mentor:** Oualid Merzouga

Understanding Optimization Algorithms for Graph Alignment Problem

**Mentee:** Yuntong Fu

**Mentor:** Xinrui Yu

Logic and Set Theory - Beyond Infinity

**Mentee:** Lex Giglio

**Mentor:** Xiangrui Luo

Computation and Complexity with Category Theory

**Mentee:** Ruxandra Icleanu

**Mentor:** Julian Gould

Algebraic Topology and the Mathematical Basis of Topological Data Analysis

**Mentee:** Elena Isasi Theus

**Mentor:** Maxine Calle

Analyzing Quantum Mechanical Systems with Harmonic Oscillators

**Mentee:** Sophie Kadan

**Mentor:** Christopher Bailey

Causal Inference and Its Applications

**Mentee:** Chenxi Leng

**Mentor:** Miaoqing Yu

Recurrence Behaviors

**Mentee:** Yiyang Liu

**Mentor:** Tianyue Liu

During this semester, we will continue to study topics in symplectic geometry by reading and solving practice problems from the lecture notes by Ana Cannas da Silva. We will focus on fixed point theorems and recurrence theorems, especially the Poincaré Recurrence Theorem and the physical example of the game of billiards.

Matroids and Combinatorial Optimization

**Mentee:** Vibha Makam

**Mentor:** Zoe Cooperband

Category Theory

**Mentee:** Eric Myzelev

**Mentor:** Marc Muhleisen

Chaos, Fractals, Dynamics and their Real Life Applications

**Mentee:** Tise Ogunmesa

**Mentor:** Yi Wang

The project will focus mainly on providing a summary of chaos theory, dynamics, and fractals. Then, real life examples ranging from fun and silly to significant and consequential will be touched upon. Thus, the project will focus on Chapters 1, 2, 5, 9, and 11 (subject to change) of the textbook: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz.

Probability Theory

**Mentee:** Evan Qiang

**Mentor:** Jae Choi

Introduction to Model Theory

**Mentee:** Elan Roth

**Mentor:** Jin Wei

Optimal Control Theory and Dynamic Programming

**Mentee:** Ayanav Roy

**Mentor:** Hangjun He

Over the semester, we hope to learn the basics of optimal control theory and use dynamic programming to solve the problem. We will first begin with linear time optimal control and talk about the Pontryagin maximum principle. Then, we will learn dynamic programming – a widely used approach to solve both linear and nonlinear control. Some interesting applications in financial economics will be introduced if time permits.

Riemannian Manifolds

**Mentee:** Arjun Shah

**Mentor:** Elijah Gunther

Harmonic Analysis and Transforms

**Mentee:** Ekaterina Skorniakova

**Mentor:** Travis Leadbetter

A Variety-Focused View of Algebraic Geometry

**Mentee:** Ethan Soloway

**Mentor:** Avik Chakravarty

We plan to study some fundamental topics in algebraic geometry with a particular emphasis on varieties. We are beginning by studying affine varieties, their inherent connections to both Algebra and Geometry via the Nullstellensatz, and the Zariski Topology defined on them. From there we will spend some time studying the sheaf of regular functions and morphisms between varieties, before defining varieties generally separate from an affine context. Time permitting, we plan to study specific varieties including projective varieties and Grassmanians. Additional topics include birational maps and an introduction to schemes. The fundamental goal is to understand varieties as objects and the type of morphisms relating them before diving into motivated examples and specific uses.

We plan to use Andreas Gathmann's Notes on Algebraic Geometry as a guide, with other textbooks such as Algebraic Geometry by Hartshorne and Commutative Algebra by Atiyah and Macdonald as supplementary reading.

Linear Algebraic Groups

**Mentee:** Santiago Velazquez Iannuzzelli

**Mentor:** Yidi Wang

Modular Forms

**Mentee:** Chenglu Wang

**Mentor:** Souparna Purohit

Exploring Algebraic Geometry

**Mentee:** Max Wang

**Mentor:** Deependra Singh

Information Theory with Focus on Quantum Entanglement

**Mentee:** Shuyi Wang

**Mentor:** Christopher Bailey

Concepts and Theories of Ricci Flow and its Consequences

**Mentee:** Ling Xu

**Mentor:** Jacob Van Hook

Foundations of Differential Geometry and Topology

**Mentee:** Eric Yu

**Mentor:** Benjamin Keigwin

Safety for Deep Learning: Conformal Prediction

**Mentee:** Boya Zeng

**Mentor:** Leonardo Ferreira Guilhoto

This semester, we plan to focus primarily on the textbook "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville.

In recent years, Deep Learning-based algorithms have been widely used in numerous life-

critical applications, such as autonomous driving and medical imaging. Therefore, it is important

for us to quantify the uncertainties of these algorithms in a provable way. Conformal Prediction

has emerged as a powerful tool to generate, instead of a single prediction, highly probable

output sets with coverage guarantees. In my presentation, I will explain the basic intuitions and

workflows of Conformal Prediction, outline the proof for its theoretical guarantees, and briefly

discuss variants and applications of this method to different settings.

A1 Milnor Numbers

**Mentee:** Zhong Zhang

**Mentor:** Thomas Brazelton

Understanding Generating Functions

**Mentee:** Darren Zheng

**Mentor:** Xinxuan (Jennifer) Zhang

Zeros of a Recurrent Sequence

**Mentee:** Mike Zhou

**Mentor:** Andrew Kwon