Embedded Files
Fall 2026 Project
Fall 2026 Project
Each project is led by a graduate student mentor and matched with one or more undergraduate participants. Projects may be based on textbooks, lecture notes, expository articles, or research papers, depending on the topic and participants' backgrounds.
Below are the current projects and participants for Fall 2026.
Simple Random Walks and the Heat Equation
Simple Random Walks and the Heat Equation
Mentor: Roan James
Participants: Logan Singh · Morgan Townley · Jiajun Du · Rosie Deng · Ruiying Ma · Nguyen Khanh Linh · Samson Lombard
Area: Probability
About the Project
About the Project
We will talk about the basics of simple random walks (SRW's) and the heat equation. We will follow a book by Greg Lawler, which I have a copy of. If there is time, we may discuss more advanced topics like Brownian motion.
Prerequisites
Prerequisites
Math 201 - Probability
Reading Materials
Reading Materials
Random Walk and the Heat Equation
What You Will Learn
What You Will Learn
Recurrence and transience of simple random walks; applications of PDEs.
Brownian Motion and Stochastic Calculus
Brownian Motion and Stochastic Calculus
Mentor: Chongzi Chen
Participants: Logan Singh · Morgan Townley · Jiajun Du · Rosie Deng · Ruiying Ma · Nguyen Khanh Linh · Samson Lombard
Area: Probability
About the Project
About the Project
The plan is to introduce the basic ideas of continuous-time stochastic processes through Brownian motion and stochastic calculus. We will begin by viewing Brownian motion as a continuous analogue of random walk and studying some of its basic properties, including martingales and quadratic variation. We will then introduce the Itô integral and Itô’s formula, and use them to study some basic stochastic differential equations. If there is time, we can discuss applications such as the Black–Scholes model and the Feynman–Kac connection between stochastic processes and partial differential equations.
Prerequisites
Prerequisites
Familiarity with calculus and basic probability is essential. Some experience with rigorous mathematical arguments or real analysis would be helpful but is not required.
Reading Materials
Reading Materials
Stochastic Calculus for Finance II: Continuous-Time Models — Steven E. Shreve
Markov Chains for Sampling and Generative Modelling
Markov Chains for Sampling and Generative Modelling
Mentor: Aryan Dugar
Participants: Joseph Tran · Ruiying Ma
Area: Probability, analysis
About the Project
About the Project
I am interested in topics broadly related to probability and stochastic processes. While the precise topic would be decided after we understand the group’s background and interests, there are several directions we can pursue/interweave:
1. Discrete-time Markov processes: such processes arise in various domains and as such form a general class with interesting properties. Topics of interest include the Markov property, stopping times, invariant measures, couplings, total variation metric, etc., across toy models such as the random walk and the Ising model.
2. Stochastic calculus: this area is the study of continuous-time stochastic processes. Topics of interest include construction of Brownian motion, definition of the Ito integral, the Ito lemma, the Fokker-Planck equation, etc.
After developing relevant tools from the above readings, we can also spend time to work on the recently buzzing field of…
3. Applications to modern statistical learning theory, sampling and generative AI: there are several theoretical guarantees supporting the algorithms underpinning generative AI, sampling and deep learning. Representative examples include MCMC methods, diffusion models and neural networks. Along this direction, we will go through the proofs of several such results. Interested students can also do simulations/build their own models!
Prerequisites
Prerequisites
I would suggest students to have some background in proof-based mathematical analysis. A course in undergraduate-level probability would also help you. While these are *not* strictly required, it would help you to maximally benefit from the project given our time constraints.
Reading Materials
Reading Materials
Markov chains and mixing times - Levin and Peres (available online)
Log-concave sampling - Sinho Chewi (book draft available online)
Stochastic differential equations- Oksendal
A mathematical introduction to diffusion models - Jianfeng Lu (available on ArXiV)
Shai and Shai - understanding machine learning
What You Will Learn
What You Will Learn
I hope that students develop a feel for useful concepts and proof strategies in the project, and are able to have a perspective on the kind of mathematics that is required by modern algorithms.
Introduction to Group Representation Theory
Introduction to Group Representation Theory
Mentor: Lippus Wenao Liu
Participant: Zekuan Guo
Area: Algebra
About the Project
About the Project
We will learn about the theory of finite group representations. More specifically, we will learn about the character theory and use it to prove the famous Burnside's p^a q^b theorem.
Prerequisites
Prerequisites
Linear algebra, basic group theory.
Reading Materials
Reading Materials
Serre, Linear Representations of Finite Groups
From Categories to Algebraic geometry
From Categories to Algebraic geometry
Mentor: Akshay Sant
Participants: Zekuan Guo · Showmee Zhou · Logan Singh
Area: Algebra, Number theory. Algebraic geometry, Commutative Algebra
About the Project
About the Project
This project will introduce some of the basic ideas of category theory and explore how they naturally appear in algebraic geometry. We will begin with categories, functors, natural transformations, and universal properties, and then study basic algebraic geometry through affine varieties, polynomial equations, coordinate rings, and morphisms. Depending on the time and how we make progress, we might also cover some basics of schemes(Hartshorn chapter 2).
Prerequisites
Prerequisites
MATH 236/236H (Abstract Algebra I) or equivalent is recommended. Students should be comfortable with mathematical proofs and basic linear algebra. Motivated students with MATH 235 and suitable proof experience are also welcome. No prior knowledge of category theory or algebraic geometry is expected.
Reading Materials
Reading Materials
The main reference will be Ravi Vakil’s The Rising Sea: Foundations of Algebraic Geometry and Hartshorne's chapter 1 and if time permits, some content from chapter 2
( https://link.springer.com/book/10.1007/978-1-4757-3849-0).
What You Will Learn
What You Will Learn
Students will become familiar with the basic language of category theory and with fundamental examples from algebra and geometry. They will learn how polynomial equations can be studied through algebraic objects such as rings and ideals, and how categorical ideas such as functors and universal properties arise naturally in this setting.
Quantum Information
Quantum Information
Mentor: Xingfa Liu (Alex)
Participant: Autumn Weaver
Area: Linear Algebra, Fourier Analysis, Quantum Mechanics (from Physics)
About the Project
About the Project
Establish the fundamental language of quantum information sciences, including topics such as 1. Unitary dynamics and quantum gates 2. Quantum information theory 3. Quantum optics and quantization of the electromagnetic field and 4. Open quantum systems
Prerequisites
Prerequisites
Some formal study of linear algebra (basis, operators, matrix representations, etc.) and basic understanding of Fourier series. At least a course in quantum mechanics is also recommended. You don't have to be a master at these topics. It just helps a lot if you have seen it (even if you have forgotten the majority), and we will review these together.
Reading Materials
Reading Materials
I plan to use a combination of Professor Gabriel Landi's lecture notes for PHYS 532 (a grad course for quantum information and optics) and Nielsen and Chuang's Quantum Computation and Quantum Information
Page updated
Google Sites
Report abuse