Current DRP Projects

Mentor: Chenxin (Derrick) Lyu

Mentees: Silvia Ban and Eseosa Belo-Osagie

Description: 

This project introduces students to stochastic differential equations (SDEs) and their applications in finance. SDEs help model the random behavior of stock prices, interest rates, and financial derivatives. We will explore basic concepts like Brownian motion and Ito's calculus before studying numerical methods used to solve these equations, such as Euler's method and more advanced techniques like Fourier-based discretization. The goal is to understand how these mathematical tools are used in option pricing and risk management. No prior knowledge of stochastic processes is required, but familiarity with calculus and probability will be helpful.

Mentor: Saranya Varakunan

Mentees: Shahla Eslami, Jessica Yu, Charelle Fon, and Jiya Ingale

Description: 

Cancer cells can adapt how they use energy, allowing them to survive and grow when resources are scarce. A well-known example is the Warburg effect, where cancer cells rely mainly on a fast but less efficient energy pathway even when oxygen is available. In this project, we will explore how mathematics can be used to study normal cell energy use (metabolism) and how cancer cells alter this to allow growth. We will start with an overview of energy metabolism, and discuss how these biological ideas translate into mathematical models. We will look at a few introductory differential equation models to build intuition, then focus on spatial models (PDE systems) that describe how nutrients spread and are consumed within tumors. Mentees will use MATLAB to simulate such systems, and possibly replicate results from the literature. Additionally, mentees will practice scientific communication by learning how to justify models and explain results clearly. No prior biology background is required.

Mentor: Harper Niergarth

Mentees: Tianyi Zhou, Erin Guerard, and Erika Ke Yu Shan

Description: 

Bijections are an important tool in combinatorics, used to count sets and prove algebraic identities. In this project, we will learn about three generalizations of bijections. First, weight-preserving bijections, where objects are assigned monomial weights, such as q^(number of cycles in a permutation), and bijections are used to prove equality of polynomials. Second, sign-reversing involutions, where objects are also assigned a sign (+/-) and we must “cancel” pairs, leaving only the meaningful objects. Third, probabilistic bijections, where our underlying sets no longer need to have the same size or weights, yet we can still prove interesting algebraic identities. There will be a heavy focus on examples and at-the-board, collaborative work.

Mentor: Zeus Dantas e Moura

Mentees: Minyang Li, Daniel Singh, Amanda Xi, and Maidah Amjad Waheed

Description:

This project explores the asymmetric simple exclusion process, a Markov chain model used to describe how particles move and interact. While it has roots in biology and physics, our focus will be on the elegant ways that probability and combinatorics intersect to explain these systems. We will look at how particles hopping on a line or a circle can be described using exact counting formulas, showing specifically how the steady-state probabilities are related to Catalan numbers and their various combinatorial interpretations.

Mentor: Saba Molaei

Mentees: Elena Thai, Aditi Raval, and Vaibhavi Agarwal

Description: 

In this project, we will study the mathematics behind origami and try to understand how a flat sheet of paper can fold into complex three dimensional shapes. We will look at the rules that decide whether a crease pattern can actually fold, and how we can describe these patterns using geometry, combinatorics, and graph theory. We will base our reading mainly on parts of Erik Demaine’s book Geometric Folding Algorithms and related lecture notes and online materials. Through guided reading and discussion, we will explore both classical results and modern algorithmic ideas, and see how mathematical reasoning helps us design and analyse foldable structures.

Mentor: Tony He

Mentees: Ming Zhe Li, Atta Amoah-Bediako, Lia Moradpour, and Dora Zang

Description: 

AI assistants like ChatGPT can now browse the web, manage your emails, and even shop online for you. But there's a problem: hidden instructions on a webpage or inside an email can trick the AI into doing things you never wanted, like leaking your private info or making unauthorized purchases. This vulnerability is called prompt injection, one of the frontier security challenges in AI. In this project, you'll learn how these attacks work and explore what researchers are building as defenses: instruction hierarchies, automated monitors, red-teaming (via reinforcement learning), etc. You'll read recent papers, follow the attacker-defender battle, and see how this connects to the broader landscape of Trustworthy AI: alignment, robustness, and safety. As AI becomes part of our daily routines, understanding these vulnerabilities and their potential solutions matters. This project gives you hands-on experience with a real, unsolved problem that affects the tools millions of people use.

Mentor: Nesa Abbasimoghaddamniasar

Mentees: Asia Celeste Mitchell, Ted Morovati, and Lennon Mercer

Description:

Have you ever wondered how critical software, like airplane autopilot systems or medical devices, can be trusted to work correctly every single time? Testing helps, but it can't check every possible scenario. In this research program, we'll explore the fascinating world of program verification: how computers can use mathematical logic to prove that a program does exactly what it's supposed to do, with no bugs hiding in edge cases. We'll read accessible articles and papers to understand how tools can automatically check programs for correctness, catching bugs that traditional testing might miss. Through guided discussions, you'll learn how simple logical statements can describe program behavior and how automated reasoning tools verify these properties. Think of it as teaching computers to be their own fact-checkers! No prior experience with logic or verification is required. Just curious about how we can make software more reliable.

Mentor: Artemiy Vishnyakov

Mentees: XinYi Ye, Mrunal Kankarej, and Emily Chu

Description: 

Every time you use your phone or laptop you’re sharing more information than you might realize. Even before you click “submit,” your device and the apps you use can collect clues about how you interact: how fast you type, how often you pause, how you move your mouse, or how long you hesitate before finishing a sentence to create patterns unique to you. These patterns -called behavioural biometrics- are increasingly used in real systems for security, health monitoring, personalization, and more. This project explores how the same behavioural data can be used in very different ways: to protect accounts from fraud, to help detect early signs of neurological conditions like Alzheimer’s disease, or to silently track and profile users without their knowledge. We’ll look at keyboard, mouse, and phone-based behavioural data, asking when these technologies are helpful, when they’re harmful, and how design choices shape their impact.

Mentor: Shufan Zhang

Mentees: Lahari Bonam, Johnathan Xu, Yunxing Teresa Zhang, and Susie Cao

Description: 

Imagine you have access to a huge trove of valuable information—say, everyone’s movie-watching habits on a streaming service. What if you could learn fun facts from that data (like which genres are booming) without anyone being able to tell what you personally watched last night? That balancing act is the heart of differential privacy for database query processing. In this project, we explore techniques that add just enough carefully-tuned “noise” to the answers a database returns so that individual users remain hidden in the crowd, while the overall trends stay clear and trustworthy. Mentees will see how computer science, statistics, and ethics come together to let organizations share insights without oversharing personal details. By the end, you’ll understand how to design and evaluate privacy-preserving queries—skills that are increasingly in demand everywhere from tech companies to public-health agencies.

Mentor: Bing Hu

Mentees: Pei Lin He, Parsa Bagheri, Suri Tian, and Jie Yang

Description: 

In this reading group, we will take an overview of drug discovery and how artificial intelligence can be used. We will look to cover the following topics and learning goals: 1. What is drug discovery? (Month 1) In this section, we will learn about both pre-clinical and clinical drug discovery - how they relate to each other and current challenges. 2. What are data sources in drug discovery? (Month 2) In this section, we will learn about types of open-source drug discovery data as well as any data challenges. 3. What are the artificial intelligence models applied in drug discovery? (Month 3) In this section, we will learn at a high-level about different types of artificial intelligence models that are applied in drug discovery. 4. What are future research directions for AI for drug discovery? (Month 4) In this section, students will begin designing their own unique research questions and directions. By the end, the goal is for students to learn to conduct independent research.

Mentor: Jiahui Huang

Mentees: Aidan Maybank and Ilmie Abeywardane

Description: 

Are you tired of working with real numbers? Do you like how much easier life becomes when you have i? Learn about complex surfaces and spaces and study their topology using Hodge theory. Hodge theory brings so much more structure to a complex space compared to a real space. We will be looking at definitions of applications of Hodge theory, the most important theory in complex analysis, algebraic geometry, even number theory. If people ask you about its applications in real life, you can say "string theory" and sound very smart, or say "quantum field theory" and sound even smarter. Wow.

Mentor: Shintaro Fushida-Hardy

Mentees: Paige Schneider, Changhong Li, Kiana Ordoukhani, and Siqi Yao

Description: 

The Poincaré conjecture is the only Millennium Problem which has been solved. It roughly states that "if all loops in a three-dimensional space can be contracted to a point, then that space is a sphere". The first goal of this reading project is to understand the statement of the Poincaré conjecture: what is a sphere? what is a space? what is a loop? The second goal of the project is to learn why the Poincaré conjecture was difficult to prove (or at least why it is not intuitively clear that it should be true.) Depending on progress and your interests, we may then consider higher dimensional analogues of the Poincaré conjecture, learn about Perelman-Hamilton's proof of the Poincaré conjecture, or investigate some other related directions.

Mentor: Lilian Gardner

Mentees: Amelia Thomas, AJ Carson, and Abdul Haseeb

Description: 

Group theory has many applications to various fields of math. This means we can study groups in a variety of contexts, including within analysis (something that isn't really touched upon in a PMATH degree). These usually come from considering group actions of infinite groups onto certain kinds of vector spaces. A natural example are amenable groups, which can be thought of as groups which don't have a "paradoxical decomposition" as in the Banach-Tarski paradox. Another example is called Property (T), which determines conditions for when a group "fixes" a subspace of the vector space it acts on. This project will study these two classes of groups, their properties, and their applications in analysis.

Mentor: Diribsa Tsegaye Bedada

Mentees: Abby Tessema, Paige Li, and Smridhi Bawa

Description:

How can we use probability to represent what we believe about the world — and update those beliefs when we see data? This is the essence of Bayesian statistics. In this project, we will explore how Bayesian ideas help us understand hierarchical models, which are used to study variability across groups (such as hospitals, schools, or experiments). We will start from simple examples — learning how priors, likelihoods, and posteriors interact — and then see how hierarchical models extend these ideas to multiple levels. Students will gain hands-on experience with Bayesian reasoning, simulation, and visualization using R. The focus will be on intuition and communication: understanding how the Bayesian framework helps quantify uncertainty and improve real-world decision-making in research.

Mentor: Zirui Wang

Mentees: Irene Chung and Julia Jia He Miao

Description: 

Financial markets trade “options,” contracts whose value depends on how a stock price moves in the future. The famous Black–Scholes model is often taught as the standard way to price options, but it relies on simplifying assumptions that don’t match real markets—especially the idea that market volatility stays constant. In reality, markets go through calm and stormy periods, and volatility tends to “cluster” (big moves tend to be followed by big moves). This project builds a more realistic option-pricing toolkit by combining two ideas: (1) a model that lets volatility rise and fall over time in a way that better reflects real data, and (2) modern machine learning to decide when it is optimal to exercise an option that allows early exercise (think: “exercise now vs. wait”). We will focus on basket-style options that depend on multiple stocks, where traditional methods often become too slow or inaccurate as the problem grows.

Mentor: Yixuan Zeng

Mentees: Dhruvi Patel and Isabella Di Nino

Description: 

Have you ever wondered how long a treatment helps patients live — and how we can know if it really works? In this project, we’ll explore how to analyze time-to-event data, like how long people stay healthy after treatment. We’ll also learn how to make fair comparisons between groups by adjusting for other factors that might affect the outcome. Using real-world data and simple statistical tools, you’ll learn how to answer questions about cause and effect in survival settings — no heavy math required!

Mentor: Aelita Huang

Mentees: Jessica Ni and Aditi Jha

Description: 

Many research questions can’t be answered well with a one-time (cross-sectional) snapshot. Longitudinal data, repeated observations on the same individuals over time, lets us study both how outcomes change and when events occur. This reading project introduces a unified way to think about time-based data through two complementary questions: (1) Change over time (e.g., how a continuous outcome such as symptoms, stress, or performance evolves, where time is a predictor), and (2) Event occurrence over time (e.g., when someone starts a behaviour, relapses, or drops out, where time is the outcome). We will read selected chapters from Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence (Singer & Willett) and focus on intuition, interpretation, and how the two approaches connect.

Mentor: Yan Yu

Mentees: Chloe Jiang, Jessica Lu, and Davina Yang

Description: 

Discover the fascinating world of hotel operations and customer experiences through data! In this project, you'll explore the exciting intersection of data science and business management by analyzing real-world hotel data. Gain valuable insights into customer behaviour, such as what influences their decisions and satisfaction, identify peak demand times to optimize hotel resources, and understand room availability issues that hotels commonly face. You'll also get hands-on experience with powerful predictive tools like LASSO and Random Forest models, learning how these methods help businesses anticipate trends and enhance operational efficiency. This project is perfect for undergraduates who are curious about how data-driven strategies shape business decisions and who want to develop practical skills highly valued in today's data-oriented industries.

Mentor: Yixing Zhao

Mentees: Raiya Minhas and Kasey Yin

Description:

Bayesian statistics is about learning from new evidence. It combines what we already believe with fresh data to update our understanding and make better decisions. This project introduces the core ideas and real-world uses of this powerful way of thinking.

Mentors: Téa Fazio and Gaia Noseworthy

Mentees: Inayah Chaudhary, Cindy Jung, Ethan Tran, Lirong Wang, and Elisa Liang

Description: 

In this project, we will explore different clustering methods and test their applications with real world data sets, such as physical ocean or disease symptom data. We will learn about how various clustering methods work, and review the mathematics and theory that define them. Then we will program and apply these methods to real data sets, visualize results and compare their performance for different types of problems. Clustering algorithms use statistics to group data based on similarities and patterns. There are many different methods, each with their own strengths and weaknesses. Clustering is widely used to simplify large datasets, identify outliers, visualize patterns, and support analysis across fields like Machine Learning, Big Data, Computer Graphics, and Bioinformatics. In studying and applying various clustering methods, you will gain skills in both programming and statistics, as well as experience in key tools within applied mathematics and computer science.

Mentor: Betty Zhang

Mentees: Katherine Chambers, Lily Li, and Aster Qianyi Huang

Description: 

In earlier work, we used ordinary differential equation (ODE) models to study epidemic dynamics at the population level. In this project, we focus on immunity-related mechanisms that drive post-pandemic dynamics. We study long-term epidemic behaviour arising from waning immunity, motivated by diseases such as COVID-19 that exhibit multiple waves. Classical epidemic models are extended to allow immunity gained through infection or vaccination to decay over time, enabling reinfection. This framework allows us to examine how the duration of immunity affects effective transmission and the timing of repeated outbreaks. We extend the model by allowing transmission rates to vary across COVID-19 variants. Selected parameters, such as incubation rates, differ between male and female subgroups. Vaccination is incorporated by reducing susceptibility and transmission. Numerical simulations of the ODE system are used to analyze epidemic trajectories in MATLAB with ode45 solver and other tools.

Mentor: James Dufresne

Mentees: Molly Xie, Zhengyu Zhu, Addysen Stewart, and Linxi Fan

Description: 

Using real stock and ETF data from Yahoo Finance's API, we will compute multiple different indicators (i.e. SMA, EMA, RSI, MACD, etc.) which we will backtest using their buy/sell signals over previous time series data and measure performance using key metrics such as average return, Sharpe ratio (return per unit risk), and maximum drawdown (largest loss from a peak), etc. We will then implement a variation of strategies that incorporate one or more indicators and test possible improvements such as adding trend or volatility filters to reduce false signals as well as adjusting different indicator parameters to improve performance. We will statistically and visually compare our approach to baseline strategies such as "buy and hold" using Python, providing insight into when and where each performs best.

Mentor: Jeronimo Valencia

Mentees: Nitya Kandadai, Hope Appelmans, Vedika Gupta, and Evan Rosen

Description: 

The standard 3×3×3 Rubik’s Cube has nearly 43 quintillion valid configurations—that is, arrangements of its pieces that can be solved using legal moves. But where does this number come from, and how can it be computed? One way to answer this question comes from group theory, which provides a mathematical framework for modeling the moves of the cube and understanding the structure of its configuration space. In this project, we will explore how group-theoretic ideas can be used to model this kind of puzzles and uncover their mathematical properties. In particular, we am to apply these techniques to count the number of valid configurations of several Rubik’s Cube variants, such as the Skewb, the Pyraminx, and the Megaminx.

Mentor: Maryam Yalsavar

Mentees: Yurim Song and Sharvi Khambaswadkar

Description: 

Many problems in science and engineering—from predicting weather to modeling how heat spreads through a material—are described using equations called partial differential equations (PDEs). Solving these equations accurately and efficiently is essential, but often very computationally expensive. In recent years, new kinds of machine learning models have shown promise as faster alternatives to traditional simulation methods. Two important examples are Transformer-based models (widely used in language and vision tasks) and State Space Models, such as Mamba, which are designed to efficiently handle long and complex sequences. This project explores how well these two types of models can learn to solve PDEs. You will compare their accuracy, speed, and reliability on a range of example problems, and investigate where each approach works best—or struggles.

Mentor: Helen Weixu Chen

Mentees: Rachel Spanglet and Gaurika Gupta

Description: 

Programming today is moving away from just typing lines of text; it’s becoming a conversation between humans and AI. However, current tools often feel rigid and disconnected from how we actually think. This project explores a more human way to build and fix software: using sketches or direct manipulation to talk to code. We are reimagining the programming experience as a creative, visual process. Imagine sketching a bug to fix it, or directly manipulating code blocks to guide AI in generating complex code.

Mentor: Paul Cusson

Mentees: Jake Edmonstone and Matthew Tchouikine

Description:

A simple sounding analysis problem with a tricky solution is the following. If f(x) is a smooth real function such that for every real number x_0, the sequence of n^th derivatives of f, evaluated at x_0, converges, is the limiting function also smooth? Give the problem a try, then take a look at the proof (from Terence Tao!) in the link below. Hint: f(x)=e^x is an example where this trivially works. https://mathoverflow.net/questions/413165/does-iterating-the-derivative-infinitely-many-times-give-a-smooth-function-whene In this research project we will work to extend the above result in two directions. First, we will generalize to iterating a linear differential operator over R^n, rather than just the standard derivative, and see when we get an analogous result. Second, we will study the dynamical properties of this iterative procedure. In the first example, e^x is a fixed point, but cos(x) and sin(x) belong to orbits of size 4. What more can be said?

Mentor: Nathan Pagliaroli

Mentees: Milagro Chen and John Kim

Description: 

Random matrices are matrices whose entries are random variables. The moments of random matrices can often be written as integrals over a group of matrices. Such integrals are computed by applying techniques involving orthogonal polynomials: collections of polynomials that satisfy an orthogonality relation. Such polynomials can satisfy a recurrence relation, whose solution can be used to compute the associated matrix integral. In this project, we would aim to extend the usual occurrence relations for orthogonal polynomials associated to random matrices by extending them for "mulit-tracial ensembles". We will start with example integrals and attempt to develop a general approach. Time permitting, we will study various applications to theoretical physics and probability theory.

Mentor: Tyler Verhaar

Mentees: Amy Xu, Sherry Xi, and Yuner Chen

Description: 

Bookmakers publish odds that can be converted into implied probabilities, but how useful are these probabilities? “The house always wins” by embedding a bookmaker margin (“vig”) into its odds, which systematically distorts the implied probability. In this project we will build a simple Bayesian model to separate two things: (1) the fair win probability (what the odds would imply without distortion) and (2) the book’s built-in edge and pricing tendencies. Using historical odds from multiple sportsbooks alongside game results, we will estimate fair probabilities and quantify how different books consistently lean high or low relative to outcomes, as well as how reliable their probabilities are. The goal is a practical, interpretable method for turning odds into better probability estimates. Materials: Bayesian inference, hierarchical models, MCMC (Stan/PyMC), proper scoring rules, and calibration plots.