Current DRP Projects

Mentor: Gordon McNicol

Mentees: Kris Zhang and Margaret Puzio

Description: 

Cells don’t just respond to chemical signals — they can also “feel” their environment. For example, they sense whether their surroundings are soft or stiff, and this affects how they move, divide, and change shape. This process, called mechanotransduction, is key to functions like touch, hearing, and balance, and is also linked to diseases such as heart problems and cancer. In this project, students will use mathematical models to explore how cells build structures that sense and respond to their environment. These structures give cells their shape, stabilise them, and transmit forces that trigger chemical signals. Depending on student interests, the project can focus on formulating differential equation models to describe the development of these structures, studying the mechanics of how cells sense forces, or running simulations to test how different parameter choices or modelling assumptions influence cell behaviour. No prior biology knowledge is required.

Mentor: Chenxin Lyu

Mentees: Jackie Liu and Anne Zhang

Description: 

This project introduces students to stochastic differential equations (SDEs) and their applications in finance. SDEs help model the random behaviour 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. 

Mentor: Victoria Ralph

Mentees: Julia Qiu, Hanin Shamsheet Abdul Rehiman, and Lily Li

Description: 

Medical imaging (e.g. MRI, CT, or PET scans) produce detailed images of what's happening inside the body and are widely used to diagnose and monitor numerous health conditions. However, the raw images are often noisy, blurred, or difficult to interpret. There are many different mathematical approaches to processing and cleaning images, identifying important features (like tumour borders or organs), extracting measurements, and combining different scans into one model. We will investigate one or several of these techniques.

Mentor: Anand Karki

Mentees: Dhruvi Patel, Kim Shin Tyler Ah Von, and Haniya Nadeem Raja

Description: 

The way we solve problems in science often does not use all the information we have. We typically approach real-world systems by positing a differential equation and then solving it analytically or numerically. However, in the modern world we also have data about these systems. This project brings those strands together, using data to inform and validate the models we write. Rather than leaning on the overused deep learning methods, we'll focus on ideas that come straight from linear algebra, statistics, and calculus—things like least squares, principal components, dynamic mode decomposition, and sparse regression. These methods are lightweight, interpretable, and mathematically sound, so you see the "why," not just the "what."

Mentor: Theodore Morrison

Mentees: Sara Abdella, Vedika Gupta, and Mahek Patel

Description: 

Shuffling a deck of 52 cards is an example of a random walk on a finite set. Each time we shuffle, we randomly move from one ordering of the deck to another. We might naturally ask: what order are the cards likely to be in after many rounds of shuffling? One of the remarkable properties of this process is that the random ordering of the cards abruptly approaches a limiting distribution. The time that the process takes to reach this distribution is called the "mixing time." In this project we will explore the mixing time phenomenon through multiple examples, including card shuffling (which is a random walk on the group of orderings) and other examples of random walks on a finite group. We will also try to relate these examples together under a general theory.

Mentor: Hidde Koerts

Mentees: Soham Pande, Wanqing Zhao, and Sally Ann Hui

Description: 

Like for other mathematical objects, many operations have been developed for graphs. These graph operations take in one or more graphs and modify and/or combine them to produce a new graph. One such operation is taking the power of a graph: the k-th power of a graph G is the graph obtained from G by adding edges between all pairs of vertices at distance at most k in G. In this project we will explore how various graph properties behave under taking graph powers, as well as which classes f=of graphs can be constructed using graph powers.

Mentor: John Smith

Mentees: Melanie Foltak, Sherry Xi, and Briana Peng

Description: 

Can one use computers to rigorously and automatically prove mathematical theorems? In certain areas, the answer is yes. Over the past 30 years, advances in computer algebra have produced methods which simplify complex sums of discrete functions—polynomials, rationals, factorials, binomial coefficients, and other standard objects of univariate combinatronics, or else prove that no such simplification exists. There are also dual version of these algorithms for continuous functions, replacing sums with integrals and adapting the function classes. We will study this duality and the theory behind such "creative telescoping" algorithms, as well as software packages that implement them and manipulate  general P-recursive sequences / D-finite functions. This project introduces students to an active application-rich area of computer algebra research with deep theoretical consequences.

Mentor: Mathieu Rundstrom

Mentees: Hanyu Liu, Riza Qin, and Nignzi Chen

Description: 

This project is in combinatorial geometry. This project is perfect for someone who likes combinatronics or geometry, and would like to see a way these two areas of math interact. We will consider Helly-type statements, statements of the following type: "if every n members of a family of objects have property P then the entire family has the property P." Perhaps the most basic statement of this type is: "if for any two segments of a family segments have a common point, then all segments in the family do (try to prove this!)." We will study such geometric questions in the plane, which often have pretty consequences, and potentially general statements about convex sets in higher dimensional Euclidean space. Depending on the mentees preferences, we can also focus on more abstract generalizaitons, such as fractional Helly theorem, or the colourful Helly theorem.

Mentor: Gabriela Bourla

Mentees: Gul Rukh, Kiera Mitchell, and Shruthi Konduru

Description: 

This project explores the idea of representing social networks through c-closed graphs, first introduced in a paper published in 2020 by Fox et al. titled "Finding Cliques in Social Networks: A New Distribution-Free Model." They considered the idea that two peoples with mutual friends are more likely to be friends with each other, and they defined a family of graphs with a property that mimics this idea: c-closed graphs. In this project, the DRP students will learn some basics of graph theory and use this to read through the Fox et al. paper. The main objectives of the project are to become more comfortable with graphs as well as to learn how to read a math paper.

Mentor: Sahab Hajebi

Mentees: Walker Stie, Shichen Gou, and Flora Wang

Description: 

In today's world, we encounter many mathematical problems that scientists attempt to solve by designing algorithms. For a small number of these problems, scientists have successfully developed efficient (reasonably fast) algorithms. However, for many others, no efficient algorithm is known. Such problems are called hard problems. One common approach to dealing with a hard problem is to restrict attention to a specific subset of its inputs, where a certain parameter k is bounded. We call this class of inputs k-bounded. The central question then becomes: Can we solve the hard problem efficiently for k-bounded inputs? For some parameters, the answer is yes (we can solve the problem efficiently), and for some parameters, the problem may remain hard. In this project, and more broadly in the field of Parametrized Complexity, we aim to understand how to address this question for a given hard problem and a specific parameter associated with it.

Mentor: Shufan Zhang

Mentees: Xin Yi Ye, Ruohan Jin, Ava Yang, and Anandi Jawkar

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 bale 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-tune "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: Shanza Shanza

Mentees: Juliann Zhu and Yuyeon Kim

Description: 

AI tools like ChatGPT and Gemini are becoming popular for answering everyday questions, including how to stay safe and protect privacy online. But how trustworthy and useful is the privacy advice they give? And does it work equally well for young people in different parts of the world? This project explores how people use AI chatbots for security and privacy advice. What kinds of privacy or security questions young people ask AI chatbots. How clear, relevant, and culturally appropriate the AI's answers are. Whether people trust and follow this advice, or prefer other sources like Google or online forums. What risks or mistakes may happen when people rely on AI for privacy guidance (for example, unsafe settings or weak passwords). The study combines survey and interviews with students to learn about their experiences.

Mentor: Pouya Sadeghi

Mentees: Fiona Cai, Kiana Ghomizadeh, and Haardik Garg

Description: 

Large language models like ChatGPT are everywhere. They can write code, solve puzzles, and even explain math. But if you've ever seen a chatbot confidently invent a fake reference, stumble on a simple logic problem, or give a different answers to the same question, you've noticed that their reasoning isn't always reliable. This reading group explores why those failures happen and the cutting-edge techniques researchers use to improve LLM reasoning at inference time without retraining it. Each week we'll read a paper or blog post, discuss the ideas in plain language, and practice giving constructive feedback on the research by asking: What works well? What seems missing? Where could it be stronger? Along the way, we'll cover prompting strategies, chain-of-thought reasoning, majority-vote/self-consistency, iterative refinement, and other methods that make models "think" more clearly. 

Mentor: Vecna Vecna

Mentees: Amalia Morarian, Veronica Gebura, and Sherry Zhang

Description: 

In secure messaging, each party has a special number called a "public key" that is unique to them. If Alice and Bob obtain each other's public keys, they can communicated securely with each other. However, an attacker can comprise their security by swapping out their public keys with the attacker's own public key (impersonating Alice to Bob and vice versa). Alice and Bob can detect these attacks by, for example, comparing their public keys in person, but doing so is inconvenient at best and does not scale well. Key transparency (KT) aims to automate this detection process so that Alice and Bob can be sure they are securely communicating with each other without having to manually verify each other's keys. Multiple real-world applications, including iMessage, WhatsApp, and Signal, have begun implementing KT. In this project, we will do a deep dive into CONIKS, the first KT system for end-to-end encrypted messaging.

Mentor: Anthony Maocheia-Ricci

Mentees: Angela Li, Erin Walshaw, and Alexandra Roszczenko

Description: 

Digital civics as a research field bridges the gap between political science and computer science. Much work in this field explores how digital technologies can be used to make public engagement more accessible to a variety of groups for a variety of purposes: discussions about public policy, organizing activist groups and protests, and beyond. Most of these engagements are of the form of a deliberative process, where community members are engaging in a free and open-ended discussion about a problem instead of trying to persuade the other. This project will give students an understanding in the current work in digital civics and community engagement through technology by surveying literature and projects from the top conference venues in Human-Computer Interaction and similar studies.

Mentor: Andrew Luo

Mentees: Aubrie Chan, Shuxiao Zhang, and Lydia Zhuo

Description: 

When one thinks of a polynomial, the idea of a circle, a hyperboloid, or even a figure-8 probably doesn't come to mind - until you start including more variables! Multivariate polynomials systems are not only interesting but have a wide range of applications. While algorithms for solving these systems exist in general, there is a large discrepancy between the (very large) worst-case run-time bounds and how they perform in practice. In this project, we will explore different techniques for solving systems of polynomial equations, starting from the necessary mathematical background and working our way up to more recent breakthroughs. If time permits, we will create the first implementations of some of these algorithms in a computer algebra system.

Mentor: Michelle Molino

Mentees: Michelle Jeon, Karen Lin, Maidah Amjad Waheed, and Mrunal Kankarej

Description: 

Mentor: Shri Lal Raghudev Ram Singh

Mentees: Alice Shimiao Zhang and Elisa Lang

Description: 

Differential equations describe how quantities change, but solving and analyzing them can be very challenging , especially beyond the familiar finite-dimensional setting of ODEs. Semigroup theory offers a unifying, abstract framework: it allows us to view a differential equation as the generator of a semigroup of operators, a direct analogue of how the exponential function governs solutions of linear ODEs. Semigroup theory provides a powerful framework to study differential equations by treating them as abstract dynamical systems. At its core, it helps us answer fundamental analytic questions: Does a solution exist? Is it unique? Is it stable under small changes? These well-posedness, questions are central to the modern analysis of differential equations, especially PDEs. In this project, we will explore the basics of semi-group theory, including its key definitions, fundamental properties, characterizations, and semi-group methods for analysis of evolution equations.

Mentor: Aareyan Manzoor

Mentees: Tina Liu, Easty Guo, Gisele Huan, and Erin Guerard

Description: 

In analysis, we often talk about things getting "infinitely close" or "going off to infinity." But the real numbers themselves don't actually contain infinitesimal or infinite elements — instead, we handle these ideas through careful epsilon-delta arguments. Non-standard analysis takes a different approach. It introduces a new number system, the hyperreals, which truly contain infinitesimal and infinite numbers. A key feature is the transfer principle, which ensures that any statement true for real numbers also holds for hyperreals. This allows us to reframe analysis in a way that often feels more intuitive, while remaining rigourous. The hyperreals can be constructed using a method called an ultraproduct, a tool from logic. Beyond analysis, these methods have proven useful for simplifying arguments and solving problems across diverse areas of mathematics. The plan is to go through all of basic analysis in this framework. 

Mentor: Faisal Al-Faisal

Mentees: Natalia Weber and AJ Carson

Description: 

Modular forms are functions that exhibit interesting symmetry. Their study involves ideas from analysis, geometry, and number theory, and as such modular forms often create bridges between different areas of mathematics. In this project, we will: (1) learn what modular forms are, and (2) explore some ways they've been applied to establish famous results, such as Fermat's Last Theorem and the optimiality of sphere packing in certain dimensions. 

Mentor: Faisal Romshoo

Mentees: Smridhi Bawa, Charles Qiu, and Caroline Knoke

Description: 

In your mathematical journey so far, you ay have encountered the standard cross product in the three-dimensional space that we all live inside of. You take two vectors in three-dimensional space, take their cross product, and obtain another vector which is perpendicular to both the original vectors. Turns out, that this is not the only cross product out there. We can also define a vector product on seven-dimensional space which is known as the exceptional cross product. We will learn about this cross product and how it leads to very interesting geometries.

Mentor: Kyu Min Shim

Mentees: Drishti Handa and Jie Yang

Description: 

What if you could test hypotheses faster—stopping early when the answer becomes obvious? The Sequential Probability Ratio Test (SPRT) is a method developed in WWII to dynamically make decisions as you collect your samples. Today, it powers A/B tests of tech giants and clinical trials for new drugs.

In this project, you’ll: 1. Simulate SPRT (e.g., ‘Is this medication effective?’ or ‘Does this feature improve clicks?’); 2. Race it against traditional tests—see how SPRT often cuts sample sizes in half compared to the traditional fixed-sample tests, and; 3. Code a live demo in Python/R (no advanced math, just stats intuition!).

By the end, you’ll understand why tech giants and scientists rely on sequential testing. Perfect for stats/data science students who want to merge theory with real-world impact!

Mentor: Hao Quan

Mentees: Gabrielle Wu, Xinyi Nie and Violet Song

Description: 

We’re building a smart system that can predict the prices of different financial assets—like stocks, gold, and currencies—at the same time using a technique called multitask learning. This helps the model learn more effectively by finding patterns shared across markets. The project is a hands-on blend of finance, data analysis, and AI. You’ll work with real-world financial data using Python, and a basic background in machine learning or programming is preferred.

Mentor: Rhoda Dadzie-Dennis, Minh Chau Nguyen

Mentees: Shreya Jain, Suneet Kaur Mahal and Milagro Chen

Description: 

This project explores the impact of climate risk on both the asset and liability sides of financial products.

On the asset side, students will learn to build climate-resilient portfolios using tools from modern portfolio theory. They will integrate climate proxies such as carbon intensity, ESG scores, and Disaster Risk Index (DRI) metrics into portfolio models to assess how these risks influence investment decisions.

At the same time, the project examines how climate change impacts the liability side of financial products like insurance, pensions, and catastrophe bonds. For example, increased extreme weather events raise claims in Property & Casualty (P&C) insurance, while shifts in temperature may alter mortality and morbidity in Life and Health insurance.

Depending on students' interest, readings and exercises may focus on either P&C or Life. Topics like Extreme Value Theory or Generalized Linear Models may be introduced or reinforced based on students' background.

Mentor: Yan Yu

Mentees: Briana Peng and Iris Mo 

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: Joshua Joseph George

Mentees: Sri Meghana Yarlagadda, Madison Han, Chaozhong Wang and Parithy Senthamilarasan

Description: 

In this reading project, we will discuss some cool topics in probability theory. We will start with the basics in Probability (random variables, expected values, and moments) and cover the necessary content required for our readings. Since probability is all about studying uncertainty, we will explore this uncertainty and see how certain outcomes tend to concentrate around their expected values. This is done through the study of concentration inequalities. Then we will explore random matrices and even see how randomness shapes machine learning through concepts like Vapnik–Chervonenkis dimension. Students are expected to linear algebra, and calculus (should be comfortable with matrices, eigenvalues, eigenvectors, sums, series, integration, and differentiation). Some basic probability background, i.e, dealing with random variables and distributions, is preferred.

Mentor: Sita Gakkhar

Mentees: Cristian Moretto, Bowen Dai and Kris Zhang

Description: 

Automated theorem proving deals with formalizing mathematics so results can be verified using computers. Additionally, it makes it possible to use machine learning methods to search for proofs of conjectures and to find theorems bridging different parts of math. This reading is meant to tap into the anticipated usefulness of machine learning to accelerate mathematical research by introducing Lean, an automated theorem proving system and a functional programming language. Terry Tao (at UCLA) in his blog, https://terrytao.wordpress.com/2025/05/31/a-lean-companion-to-analysis-i/ has put forward a project to formalize his analysis textbook, Analysis I, using Lean. The goal for this term’s reading is to come up to speed with Lean, and following Terry’s lead, hopefully start contributing to the github repository for the project https://github.com/teorth/analysis . Time permitting we will attempt a Lean implementation to go along with Sussman et al’s text “Functional Differential Geometry.”

Mentor: Maryam Yalsavar

Mentees: Xena Jiang and Sandy Banh

Description:

Large language models like ChatGPT learn by focusing on the most important parts of text — this is called the attention mechanism. In this project, we will apply this same idea of "paying attention" to help computers solve partial differential equations (PDEs), which are mathematical tools used to describe things like how heat spreads or how fluids flow. By combining attention with these equations, we aim to create faster and more flexible ways to simulate complex systems in science and engineering.

Mentor: Harper Niergarth

Mentees: Vixail Hadelyn, Wenhui Li and Weiyou Li

Description: 

What do you get when you plug an integer partition into a symmetric function?

Given a partition λ, there’s a natural way to construct a new, larger partition f(λ) by evaluating a particular symmetric function—called an elementary symmetric function—on the parts of λ. Recently, an interesting question was asked: can two different partitions λ and μ give the same result, so that f(λ)=f(μ)? In this project, we will explore this question with the following twist: what happens if we replace “elementary symmetric function” with other kinds of symmetric functions? No prior experience with symmetric functions is required, we will build up the background together!

Mentor: Fernanda Rivera Omana, Noah Weninger 

Mentees: Richard Zhang, Elaine Zhao and Betty Zhang

Description: 

Verifying the correctness of mathematical proofs by hand can often be tedious and difficult. Using computer proof verifiers has increased in popularity in recent years, not only to check the correctness of new research but also as a teaching tool. Using proof assistants can help students learn mathematical rigour since the computer gives instant feedback on the correctness of the proof while it is being written. Lean4 is one of the most widely adopted proof assistant environments, and many results in graph theory have already been formalized in Lean4. However, many fundamental theorems remain to be formalized. One area where formalization of basic results is not complete is the study of Hamiltonian cycles. A Hamiltonian cycle of a graph is a cycle that covers all the vertices. The objective of this project will be to formalize results on Hamiltonian cycles in multigraphs.

Mentor: Matthew Hough, Viktor Pavlovic

Mentees: Eason Li and Qinyi Guo

Description: 

The Minimum Volume Ellipsoid (MVE) problem appears in data science and statistics. Given m points in R^n, the goal is to find the smallest ellipsoid that contains them all. This has practical applications in clustering, anomaly detection, and machine learning.

While the MVE problem is convex, solving it becomes computationally challenging as datasets grow. This project explores using the Primal-Dual Hybrid Gradient (PDHG) method—a fast, first-order optimization algorithm—to solve the MVE problem more efficiently.

Students will develop both the theoretical framework for applying PDHG to MVE and implement experiments to evaluate its performance compared to existing solvers.

Mentor: Alex Cowan 

Mentees: Sally Ann Hui, Peter Yang and Suhao Hu

Mentor: Jesse Huang

Mentees: Elizabeth Cai and Kenneth Xiao

Description: 

Investigate how the Kasteleyn matrix valued in Laurent monomials associated with a zigzag consistent dimer model on a torus changes as the zigzag paths deform in a Seiberg duality patterns or urban renewal moves.

Investigate how polynomial invariants of the Kasteleyn matrix change under these dimer transitions and prove that the associated toric Landau-Ginzburg models under dimer moves are related under cluster transformations of Laurent polynomials.

Mentor: Jingcheng Yu

Mentees: Nyra Rodrigues, Yurim Song, Cindy Yang and Wincy Huang

Description: 

We all make choices under uncertainty, whether it’s deciding if you should bring an umbrella, picking a new phone plan, or figuring out how much to save for a rainy day. But how do we actually think about risk—and how do banks, insurers, or even game designers use math to make sense of the unknown? In this project, we’ll explore the surprisingly fascinating world of risk and uncertainty. We’ll talk about real-life scenarios (no finance background needed!), dive into simple probability, and gradually build up to how these ideas power big decisions in finance and insurance. If you’re curious about why people play the lottery, how insurance works, or just want to see math applied in new ways, you’ll find something to enjoy here. Bring your questions and your curiosity—we’ll make sense of risk together!

Mentor: David Awosoga

Mentees: Anna Takegawa, Yushi Liu and Allie Dong

Description: 

Meta-analytics (Franks et al, 2017) can be used to evaluate the quality of metrics that are used to assess player ability in sports. This is important for effective comparisons to be made between them so that stakeholders are not overwhelmed with numerous metrics that lack practical significance, technical depth, or appropriate data considerations.

Meta-analytics are based on three criteria:

(1) stability: does the metric measure the same thing over time?

(2) discrimination: does the metric differentiate between players and

(3) independence: does the metric provide new information?

In this project we will apply meta-analytics to evaluate the quality and underlying properties of a set of volleyball metrics, so that we can identify those that provide the most unique and reliable information for the wider volleyball community, including coaches, athletes, and fans.