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: 

The p-adic numbers were invented by Kurt Hensel in the late 1800s as a new way of thinking about numbers—and they turned out to be incredibly useful. These numbers behave in strange and interesting ways, and today they are a key tool in number theory, cryptography, and more. In this project, you'll get a feel for how p-adic numbers work, what makes them different, and why mathematicians care about them.

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: Yixuan Zeng

Mentees: Anwesha Bali and Maheen Malik

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: David Awosoga

Mentees: Maria Gharabaghi, Lillian Mo, and Victoria Gao

Description: 

An Integrated Support Team (IST) is a collaborative group of individuals who support athletes, coaches, and teams by consolidating expertise from their respective domains to ensure that athletes are optimized for performance. The demand of data analytics support in these groups have proliferated in recent years. This project will be a term-long investigation, tackling a sport-specific question of interest to coaching staff, sport scientists, and support personnel. Throughout this experience mentees will learn to understand and apply the fundamentals of sports performance analysis, including: player evaluation, prospect assessment, roster construction, asset valuation, and on-field strategy. Mentees will also learn to conceptualize the components of the data science workflow and related techniques, as well as the problems to which they apply in amateur and professional sport. Finally, mentees will gain an appreciation of sports analytics in action via relevant case studies.

Mentor: Edward Chang

Mentees: Ruby Liu, Xing Liu, and Sara Pace

Description: 

Have you ever wondered how your email app knows which messages are spam, or how photo apps can tell cats apart from dogs? These are examples of classification, a key idea in Machine Learning and Statistical Learning. In this project, you'll explore how computers can learn from data to make decisions, much like how we learn from past experiences. We'll start with simple, theoretical treatment to classification, then apply what we learned to analyse some real-world data. By the end, mentees will 1. Understand what it means to "train" a model 2. Experiment with basic classification methods 3. See how to test whether a model is actually working. 

Mentor: Yan Yu

Mentees: Melissa Sham, Ayushi Negi, and Lina Kim

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. Mentees also get hands-on experience with powerful predictive tools likes LASSO and Random Forest models, learning how these methods help businesses anticipate trends and enhance operational efficiency.

Mentor: Yun Su

Mentees: Shruti Dua and Rui Wang

Description: 

Have you ever wondered how math can help us understand the ups and downs of financial markets? In this project, we'll explore how randomness influences stock prices, interest rates, and investments by building simple mathematical models. We'll use ideas from probability and calculus to describe how financial systems evolve over time, then test these models through numerical simulations. Along the way, we'll ask questions like: What makes an investment strategy stable or risky? How can we model sudden shocks in the market? Can we predict whether a portfolio will grow steadily or collapse? This project combines mathematical modelling, probability, and a bit of coding to see how abstract math connects to real-world finance.

Mentor: Hassaan Qazi

Mentees: Yurim Song, Lorena Brito Maia, and Ningya Shen

Description: 

In this study, we will investigate how to apply graph-theoretic tools to achieve consensus and formation control of a swarm of flying drones. We will start with the fundamentals of graph theory and will read some literature available on the coordination and cooperation of Unmanned Aerial Vehicles (UAVs). To make the project more realistic, we will impose assumptions on the agents' communication capacity with the centralized control hub. Furthermore, we will try to incorporate the strategies to deal with the delays that happen in the communications and prepare the overall scheme for fault tolerance. The emphasis will be to design a control law that tracks the time-varying target. The project has a wide range of applications from monitoring the environment in agriculture to capturing images of airplanes frm various angles in the aeronautics industry.

Mentor: Betty Zhang

Mentees: Raluca Rusu, Theodora Turda, Khushi Adukia, and Shreya Jain

Description: 

In our earlier model, we used ordinary differential equations to link epidemic dynamics with our economics outcomes, focusing on school closures and their impact on women's workforce participation. We found a trade-off: more closures led to gentler outbreaks but slower economic recovery, while fewer closures caused sharper outbreaks but faster economic recovery. As future work, we propose modelling the household effect, since transmission within families and shared childcare responsibilities influence both infections and labour participation. We also extend the framework to consider other industry closures beyond schools. Using a Susceptible-Exposed-Infected-Quarantined-Recovered (SEIQR) disease model linked with economic growth equations, this ODE approach will let us study how closures, household, and workforce participation affect both the epidimec curves and economic health.

Mentor: James Dufresne

Mentees: Deborah King and Jessie Deng

Description: 

The Moving Average Convergence Divergence (MACD) indicator is one of the most widely used tools in financial markets for deciding when to buy or sell. In this project, you'll study how well MACD actually performs as a trading signal, using real stock of ETF data. We'll start by implementing the basic MACD crossover rule and testing it over historical prices. Then we'll measure how effective it is using metrics such as average return, Sharpe ration (return per unit of risk), and maximum drawdown (largest loss from a peak). Finally, we will explore improvements like adding trend or volatility filters to see whether they reduce false signals in sideways markets. 

Mentor: Jun Kwon

Mentees: Hannah Lee Zhong and Richard Zhang

Description: 

Verifying the correctness of mathematical proofs by hand can often be tedious and difficult. Using computer proof verifies 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 Matchings. The objective of this project will be to formalize results on Matchings in multigraphs.

Mentor: Hanna Dodd and Carolyn Wang

Mentees: Mahi Rathi, Rachel Spanglet, and Alice Qi

Description: 

Previous work in human-computer interaction (HCI) has established that video games can profoundly influence players outside of the gaming context, influencing emotions or educating players on important topics. As such, social impact games have become increasingly popular, leveraging persuasive game mechanics to effect opinion formation and opinion change on a range of social issues. In this project, we will examine how developers leverage game design to influence player behaviour. By closely examining existing HCI and game studies literature, we will develop a game prototype that aims to combat mis/disinformation as our chosen social issue. We will also review literature in online mis/disinformation to further ground our game design. 

Mentor: Maryam Yalsavar

Mentees: Yana Jakhwal and Xiao Lu

Description: 

Many existing deep learning models that solve partial differential equations (PDEs) are designed to handle only a very specific version of a PDE with fixed parameters. When these parameters change, the models often perform poorly. In this project, we will explore how to build more general and flexible PDE solvers that can still perform well even when the equation's parameters vary. The goal is to keep learning-based solvers more adaptable and useful in real-world scientific and engineering problems.

Mentor: Mohammad Abolnejadian

Mentees: Christal Xing, Le Kong, Ethan Tran, and Hana Rashad

Description: 

In this Human-Computer Interaction (HCI) and Applied AI research project, we will explore how to support synchronous conversations with the help of AI. Collaborative decision-making occurs in various settings (e.g. a board debated or planning a trip). While background information can help decision-makers make more informed decisions, often times they go underutilized due to the ongoing nature of these conversations. We will explore the proactive AI prototype built on RAG architecture, and more importantly, assist in conducting and administering user studies to collect, process, and finally analyze participants' inputs on the prototype. Finally, as the output of this project, we will learn about how results are presented in HCI papers by presenting the results from the user studies.

Mentor: Mehrad Haghshenas

Mentees: Jeannie Shi, Manvi Sharma, Sarah Nurse, Tianyi Zhou, and LInxi Fan

Description: 

This project aims to bridge the gap between assistants and SAT solvers by developing a tool to verify the correctness of SAT encodings in Coq Formal verification provides correctness assurance to programs, while SMT solver (think of it as enriched SAT solvers) are at the backend. However, applications are not expressed natively in SAT and must be encoded into SAT, which is error-prone. The project will consist of constructing Coq proofs for SAT encodings. By ensuring that the encoding is formally verified, the project will increase the trustworthiness of SAT-based proofs. Inspired by methodologies presented by Codel et al. for Lean, this work will construct an independent framework in Coq and try to enrich it to overcome the limitations in general cardinality, pseudo-boolean constraints, or symmetry-breaking predicates. Please read the paper "Verified Encodings for SAT Solvers" for more information.

Mentor: Saba Maloaei

Mentees: Alia Cai, Zunairah Shahzad, and Zainab Mohamed

Description: 

Math and computer science can do more than solve problems. They're also about patterns, beauty, and structure. In this project, we'll explore how simple geometric ideas can be woven together to create intricate visual designs and sculptural forms. Inspired by the Bridges paper "Interwoven Geometric Patterns and Two-Layer Sculptures," we'll look at how algorithms can build patterns that feel both artistic and mathematical. Together, we'll read a few short papers, experiment with digital tools or simple programs to design our own woven structures, and work toward writing a short survey or expository paper about what we discover. Along the way, we'll gain an intuitive sense of how geometry underlies visual harmony and how mathematical thinking can drive creative design. 

Mentor: Nathan Pagliaroli

Mentees: John Kim, Milagro Chen, Tanya Lu, and Mohi Kambo

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 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 extent the usual occurrence relations for orthogonal polynomials associated to random matrices by extending them for "multi-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: Hao Quan

Mentees: Hany Jiang, Momina Butt, and Bhavika Dindyal

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. We will work with real-world financial data using Python, and a basic background in machine learning or programming is preferred.