Current DRP Projects

Mentor: Victoria Ralph

Mentees: Meg Tobin, Isabel LI, Citlali Belmont-MacGregor, and Jinyu Sha 

Description: 

How can mathematics help the healthcare sector improve patient access to care, reduce wait times, and make better use of limited resources? Across healthcare systems, decision makers face difficult questions about how to distribute resources, allocate staff, respond to changing patient needs, and design services that are effective and equitable. Mathematical models are tools for exploring these problems and for identifying and testing solutions. Health care economics applies traditional economic theories to the health sector. It evaluates how government policies, and financial incentives affect actions of patients, providers, and stakeholders to maximize healthcare access, equity, and cost-effectiveness. By learning how researchers use mathematical models to inform healthcare decisions, students will gain insight into how mathematics can have a direct impact on real-world problems, and contribute to better outcomes for patients, providers, and communities.

Mentor: Scott Holtshousen

Mentees: Elyn Huang, Wenling Zou, Elaine Qian, and Kaius Jin

Description: 

Modern engineering relies heavily on numerical simulations of partial differential equations to test designs before manufacturing. However, this simulation pipeline is expensive, consuming a massive chunk of the time and money allocated to the engineering design and optimization cycle. The main bottlenecks are preprocessing geometric computer-aided design models, generating computational meshes, and solving physics equations over those meshes. Traditionally, this requires expensive algorithms and constant manual intervention to ensure accuracy. Recent research, however, shows that deep neural networks can revolutionize this workflow. This reading program will investigate the most powerful class of solutions to these problems: graph neural networks and Transformers. We will explore how these architectures learn directly from 3D shapes and physical meshes, studying how they are used to encode complex geometry, automate mesh generation, and act as ultra-fast, learned physics simulators.

Mentor: Elnaz Hessami Pilehrood

Mentees: Sarah Nurse, Tianyi Zhou, Jiatong Li, and Aditya Menon

Description: 

Modern digital security relies on mathematical problems that are believed to be difficult for computers to solve. However, the development of quantum computers threatens many of the cryptographic systems used today. Lattice cryptography is one of the leading approaches to building secure communication systems that are expected to remain safe even in the presence of quantum computers. In this project, we will explore the mathematical ideas behind lattice-based cryptography. We will begin by studying lattices as discrete geometric structures in Euclidean space and examining computational problems associated with them, such as finding short vectors. We will discuss why these problems are believed to be computationally hard, and how this hardness can be used to build cryptographic schemes. We will also look at examples of standardized schemes such as Kyber and Dilithium to see how the mathematical concepts developed throughout the project are applied in practice.

Mentor: Bella Chen

Mentees: Raina Jasleen Singh, Valeria Lucia Aguilera, Lily Ryan, and Alishba Rizwan

Description:

Games research is increasingly concerned with designing games that serve specific populations — including children with health and developmental needs. However, many healthcare games developed for children fall short in practice: children often lose motivation quickly, disengage, or fail to experience meaningful intervention outcomes. Understanding why these games fail is just as important as knowing how to build better ones. In this reading project, you will explore the current landscape of games research through a curated set of academic papers. You will examine how existing work has approached — and sometimes missed — the needs of child players, learn about foundational principles of game design for children, and understand how researchers measure whether a game is actually effective. By the end of this project, you will have a solid foundation for thinking critically about how to design healthcare games that genuinely engage children and support meaningful outcomes.

Mentor: Pouya Sadeghi

Mentees: Erin Guo, Myra Gupta, and Yutika Garg

Description: 

We've all used ChatGPT, a chatbot that answers a question in a single reply. But the newest AI systems can do something more interesting: they can plan, look things up, and take action across several steps to accomplish a goal. They behave less like autocomplete and more like an assistant. These systems are called "agents," and this reading group is about understanding how they actually work. In this group, we'll look more closely at these agents to understand two things: (1) what actually turns a large language model into an AI agent, and (2) what the best practices are for using one well. We'll read a mix of articles and papers and meet weekly to discuss them, building up from "what is a language model?" to "how do today's research agents explore a topic on their own?"

Mentor: Shufan Zhang

Mentees: Saima Siddique, Juliann Zhu, Lesley Wang, and May 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: Sreepriya Pulyassary

Mentees: Dora Zang and Adya Bhardwaj

Description:

Ask an LLM the same question twice and you might get two different answers. How do we build AI systems that behave in predictable, controllable ways? A recent and exciting idea in AI research is to leverage micro-economic theory. Specifically, there exist decades of work studying how one person can design incentives to get another to act in their interest, and how to tackle this problem in a principled way. The catch is that most of this literature assumes we know what the other person is trying to optimize for. This is not true for LLMs. Fortunately, there exist tools from economics that can still be applied as long as we can uncover certain mathematical properties of how LLMs make decisions. This project asks: do LLMs have those properties — and if so, can we use them to make AI systems more predictable and controllable?

Mentor: AJ Fong

Mentees: Minna Choi, Edward Chang, Jude Tuladhar, and Emily Jiang

Description: 

Imagine two mirrors joined at a right angle. They reflect an object in a square pattern. The reflections and rotations of that square form a group. Coxeter groups are a beautiful generalization of these. They take these two-mirror, two-dimensional symmetries and extend them into three, four, or even eight dimensions. In two dimensions, you can build infinitely many different kaleidoscopes. But in three dimensions, there are only three valid mirror arrangements. Amazingly, these correspond exactly to the Platonic solids: the tetrahedron, the cube, and the icosahedron. Pushing into higher dimensions reveals deeper magic: valid mirror arrangements become incredibly rare. Instead of wrestling with high-dimensional geometry, we will translate these complex mirror patterns into graphs. Using basic linear algebra, we can look at one of these graphs and know if a kaleidoscope can exist in any dimension.

Mentor: Faisal Al-Faisal

Mentees: Danielle Rodgers and Gabriela Lima

Description: 

When can a prime be written as a sum of two squares? The first few examples are 2=1^2+1^2, 5=2^2+1^2, 17=4^2+1^2, 29=5^2+2^2, 37=6^2+1^2 and 41=4^2+5^2. Besides p=2, all the others in the list are 1 mod 4. It turns out that the set of all such primes is precisely those that are 1 mod 4 (and p=2). This fact was first observed by Fermat and later proved by Euler. In this project, we will look at a proof that involves enlarging our universe of integers by allowing certain complex numbers into the fold. Afterwards we will explore analogous questions about primes p of the form p=x^2+ny^2, where n is a positive integer. This will involve some surprisingly deep ideas that belong to the domain of Class Field Theory---one of the great mathematical achievements of the 20th century.

Mentor: Laindon Burnett

Mentees: Mrunal Kankarej, Kate Pan, Akshata (AK) Achal Kumar, and Irene Hu

Description: 

An automatic sequence is one that can be computed using a simple machine with a finite number of states, and with no memory required. Despite how simple this sounds, these sequences have a wide variety of interesting properties associated with them. We'll study these sequences from the perspectives of number theory, combinatorics on words, and logic. Even better, we'll be able to use a computer program called Walnut, which can tell us when certain statements about automatic sequences are true or false! Who knows, you might even be able to find some new facts about these sequences that nobody's ever seen before!

Mentor: Michelle Molino

Mentees: Katie Lau, Frank Xu, Wania Latif, and Suri Tian

Description: 

Have you ever wondered why some lengths can be drawn exactly with only a ruler and compass while others cannot? This project explores the world of constructible numbers, where geometry and algebra meet in a surprising way. We start with simple geometric constructions—drawing lines, circles, and building new points from old ones—and translate these steps into algebraic rules about numbers. This naturally leads to the idea of field theory, a branch of advanced algebra that studies how number systems can be built up step by step. In this language, each construction corresponds to extending a field by adding new elements in a controlled way, which helps explain exactly which lengths are possible to construct. Along the way, we uncover why famous problems from ancient Greece, like angle trisection or squaring the circle, turned out to be impossible using only classical tools. The project gives an accessible introduction to how modern algebra provides a powerful framework for understanding

Mentor: Sarah Kluchky

Mentees: Amber ZHANG, Grace Qi, Ibrahim Gnessien, and Karlie Deng

Description: 

Imagine you are standing in the 2-dimensional lattice of integers. Starting from the origin, at each point you pick a random number between 1-4 and then proceed in the direction represented by that number. Will you ever make it back to the start? Will you find yourself forever trapped in one quadrant? This is an example of a random walk on Z^2, an infinite group, albeit one that provides an intuitive picture. In this project we will explore these random walks and what properties we're able to deduce about them. Furthermore, we will explore what type of problems we can model using random walks. An example of what we will aim to explore is how card shuffling can be represented using a random walk, as well as the idea of a "cut-off point" which will help us determine how many iterations of shuffling we have to do to get a "well-distributed" deck. If you love probability, this is (in my opinion) an interesting pure mathematical problem which makes good use of it.

Mentor: Diribsa Tsegaye Bedada

Mentees: Kasey Yin, Susan Ho, Allison Trieu, and Shonima (Shoni) Shridhar 

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: Hao Quan

Mentees: Lian Qian, Myla Khan, and Ximo Wang

Description:

Many retirement products promise investors that they will receive at least a minimum amount of money in the future, even if the stock market performs badly. These promises are valuable to customers but risky for insurance companies, especially during crashes or long bear markets. In this project, students will study a simplified variable annuity with a guaranteed minimum accumulation benefit, meaning the insurer may need to cover the shortfall if the investment account falls below a guaranteed level. The project compares a standard “smooth market” model with models that allow sudden crashes and changing market conditions. Students will use Python simulations to estimate the cost of the guarantee, visualize possible investment paths, and measure how much risk is missed by simpler assumptions. The goal is to connect financial modeling, insurance risk, and practical risk management in a project that is quantitative but accessible.

Mentor: Sijie Chen

Mentees: Maheen Malik, Hanin Shamsheer Abdul Rehiman, Marlyne Louis, and Feier Liu

Description: 

A common saying in statistics is that "association is not causation", just because two things move together doesn't mean one causes the other. So how do we figure out whether A truly causes B? This is the central question of causal inference. In this project, we'll explore the core ideas behind it and why establishing causal effects from real-world data is so surprisingly hard. We'll introduce a powerful tool called instrumental variables, which can recover causal relationships even when important variables go unobserved. We'll use What If by Hernán and Robins as our guide, and you will gain hands-on experience with statistical reasoning and coding in R.

Mentor: Tyler Magnus Verhaar

Mentees: Haniya Nadeem Raja and Jiaying Zeng

Description: 

Object tracking means following something as it moves, such as a robot, drone, athlete, storm, or object in a video. There are several ways to do this: some methods use simple rules about how objects move, some use machine learning to find objects in images, and some use probability to handle noisy or incomplete data. In this project, we will focus on one especially useful probability-based method called the Kalman filter. A Kalman filter combines what a model predicts will happen next with what a noisy sensor actually sees, then updates its best guess of the object’s location, speed, and uncertainty. Students will learn the basic Bayesian ideas behind this process, review multivariate normal distributions and simple motion models, and use Python simulations to track moving objects and visualize uncertainty.

Mentor: Xinlei Xu

Mentees: Koena Mukherjee and Catherine Armitage

Description:

Have you ever wondered how researchers decide which treatment works best when there are many competing options? Network meta-analysis is a modern statistical method that combines evidence from multiple studies to compare and rank treatments simultaneously, even when some treatments have never been evaluated head-to-head. In this reading project, we will investigate the ideas and methodology behind network meta-analysis, one of the most widely used evidence-synthesis tools in modern healthcare research. Students will learn how treatment networks are constructed, how direct and indirect evidence are combined, and what assumptions are required for valid conclusions. Through guided discussions of research papers and real-world case studies, participants will develop skills in statistical thinking, critical reading, and scientific communication while gaining exposure to an active area of statistical research.

Mentor: Yan Yu

Mentees: Eike Zhao, Pei Lin He, and Shenchen Lei

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: Yixuan Zeng

Mentees: Kelly Situ and Melinda Caringal

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!

Mentors: Alain Gervais

Mentees: Alice Qi, Zainab Mohamed, and Urvi Bhardwaj

Description: 

The singular value decomposition (SVD) is a well known matrix decomposition used to uncover the dominant components of any dataset that can be represented as a matrix. Inspired by advances in tensor analysis by contemporary mathematicians Elizabeth Newman and Tamara Kolda, this project will contribute to a post-processing tool to analyze fluid simulations: we will extend a tensor-based decomposition, the "star-M SVD", to 4th-order tensors (applicable to 4-dimensional systems, e.g., geophysical data with 3 spatial dimensions + time, which can be difficult to analyze). Tensor decompositions offer a promising alternative to classical SVD, which can obscure spatial correlations in multidimensional data. Starting from matrices (i.e., 2nd-order tensors), we will build intuition for 3rd- and 4th-order tensors; develop a 4th-order analogue of star-M SVD; extend open-source Matlab code to compute the decomposition; and, apply the method to a 3+1 (space+time)-dimensional dataset of fluid flow.

Mentor: Betty Zhang

Mentees: Katherine Chambers, Lily Li, Aster Qianyi Huang, XinYi Ye, and Rasana Yogarajan

Description: 

In our previous DRP projects, we developed an SVEIQR-based ODE model to study epidemic dynamics with vaccination, waning immunity, and repeated outbreak waves. In this renewal project, we will extend this framework in three connected directions based on the interests of my current students: 1) develop a stochastic version of the deterministic model, focusing on how random variation in the transmission rate affects forecasting and long-term disease behaviour; 2) study human behavioural feedback, especially how fear, public opinion, and vaccination attitudes can change contact patterns and generate multiple waves; and 3) add hospitalization dynamics to connect infections with healthcare burden, inpatient care, recovery, and death. New students are very welcome to join by developing related extensions within the same framework. We are excited to continue building this as a collaborative project where students can pursue their own questions within one shared epidemic-modelling framework.

Mentor: James Dufresne

Mentees: Molly Xie, Addysen Stewart, Linxi Fan, David Oladejo, Georgina Wang, and Cindy Shui

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: Maryam Yalsavar

Mentees: Yurim Song, Sharvi Khambaswadkar, Xiao Lu, and Vaibhavi Agarwal

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: Ian George

Mentees: Akshay Arora, Pritika Lahiri, and Alfred Zhang

Description: 

This project explores partitions of partially ordered sets (posets) into convex sets. A convex subset of a poset is analogous to the usual definition of a convex set in the sense that it is closed under taking paths between points. In particular, we will focus on the problem of enumerating these partitions for specific families of posets. We will also consider the special case where each convex set is an interval. To our knowledge, this problem has not appeared in the literature other than the particular case of partitioning a product of two chains (total orders) into intervals. Even so, it is not phrased in this poset terminology, rather as the problem of tiling a rectangle into smaller rectangles. Objects and methods that students can expect to encounter include generating functions, recurrences, and bijections.

Mentor: Jeronimo Valencia

Mentees: 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: Shao Yuan Lin

Mentees: Ming Zhe Li, Noel Kerschner, and Neelesh Nayak

Description:

A symmetric function (SF) is a power series of bounded degree in countably infinite variables, which is invariant under permutations of said variables. When we set the first n variables to 1,q,q^2,..., q^{n-1}, and the rest to 0, we get the SF's principal specialization (PS). Many fascinating results in algebraic combinatorics arise from the PS of certain SFs. The chromatic polynomial of a graph tells us the no. of ways to color a graph with n colors for each n∈N such that no pair of adjacent vertices share the same color. It has a SF-valued generalization called the chromatic symmetric function (CSF), which encodes color combinations of such colorings in addition. This extra information allows us to deduce many other graph invariants. In this project, we will investigate the PS of the CSFs of special classes of graphs to see if we can conclude anything new and interesting, such as a novel algebraic/combinatorial identity, or a formula for a certain graph invariant of said graphs.

Mentors: Taite LaGrange and Gabriela Bourla

Mentees: Rita Bhowmik, Gul Rukh, Tania Frunza, and Kiera Mitchell

Description: 

This project explores the idea of representing social networks through c-closed graphs, which were first introduced in a paper published in 2020 by Fox, Roughgarden, Seshadhri, Wei, and Wein titled "Finding Cliques in Social Networks: A New Distribution-Free Model". These graphs mimic the idea that two people with mutual friends are more likely to be friends with each other. Work on c-closed graphs has looked at finding subgraphs that represent community structures, among other classical graph theory problems. An unexplored problem on c-closed graphs is vertex coloring, which is a difficult problem on general graphs. In social networks, coloring can represent forming non-overlapping sets of people who are not acquainted with each other — say, putting together groups for an ice-breaker activity at an event. The goal of the project will be to investigate how the difficulty of vertex coloring is affected by the c-closed property.

Mentor: Nafisat Ibrahim

Mentees: Olivia Comisso, Milly Chai, Brooke Crichlow, and Joyce Mai

Description: 

Hidden among billions of daily financial transactions are fraud and money laundering. The challenge is that we rarely know which transactions are suspicious. There are no labels and no ground truth, so how do we catch criminals without that information? The answer is anomaly detection: the science of finding things that do not fit. In this project, we use machine learning to flag suspicious transactions without any labeled examples, mirroring real industry conditions at banks and fintech companies. You will work with real financial datasets, study how anomaly detection algorithms work mathematically, and run experiments to understand when and why they succeed or fail. As a team, we will then propose and test improvements to existing methods. You will leave this project with practical research experience, technical skills in Python and machine learning, and a co-authored write-up you can present on your resume or graduate school application. No prior research experience is required.

Mentor: Han Le

Mentees: Yuanwen Yin and Dhiren Swayze

Description:

How does the microstructure of a complex system determine its large-scale behavior? This project investigates this question through random resistor networks. We consider connected graphs whose edges represent electrical connections and whose resistances take one of two possible values. The goal is to understand how the random arrangement of these local resistances affects global quantities such as the effective resistance between two designated nodes. Using tools from graph theory and linear algebra, we first relate the effective resistance to a matrix associated with the network, called the transfer current matrix. We then study how random assignments of edge weights give rise to random submatrices and explore the spectral properties of these matrices as the size of the network grows.

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: Liam McKelvey

Mentees: Caroline Yiyang Lu, Dana Yuan, Aditi Jha, and Danielle Alterman

Description:

Did you know that a properly shuffled deck of cards is almost certainly in an order no one has ever seen before, and no one will ever see again? There are more than 3 times as many possible card orderings as atoms in the Milky Way. But if there are so many possibilities, why can a poorly shuffled deck still feel predictable? In this project, we ask: how many shuffles are enough? Focusing on the classic riffle shuffle, we will study why the first few shuffles do very little, while around seven shuffles can make a standard deck essentially random. We will model shuffling using Markov chains, introduce stopping times to measure when randomness has been reached, and connect the problem to information theory by studying how each shuffle destroys information about the original order. We will also compare different shuffling methods, run numerical experiments, and discuss applications beyond cards, including dataset shuffling in machine learning and random number generation.

Mentor: Liam McKelvey

Mentees: Caroline Yiyang Lu, Dana Yuan, Aditi Jha, and Danielle Alterman

Description:

Did you know that a properly shuffled deck of cards is almost certainly in an order no one has ever seen before, and no one will ever see again? There are more than 3 times as many possible card orderings as atoms in the Milky Way. But if there are so many possibilities, why can a poorly shuffled deck still feel predictable? In this project, we ask: how many shuffles are enough? Focusing on the classic riffle shuffle, we will study why the first few shuffles do very little, while around seven shuffles can make a standard deck essentially random. We will model shuffling using Markov chains, introduce stopping times to measure when randomness has been reached, and connect the problem to information theory by studying how each shuffle destroys information about the original order. We will also compare different shuffling methods, run numerical experiments, and discuss applications beyond cards, including dataset shuffling in machine learning and random number generation.