Current DRP Projects

Mentor: Zoya Abbasi

Mentees: Victoria Kolb and Xuanyi Chen

Description: 

Ever wondered how scientists predict the spread of diseases like COVID-19? Our project uses the power of math and mathematical modeling to uncover these secrets. By creating and analyzing models, we can forecast infection trends, evaluate the impact of interventions like vaccines, and help shape effective health policies. Join us to explore how math can make a real difference in combating infectious diseases and protecting our communities!

Mentor: Vyom Patel

Mentees: Emma Xing and Nicole Masodje

Description: 

With global efforts ramping up in quantum technology, learning the foundations of quantum computing gives you a head start in one of the fastest-growing fields. In this project, you'll explore quantum computing with just a background in linear algebra. Using ZX calculus, a visual, user-friendly method to represent quantum operations, you'll learn how quantum computers work and how they differ from regular ones, all while using intuitive diagrams that make complex concepts easier to grasp.

Mentor: Maxwell Yue

Mentees: Amy Bui and Debra Rose Defazio

Description:

The Hodgkin-Huxley model is a fundamental mathematical model in neuroscience that provides insights into the generation and propagation of action potentials. However, certain parameters within the model are difficult to obtain through experiments and traditional optimization techniques. In the project, we aim to investigate the use of AI approaches, particularly artificial neural networks, to estimate these parameters.

Mentors: Kaixin Zheng

Mentees: Yuan Yuan and Jackie Liu

Description:

In many organisms, circadian rhythms regulate behaviors like sleep and metabolism. These daily cycles, controlled by molecular mechanisms recognized with a Nobel Prize, are the focus of this project. Ideal for students interested in applied math and biology, the project explores mathematical modeling of biological processes. Mentees will gain foundational knowledge, coding skills, and experience in interdisciplinary research, along with opportunities for review and presentation practice.

Mentor: Yusuf Aydogdu

Mentees: Joanna Jiang, Erica Han, and Katrina Wohlgemut

Description:

In the era of climate change, climate modelling and simulations are both prominent and challenging research topics. In this project, students will investigate climate models, such as the El Niño-Southern Oscillation (ENSO), which has a significant impact on climate change. They will learn important aspects of climate modelling, such as ocean-temperature interactions, and perform basic climate simulations and data generation using Python, incorporating some machine learning applications.

Mentor: Massimo Vicenzo

Mentees: Yuan Si and Silvia Zheng

Description:

A robber is on the loose and the cops are trying to catch them! Given some graph (a set of points and connections between them), the robber and cops take turns moving to adjacent points. Can the cops capture the robber? Or will the robber be able to evade the cops forever? This depends on what the graph looks like and how many cops there are. We will investigate the minimum number of cops we need to put on a graph to guarantee we catch the robber, no matter how the robber plays.

Mentor: Gabriela Bourla

Mentees: Jenna Johnston and Peter Yang

Description:

How can we represent social networks through graphs? Think of people as dots and relationships as lines between them. Whether it's in real life or online through social media, two people are more likely to be friends if they have mutual friends. In a 2020 paper by Fox et al, they introduced the concept of c-closed graphs where if two people have at least c mutual friends, then they are friends. This project will work through parts of this paper as well as results found in related papers.

Mentor: Taite LaGrange

Mentees: Sarah Nurse

Description:

This project will build up from graph theory basics to eventually understanding clique-width — a graph parameter that, roughly, measures how complex a graph’s structure is based on how difficult it is to build that graph using a specific set of operations. We’ll look at both the definition of clique-width as well as how we can use it to say interesting things about what graphs look like and to help us solve some fun graph problems.

Mentor: Vahid Reza Asadi

Mentee: Rachel Ma and Vivian Guo

Description:

Imagine Alice's program counts primes up to 1 billion. Can Bob verify her claim without having to rerun the program? This project explores cryptographic proofs, focusing on how the sum-check protocol can positively verify such claims through secure and efficient computations. Investigate these concepts to gain a deeper understanding of their crucial role in real-world cryptography, underpinning the security of digital communications and transactions.

Mentor: Juju Quartz

Mentee: Prisha Aggarwal and Abirami Karthikeyan

Description:

This DRP on the mathematics of deep learning will cover key foundations such as neural network architectures, optimization techniques, and training methods. After that, we will explore theoretical results, including universal approximation, benign overparameterization, and double descent with less emphasis on formal proofs and more on understanding. We will blend theory with hands-on coding to deepen our understanding of both the mathematics and practical implementation of deep learning models.

Mentor: Omkar Bhalchandra Baraskar

Mentees: Rachel Spanglet and Myra Gupta

Description:

Probabilistic Methods is an umbrella term for simple yet powerful tools extremely useful in graph theory and combinatorics. Let's take a simple example of a problem: Consider a complete graph and now colour its edges red or blue, how large should the graph be for it to have a monochromatic triangle? Answer is 6. This is the starting point of Ramsey Theory and probabilistic methods gives lower bounds on Ramsey numbers. We aim to learn several probabilistic methods in application oriented manner.

Mentor: Ruizhe Wang

Mentees: Siyu Huang and Mikel Osei-Owusu

Description:

Data breaches are a constant and growing threat in today's world, often leading to questions like: why can't systems simply be made more secure? The answer is complex, as enhancing computer security involves trade-offs in terms of performance, cost, system complexity, and much more. In this project, we aim to explore these trade-offs, particularly focusing on the issue of memory vulnerabilities, one of the most prevalent and dangerous types of security flaws.

Mentor: Prashant Gokhale

Mentees: Tanisha Dhami and Neva Wilson

Description:

Randomized algorithms harness the power of randomness to solve a problem (for ex - Karger's min cut algorithm) and have found widespread applications in algorithm design and machine learning. How do we analyze the runtime of randomized algorithms? One way is by cleverly using some 'concentration inequality' to bound the probability of a random variable taking a value away from the mean (for ex - Markov's inequality). In this project, we will together explore their utility in algorithm analysis.

Mentor: Henry Yang

Mentees: Katherine Liu and Freya Zhang

Description:

Being curious can be easy. One can lie on the couch and swipe through Youtube and TikTok videos and stay curious all day. But it's much harder to stay “Productively Curious”. Especially for researchers, it is a challenge to find a way to characterize and intervene in users' knowledge, interests, distraction and motivation. In this project, we will study users' browsing behaviors and cognitive patterns during information searching, and find a balance between curiosity and productivity.

Mentor: Joey Lakerdas-Gayle

Mentees: Ellie Hamer and Gisele Huang

Description:

Many common mathematical structures are infinite, like the natural numbers (0,1,2,3,...) and the real numbers (0,-5,√2,π,...). However, there are many different ways that a structure can be infinite. For example, the infinity of real numbers is larger than the infinity of natural numbers. We will consider different kinds of infinities, and study the basic properties of infinite combinatorial structures, from infinite trees to infinite-dimensional spaces.

Mentor: Paul Cusson

Mentees: Edward Chang and Betty Zhang

Description:

We will be studying vector bundles, an extremely interesting and useful object found in the fields of algebraic topology, differential geometry, and algebraic geometry, among others. In essence, it is a way of assigning to every point of a topological space a vector space, each of the same dimension, and then studying all the different ways these vector spaces can move and twist in a continuous way along the topological space. The approach to this topic can easily be adapted to your interests.

Mentor: Robert Harris

Mentees: Isabela Souza Cefrin Da Silva and Soumya Menon

Description:

Place a piece of string into your pocket. If our experience with headphones tells us anything, when we take it out it will be horribly twisted. Now, connect the two ends of the tangle together. Congratulations, you have made a mathematical knot. The question remains, what can we do with this knot? This project will serve as a hands-on introduction to the world of knot theory and the games mathematicians play to turn them into one another and how we can tell them apart.

Mentor: Zhenchao Ge

Mentees: Rachel He and Chandni Wadhwa

Description:

In number theory, there are many fundamentally useful arithmetic functions, such as the divisor function, Euler's totient function... Studying these functions help us understand how multiplication influences the integers. In particular, studying their mean values helps us understand the distribution of certain special types of integers. In project, I will lead the mentee to some basic arithmetic functions and study their mean values using elementary and combinatorial methods.

Mentor: Aleksa Vujicic

Mentees: Ziyan Chen and Khanjan Soni

Description:

Abstract harmonic analysis is a flexible term than can mean many different things. Often it refers to the study of locally compact groups by looking at certain types of functions on them. It is a beautiful theory with results that connect various areas of mathematics such as functional analysis, group theory, dynamics, measure theory, and representation theory. Possible reading topics include: the Haar measure construction, Fourier analysis on Abelian groups, L^p spaces, and Pontryagin duality.

Mentor: Erica Liu

Mentees: Jessica Ni, Bowen Dai, Xena Jiang, and Adrina Esfandiari

Description:

A line, a circle, a twisted cube, all of those interesting geometry objects can be described by a system of polynomial equations. Meanwhile, polynomials have different algebraic structures and can be studied algorithmically. In this reading project, we will embark on a journey to unravel the mysteries behind algebraic varieties, exploring the profound connections between algebra and geometry, with the help of computer algebra.

Mentor: Xianwei Li

Mentees: XinYi Ye and Jialu Sun

Description:

Ever wondered how doctors use data to make life-saving predictions? In this reading group, we’ll dive into building powerful healthcare prediction tools to help diagnose disease, therapeutic decision-making, and disease prevention. You’ll learn to construct effective predictive models tailored to your data type, choose the optimal model, apply them, and evaluate their performance -- enriched by real-world data examples and hands-on, practical challenges!

Mentor: Xipeng Huang

Mentees: Qianqian Wang and Micky Liu

Description:

Valuation of American-style options has long been a key problem in finance. These options, which allow early exercise, are used across major markets, like stocks, commodities, and currencies. This project will introduce you to the numerical methods for option pricing. We will start with the classic Least Squares Monte Carlo method, and more advanced methods may be covered depending on our progress.

Mentor: Yixing Zhao

Mentees: Xiaoyan Wang and Catherine McCulloch

Description:

Bayesian statistics is a way of thinking about data and uncertainty that combines prior knowledge with new evidence. Imagine you have a belief about something—like the likelihood of rain tomorrow. Bayesian statistics allows you to start with that belief (prior), then update it as you gather new information (data), like the weather forecast. It's useful in various fields, from medicine to finance, where understanding uncertainty is crucial.

Mentor: Jessica Xu

Mentees: Mathilda Lee, Maria Khan, Parithy Senthamilarasan, and Anika Smith

Description:

In today’s world, every groundbreaking discovery, such as curing diseases, relies on a foundation known as statistics. However, in the hopes of having significant results, misapplications of statistics are becoming more and more common. In this project we will expose common statistical pitfalls found in research like p-hacking, HARKing, and more! By the end, you will become an expert in ethical and reproducible statistical practice, making you an even stronger statistician!

Mentor: Yan Yu

Mentees: Lily Li, Anwesha Bali, and Yanqi Gao

Description:

Discover the fascinating world of hotel operations and customer experiences through data! This project offers a unique blend of data science and business management, where you'll analyze hotel operations using real-world datasets. Learn about customer behaviour, peak demand times, and room access issues, and use predictive models like LASSO and Random Forest to forecast hotel metrics. Ideal for undergrads interested in data's role in business strategy!

Mentor: Rhoda Dadzie-Dennis

Mentees: Joanna Moon, Arielle Chan, Jane Zhu, and Lena Ye

Description:

This project teaches students to create climate-resilient portfolios by integrating climate risks, such as ESG scores and other climate proxies, into traditional portfolio models. Using machine learning techniques, students will explore how these factors impact asset allocation, aiming to achieve sustainable returns. Perfect for those interested in sustainable finance, this project combines climate data and financial analysis for adaptive investment strategies.

Mentor: Zelalem Arega Worku

Mentees: Moustapha Diallo, Chloe Young, and Shraddha Shankar Aangiras

Description:

This project aims to extend a novel method developed by the mentor for deriving quadrature rules on simplicial shapes to other fundamental domains, like prisms and pyramids. Quadrature rules are vital in scientific computing, with strong interest in more efficient methods to reduce computational costs. Through this project, mentees will learn a state-of-the-art approach to creating quadrature rules, with the potential to develop new ones suitable for publication in respected academic journals.

Mentor: Fernanda Rivera Omana and Mathieu Rundstrom

Mentees: Hadas Barabash, Jiaying Li, Nitya Kandadai, and Zahra Ben Sabeur Tobar

Description:

Something like this: Menger's theorem is a fundamental result in graph theory that connects vertex connectivity with the existence of multiple disjoint paths. Extending this concept to matroids, Tutte proved a generalization now known as Tutte's linking theorem. The primary objective of this project is to formalize Tutte's linking theorem using the Lean proof assistant, a tool for verifying the correctness of mathematical proofs with the help of a computer. Proof formalization is an exiting area of mathematics which has received a lot of attention recently, for various reasons such as automatic theorem proving with the help of AI, and also for its ability to guarantee the correctness of proof.

Mentors: Kimia Shaban, Tiadora Ruza, and Ian George

Mentees: Hanyu Zhou, Alishba Malik, Lihuizi Chen, Jonathan Friedberg, Sherry Feng, and Annie Sun

Description:

This project is motivated by causal set theory (CST): a promising theory of quantum gravity where spacetime is modeled as a causal set, or a locally finite partially ordered set (poset). The causal order of events in a spacetime is the partial order of the poset.

This project explores past-future (p-f) invariants of posets, recently defined by Rafael Sorkin. We will investigate poset structures that p-f invariants cannot distinguish. We aim to determine if p-f invariants can distinguish manifold-like posets from non-manifold-like ones, and in the former case determine the underlying manifold. This research requires a basic understanding of matrices and discrete structures. We will use SageMath and Python to compute examples and gather data.

There are many avenues for exploration in this project as we are investigating several different properties. Thus, we anticipate being able to effectively have many mentees work on this project.

Mentor: Matheus Jun Ota

Mentees: Modi Liu and Yirui (Ruby) Fang

Description:

The high-level goal of this project is to accelerate state-of-the-art exact algorithms for the capacitated vehicle routing problem (CVRP). The CVRP concerns the design of a routing plan for capacitated vehicles to collect customer demands. The CVRP is one of the most well-studied problems in Operations Research, and improving the best algorithms for it may seem overly ambitious. However, we will focus on a very specific aspect: a family of cutting-planes called "rounded capacity inequalities" (RCIs).

We know that RCIs can be obtained by multiplying some valid inequalities by certain coefficients, summing them, and then applying a rounding procedure. Our aim is to discover new "RCI-like" inequalities by experimenting this procedure with different coefficients. Since RCIs are a crucial part of any efficient exact algorithm for the CVRP, even small improvements on them can have a significant impact. In my opinion, this project nicely combines theory and practice.

Mentors: Cicely (Cece) Henderson and Hidde Koerts

Mentees: Boxuan Meng, Elaine Zhao, Helena Devinyak, and Molly Wu

Description:

In 1979 Regina Tyshkevich defined an operation that allows us to compose graphs to get a bigger graph. A graph is decomposable if it is the Tyshkevich composition of other graphs. Tyshkevich's operation is special because every graph has a unique Tyshkevich decomposition. But this is not the only way to combine two smaller graphs into a new, larger graph. Over the last decades, many graph operations have been defined and studied that combine two graphs into a larger graph in such a fashion or augment one graph into a larger one. For example, we can glue two graphs together along a vertex, or add a vertex to a graph adjacent to all other vertices. In this project, we will investigate the relation between Tyshkevich decomposition and other such graph operations. What can we say about the Tyshkevich decomposition of the bigger graph resulting from a different graph operation, when we know the corresponding decompositions of the smaller graph(s)?

Mentor: Jerónimo Valencia Porras

Mentees: Fillion Ding

Description:

In 1899 Pick gave a surprising formula to compute the area of a polygon in the plane. In 1956, Reeve gave a generalization of Pick’s Theorem for the case of 3-dimensional polytopes. There are other versions of this theorem when we consider more general sets in the plane, for instance half-open polygons and polygons with holes. This theorem is also connected to Ehrhart theory, an algebraic approach to understanding lattice-point enumeration in polytopes. However, the literature regarding this topic is scattered on different papers and books. The goal of this project is to write a survey on Pick’s theorem and its generalizations that includes a good bibliographic revision and key remarks on the connections of these theorems to other areas.

Mentor: Runsheng Guo

Mentees: Shu Cong, Maggie Liu, Divya Makkar, and Anna Jia Wang

Description:

Large language models (LLMs) like GPT, Claude, and Gemini are revolutionizing content creation, research, and code generation. Yet, LLM-powered applications typically rely on high-performance GPU clusters, which are costly and in high demand. This project aims to democratize access to LLMs by developing systems for running these models efficiently on clusters of low- to mid-tier GPUs, making LLM capabilities more accessible to a wider range of organizations.

Mentor: Sina Kamali

Mentees: Lianghan Dong, Sarah Wilson, and Stella Tian

Description:

Fully encrypted protocols (FEPs) are protocols that try to hide all the important communication information. Their ultimate goal is to make every single byte of communication look "uniformly random." FEPs were originally used as a means to circumvent censorship, but nowadays, many companies and governments are considering them as normal means of secure communications.

In this project, we plan to get familiar with and create a new FEP. We will try to reason about its security, along with the soundness of its implementation. This topic is one of the hot new topics in the area of privacy-enhancing tools, and a project might lead to a publication!