Current DRP Projects

Mentor: Milad Moshayedi

Mentees: Mincy Yang and Alice Qi

Description: 

Differential equations are the most essential building blocks of theories in physics, chemistry, biology, economics, etc. Therefore, it is crucial to have a solid knowledge of differential equations if you are going to continue studies in these fields. In this project, first, you will get familiar with some theoretical backgrounds, then use this knowledge to solve some famous differential equations numerically using computers. Being familiar with Python or C++ is a wealth in this project.

Mentor: Yusuf Aydogdu

Mentees: Mabel Luo and Daisy Li

Description: 

In the era of climate change, climate modelling and simulations are hot yet challenging research topics. From the equatorial to polar regions, climate modelling requires different physical and mathematical assumptions. In this project, students are expected to learn important properties of climate modelling using atmosphere-ocean-temperature interaction models, i.e. El Nino Southern Oscillations (ENSO) and gain experience in making basic climate simulations and generating data using Python.

Mentor: Yanming Kang

Mentees: Sachi Shah and Hadeya Ikram

Description:

The transformer is now the most important neural network architecture. However, it is expensive for long sequences with its quadratic time and memory cost. In this project, our objective is to improve transformer models by optimizing its ability to process very long data sequences efficiently. We will examine existing works and try to modify the structure of the attention mechanism, the key component in transformers that determines which data points are most relevant during processing.

Mentors: Matheus Hrabowec Zambianco

Mentees: Sehar Durrani and Rem Dong

Description:

This project is designed as a comprehensive journey into the complex beauty of General Relativity. We will start with the necessary mathematical tools and physical intuition from Special Relativity, and then we will move on to understand how gravity can be described as the “geometric soul” of spacetime, and how fantastic objects like black holes can exist. The acquired knowledge will be used to study alternative versions of the famous “twin paradox” in spacetimes with gravity.

Mentor: Ruikun Zhou

Mentees: Gengyang Chen and Jiaren Yue

Description:

Stability analysis is at the core of dynamical systems, where the well-known Lyapunov stability theorems play an important role in the stability of equilibrium points. This project aims to provide the mentees with the basics of stability analysis of nonlinear dynamical systems, at the same time to learn how to analyze/provide stability of the systems using state-of-the-art machine learning techniques for nonlinear systems.

Mentor: Zoya Abbasi

Mentees: Shiyu Liu and Alishba Malik

Description:

Our project dives into the world of infectious diseases, using math as a detective tool. Imagine playing a detective game where numbers help predict how a virus spreads, who it affects the most, and how we can stop it. We'll explore simple mathematical models to understand the outbreak and control measures. Join us to uncover the math behind epidemics and learn how to fight against viruses with knowledge and strategy. No heavy math background is needed, just curiosity and eagerness to learn!

Mentor: Mathieu Rundstrom

Mentees: Heer Mehta and Sneha Elavarasan

Description:

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 of segments have a common point, then all segments in the family do. We will study such geometric questions in the plane, and potentially  general statements about convex sets in higher dimensional Euclidean space.

Mentor: Cicely Henderson

Mentees: Nitya Kandadai, Zahra Ben Sabeur Tobar, Boxuan Meng, and Elaine Zhao

Description:

A graph is a network of dots and lines between them. Regina Tyshkevich defined an operation: just as you can multiply integers, you can Tyshkevich compose graphs to get a bigger graph. A graph is decomposable if it is the Tyshkevich composition of other graphs. What makes Tyshkevich's operation so special? Just as every integer has a unique prime factorization, every graph has a unique Tyshkevich decomposition! We will study graph theory, Tyshkevich composition, and decomposability.

Mentor: Sepehr Hajebi

Mentees: Anna Henderson and Gioia De Leonardis

Description:

In this project, we explore the application of tools from one branch of math to another, seemingly completely unrelated one, from topology, analysis and algebra to combinatorics and number theory. We will cover several examples and will see in some cases how a cornerstone theorem with a misleadingly elementary statement from an area, say graph theory, not only "can" but in fact "should" be proved using relatively advanced tools from another area of math.

Mentor: Hidde Koerts

Mentees: Bandana Bajaj and Khanjan Soni

Description:

The mathematical objects used to model networks, called graphs, are fascinating structures. In this project, we will look at a family of operations of graphs called graph products. A graph product takes two graphs and creates one larger graph with its structure resembling both the earlier graphs. We will cover the basics of graph theory, and then consider how graph properties are affected by the products. Time permitting, we will also cover related deep structural theorems and/or algorithms.

Mentor: Tiadora Ruza

Mentees: Novel Peng and Joyce Chen

Description:

Design theory looks at how a given set can be arranged into a set of subsets satisfying certain properties. Many designs arose from the study of mathematical puzzles like the magic square. Today, designs have many applications like cryptography and combinatorial games. The objective of this project is to learn the basics of design theory, with a goal of reading a paper that looks at what happens when Tic-Tac-Toe is played on two designs: Balanced Incomplete Block Designs and Transversal Designs.

Mentor: Xinyue Xie

Mentees: Neha Munje and Amy Bui

Description:

Graph theory is a rich field that studies graphs, which consist of a set of vertices and a set of edges, each joining two vertices. If the edges are assigned weights from a group, such as the group of integers modulo n, additional algebraic properties emerge. By exploring an interesting class of graphs, called biased graphs, this project aims to provide the mentees with 1) an introduction to graphs and groups; 2) experience in reading math publications; and 3) exposure to open research problems.

Mentor: Punit Kunjam

Mentees: Sophia Zhu, Luna Wu, Alia Cai, and Nyra Rodrigues

Description:

Embark on a fascinating journey into the world of problem-solving and logical thinking with the Tower of Hanoi, where players engage in the ancient puzzle of moving rings between three pegs. This project introduces students to essential concepts in mathematics, honing their skills in algorithmic thinking and computational problem-solving. As students delve into the game development process using Unity, setting the stage for deeper exploration and potential joint research opportunities.

Mentor: Shashank Joshi

Mentees: Yidan Wang, Ruohan Jin, and Lily Jiang

Description:

The project explores blockchain technology, emphasizing consensus models that enable agreement on transaction validity without a central authority. It clarifies blockchain as an immutable ledger requiring unanimous transaction agreement. Consensus algorithms ensure blockchain reliability by facilitating node agreement on network state, necessitating security, reliability, and speed.

Mentor: Ronak Pradeep

Mentee: Aarushi Jain and Diliara Kaniazova

Description:

With an avalanche of information available online and the growth of large language models (LLMs), efficiently finding and leveraging relevant data has never been more critical. We delve into how the precision of information retrieval (IR) can be combined with the generative reasoning capabilities of LLMs to create systems like Perplexity or Microsoft's Copilot that leverage LLMs to generate up-to-date information free of stale facts using the best of IR and how we can evaluate such systems.

Mentor: Noble Saji Mathews

Mentee: Helen Dong and Fion Zheng

Description:

Are you fascinated by how Artificial Intelligence is being used to build tools that almost seem magical? In this project, you'll be exposed to the mechanisms behind such systems and gain hands-on experience with the latest techniques in the field. The project will primarily focus on demystifying these 'LLM Apps' by experimenting with different models of LLM interactions and based on interests and progress, focus on evaluating their effectiveness in Software Development tasks.

Mentor: Prashant Gokhale

Mentees: Shriya Kulkarni and Lynn Li

Description:

Suppose one wishes to study how susceptible a network is towards spreading of rumors (or a fire): one way is to model "rumor spreading" as a discrete time step process with some set of natural rules, and then investigate the amount of rounds it takes for a rumor to reach every vertex of the graph (every vertex is burned). The burning number of a graph is the minimum number of rounds needed. We will explore some exciting algorithmic questions related to finding the burning number.

Mentor: Spencer Szabados

Mentees: Ananya Kumar and Zayaan Mulla

Description:

Data depth measures quantify centrality of data, and are useful for nonparametric analysis. Robust to noise, they aid out-of-distribution detection, e.g. spotting anomalous EKG rhythms. However, their super-linear computational complexity in dimension limits their applicability. This project aims to combine dimension reduction techniques like variational autoencoders to extend their use to higher dimensions, facilitating robust out of distribution detection in domains with untannable dimensions.

Mentor: AJ Fong

Mentees: Bethelhem Bezabeh and Melody Tian

Description:

Wallpaper patterns that repeat have a certain amount of symmetry: we can move around, reflect or rotate them and preserve the patterns. These symmetries are measured by groups, and we will classify the groups that can occur as symmetries of wallpaper patterns. This is the two-dimensional case of crystallographic groups, or the groups of symmetries of atomic structures of crystals.

Mentor: Yash Totani

Mentees: Bryan Chen and Natalia Weber

Description:

The connection between Elliptic curves and Modular forms played a huge role in the proof of Fermat's last theorem. Modular forms are functions on the upper half complex plane that have nice transformative properties. In this project, we aim to get a basic understanding of these objects. A rough sketch of the 'how' is described in the Goals section.

Mentor: Liam McQuay

Mentees: Amelie Zhou and Manya Singh

Description:

A ring is a type of algebraic structure where we can add and multiply elements. The prototype of this structure is the integers, but this also captures real numbers, matrices, polynomials, and more. We will study how, under certain circumstances, one can “break down” a given ring into simpler parts. We shall see that each such part of this decomposition turns out to simply be a set of square matrices. In our study, we shall also consider the interesting historical development of this theory.

Mentor: Annie Li

Mentees: Sara Nayar and Marshall Cowie

Description:

Tensor products come up in almost every area of pure mathematics. But often people find that they never really know what tensor products are. We will learn tensor products that are often used in functional analysis.

Mentor: Keke Zhang

Mentees: Hope Appelmans and Hanwu Zhou

Description:

Our project explores the captivating world of symmetry and structure, focusing on Lie groups and Lie algebras. We'll uncover the rules behind sphere rotations, crystal patterns, and quantum mechanics principles, simplifying complex systems. No prior knowledge needed besides linear algebra—just bring your curiosity!

Mentor: Erica Liu

Mentees: Stephanie Penner and Stella Xiao

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 system.

Mentor: Henan Xu

Mentees: Xinyang Li, Tiffany Wang, and Zhige Chen

Description:

Learning Causal Inference is like becoming a detective in the data world, figuring out how one thing truly affects another. It's more than just seeing patterns; it's about understanding why things happen. From whether a new teaching method works to the real effects of drinking coffee, we'll explore and apply these insights to a real problem by the end, making learning both fun and impactful. Dive into this adventure to unlock the mysteries of cause and effect!

Mentor: Minh Chau Nguyen

Mentees: Aastha Hiten Kotecha and Zhizhu Meng

Description:

Climate change and climate risk have recently been a major concern for the finance industry, as investors have to suffer dire consequences of changing natural conditions, at the same time they have the financial ability to drive changes. However, there are numerous factors that need to be accounted for in a sustainable financial decision. This project will introduce some of these factors and how they are measured and applied.

Mentor: Yan Yu

Mentees: Claudia McComb and Charlotte Liu

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: Xiaoya Wang

Mentees: Katherine Aryawan and Shu Cong

Description:

Explore the intricate web of cause and effect with 'Deciphering Causality: A Primer on Causal Inference.' This guide demystifies the art of causal inference, equipping you with the tools to untangle complex relationships. From grasping the basics to navigating real-world applications, embark on a journey of discovery and empowerment. Join us in decoding the secrets of causality and unlocking a deeper understanding of the world around us.

Mentor: Rhoda Dadzie-Dennis

Mentees: Aliard Jerome and Jingyan Ji

Description:

The impact of  climate change manifest in various forms like heatwaves, wildfires, and floods. For investors, quantifying these potential impacts on portfolio returns is a challenge. To start, our focus will be on understanding the types of climate risks that investors consider. Following this, we'll examine of the diverse portfolio techniques typically used by investors to mitigate these climate risks.  If time permits, we'll conduct feasibility tests on these portfolio strategies.

Mentor: Yuliang Shi

Mentees: Shuha Sheikh and Dyuti Bhatia

Description:

In real application studies, missing data can happen in multiple ways. In many cases, either the exposure of interest or the outcome may not be fully observed, which largely affects the estimated results. For example, patients' health status may not be entirely recorded when they unexpectedly drop out from clinical studies. We will read some articles related to missing data and investigate different ways to deal with this incomplete data issue, such as regression or imputation methods.