Current DRP Projects

Mentor: Melissa Stadt

Mentees: Betty Zhang and Lily Zhou

Description: 

Potassium plays a crucial role in numerous physiological processes, including heartbeat regulation and the functioning of muscles and nerves. Dysregulation of potassium is a common side effect of pathologies, medications, and altered hormone levels. Despite its importance, our understanding of the body’s mechanisms for maintaining potassium balance, particularly in the face of pathologies, medication effects, or hormonal changes, remain incomplete.

This project aims to explore the complex system of potassium regulation through the lens of mathematical modeling. We will use an ordinary differential equation model to investigate the regulatory mechanisms of potassium balance during applications such as exercise. The objectives of this project are (i) to provide an introduction to the field of mathematical biology, (ii) to develop skills in coding ODE models using MATLAB, and (iii) enhance the ability to read and comprehend research articles in interdisciplinary fields.

Mentor: Athira Kumar

Mentees: Alice Sun and Austin Zhu

Description: 

Infectious disease models study the spread of contagious diseases through a population. These models help to understand how a disease (eg: COVID-19, Malaria, chicken pox,...) spreads in a population and how it splits the population into different non-intersecting compartments. In general, the whole population is prone to a disease but only a few of them get infected. After being infected people might recover with lifelong immunity (eg: Chickenpox) or can be prone to the same disease again (COVID). Mathematical models help to understand a system, study the various factors that might affect the system, and understand the behavior of the system. In this project, we will understand how these models are developed, what are some basic concepts used in developing these models, and also about the standard models that exist. We will also learn to analyze these models and finally write a research project which will help in developing the skills required for research writing.

Mentor: Liang Chen

Mentees: Natalia Manturowicz and Alicia Lin

Description:

Neuroscience is the scientific exploration of the nervous system, encompassing the brain, spinal cord, and peripheral nervous system. This reading project is to familiarize participants with essential concepts in neuroscience. Initially, we delve into the typical behaviors exhibited by individual neurons. It is to encourage readers to perceive neurons not only in terms of ions and channels, as often done in biology, but also as dynamic systems, aligning with the perspectives of mathematicians. Subsequently, we explore the collective activity of large groups of neurons and their interactions, forming what is known as a neural network. This level of analysis provides valuable insights into the foundational components of brain function. Understanding neural dynamics at this scale is crucial for comprehending normal cognitive functions, such as attention and working memory, as well as disorders like epilepsy or Parkinson's disease.

Mentors: Matheus Jun Ota

Mentees: Xinyue Fan and Crystal Zhou

Description:

Suppose that we need to construct the logistic network of a company. We have to set up the distribution centers in a way that we can attend to the demands of the company stores. Our goal is to construct this network with the minimum cost possible. In a deterministic setting, we might model this problem as a type of facility location or a spanning tree problem. However, real-world is often subject to uncertainties and we might not know precisely the cost of travelling between A and B or we might not know exactly the demand of some stores.

Robust optimization (RO) is a paradigm that lets us tackle problems with uncertainties like the one described above. The aim of this project is to learn a little bit of RO, see some applications of RO to discrete optimization, and (possibly) implement (in any language) an algorithm for some robust variant of a problem. This project will be somewhat related to operations research, so it would be nice to have a good balance between theory/practice.

Mentor: Cicely Henderson

Mentees: Helena Devinyak and Molly Wu

Description:

In discrete math, a graph is a network of dots called vertices and lines between them called edges that can be used to visualize combinatorial problems. In 1979, Regina Tyshkevich defined an operation on graphs similar to multiplication with integers: just like you can multiply two numbers, you can Tyshkevich compose two graphs to get a bigger graph. Once you know how to "multiply" graphs, you can also "divide" them: a graph is decomposable if it is the Tyshkevich composition of two other graphs (like a composite number), and it's indecomposable otherwise (like a prime). But what makes Tyshkevich's operation so special? Tyshkevich proved a powerful theorem analogous to prime factorization of the integers: just as every integer has a unique prime factorization, every graph has a unique Tyshkevich decomposition! In this project, we will cover basic graph theory, study split graphs (the building blocks of Tyshkevich composition), and investigate how to tell if a graph is decomposable.

Mentor: Jerónimo Valencia-Porras

Mentees: Gloria Wang and Melody Tian

Description:

A polytope is the smallest convex set containing a given finite set of points in Euclidean space. In the plane, a polytope is a convex polygon. If we restrict to polygons such that all corners have integer coordinates, a beautiful result from Pick in 1899 relates the area of the polygon with the total number integer points it contains. In 1956, Reeve gave a generalization of Pick’s Theorem for the case of 3-dimensional polytopes. These types of integer (or lattice) point counting results are the tip of the iceberg of Ehrhart Theory.

Ehrhart theory studies the “lattice point enumeration function” of a polytope. A famous result from Ehrhart in 1962 shows an unexpected property of this function, whose consequences have motivated current research and further generalizations of the theory in the last decades. 

This project will cover the basic notions of polytopes and lattice-point enumeration, aiming to understand Pick’s, Reeve’s and Ehrhart’s theorems. 

Mentor: Fernanda Rivera Omana

Mentees: Grace Terhljan and Soumya Menon

Description:

Ramsey Theory is an area of combinatorics that shows that if an object is big enough, any colouring of it with a fixed number of colours, will contain a smaller monochromatic substructure. That is, a substructure where each element has the same colour. Anti-Ramsey theory focuses on a dual problem. Given an object of fixed size, finding the minimum number of colours needed such that any colouring of it will contain a rainbow substructure. Where a rainbow structure is a structure whose elements all have different colours. This number was first introduced by Erdős, Simonovits and Sós. The objective of this project is to learn basic concepts of graph theory. We will focus on Ramsey and Anti-Ramsey theory with the objective of reading a paper that bounds the Anti-Ramsey number of diamond graphs.

Mentor: Owen Sharpe

Mentees: Daniel Liu, Tengyi Xu, and Jazlynn Leung

Description:

We will investigate a broad selection of primality testing and factoring algorithms. These algorithms are extremely important in cryptography and have intrinsic mathematical value. Specific algorithms include the sieve of Eratosthenes, the Miller-Rabin probabilistic test, the Lucas-Lehmer test for Mersenne numbers, the AKS test, trial factoring, Pollard's rho algorithm, and Shor's quantum computer algorithm. Our focus will be primarily on the number theory underlying these algorithms and secondarily on comparing their performance and use cases

Mentor: Aisha Khatun

Mentees: Kun Zhu and Qi Fang

Description:

The hype about LLMs is eating us alive. I want to use it for something but how do I get started? In this project, you will dive into the technical world of LLMs. You will use huggingface to run your own LLM, try various different LLMs, try running them in CPU, small GPU, and see how we have so many more options than just ChatGPT. Now that we have all these options, we can choose any LLM and use it to generate stories. Perhaps even compare stories produced by various models with ChatGPT, GPT4 and others. If you have a use case in mind, we can switch story generation with your use case. Your call!

Mentor: Punit Kunjam

Mentees: Sirui Wu

Description:

Embark on a fascinating journey into the world of problem-solving and logical thinking with the Tower of Hanoi! Imagine creating a captivating game using Unity, 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, they'll unlock the secrets behind this classic puzzle, setting the stage for deeper exploration and potential joint research opportunities. Let's go to this exciting venture, where fun meets fundamental mathematical principles!

Mentor: Shashank Joshi

Mentees: Yuhan Wu and Ashely Ge

Description:

The proposed project delves into the heart of blockchain technology and declutters the buzz revolves around the decentralised technology paradigm, focusing primarily on the “consensus models” which refers to the rules and processes that allow all participants in a blockchain network to agree on the validity of transactions (or operations) in absence of a central authority or a regulator. Imagine blockchain as an immutable digital ledger where transactions are recorded and for this ledger to be trustworthy, everyone involved must agree on what gets recorded. This is where consensus models come in. A consensus algorithm is a mechanism that guarantees the reliability of the blockchain and helps all connected nodes or peers to reach common ground regarding the present state of the blockchain network thus an ideal consensus algorithm must be secure, reliable, and fast.

Mentor: Sina Kamali

Mentees: Keqing Wang and Weixin Huo

Description:

In this project, we plan to familiarize ourselves with Privacy Enhancing Tools (AKA PETs) to understand their importance in today's world. These tools can range from simple password-managing apps to sophisticated ones that users use to avoid being monitored over the internet(e.g., VPNs) and even blockchain-related technologies. We plan to get a good understanding of what these tools are, how they work, why they matter, and what goes into creating one. An example would be censorship evasion tools. These tools are the only way for people living in censored countries (such as Iran, Russia, and China) to access free internet, and in their absence, countless lives and businesses around the globe would be compromised. The way researchers and companies create new and creative ways to evade the censors is astonishing.

Mentor: Vihan Shah

Mentees: Rachel Ma and Vivian Guo

Description:

In our project, we delve into the world of "Algorithms for Big Data: Sublinear and Streaming Algorithms." Imagine handling vast amounts of data so massive that it cannot fit into your computer's memory. That's where sublinear and streaming algorithms come into play! We're on a quest to develop smart algorithms that efficiently process enormous datasets without the need to store them entirely. It's like solving puzzles piece by piece instead of trying to fit the whole picture in one glance. These algorithms not only make it feasible to analyze massive streams of data but also provide clever shortcuts to get approximate answers quickly. Join me on this journey to explore the fascinating realm of algorithms that power the backbone of handling Big Data challenges. While a strong foundation in probability (and graph theory) would be advantageous, it's perfectly fine if you're at the beginning of your exploration in the field of computer science!

Mentor: Lena Podina

Mentee: Akira Yoshiyama

Description:

Transformers (a machine learning architecture) are currently state-of-the-art in many language tasks, including automated voice recognition and machine translation. The goal of this project is to apply transformers to extend the work of a recent paper "A high-performance speech neuroprosthesis" (https://www.nature.com/articles/s41586-023-06377-x). This work uses RNNs (recurrent neural networks) and language models to translate electrical signals from the brain into words. This allows a person who cannot speak to produce words through attempted speech. The authors have published their code and dataset, and encourage others to create architectures that attain better performance than theirs.

Mentor: Cameron Seth

Mentee: Aadila Ali Sabry

Description:

The graph container method is a tool that can be used to count the number of independent sets in graphs. It roughly says that the independent sets in any graph are clustered together into “containers”. This method has many applications throughout combinatorics because many problems can be rephrased as problems involving independent sets.

Recently, it has been shown that the graph container method also has applications in algorithms. In particular, it was shown that the container method can be used to make faster algorithms for some NP-Complete problems, and also has connections to an area called property testing.

We will start the project by learning about the graph container method, and then we will aim to understand these recent applications to algorithms.

Mentor: Kaleb D Ruscitti

Mentees: Elizabeth Cai and Avery Cormier

Description:

Elliptic curve cryptography (ECC) is an approach to cryptography that uses the algebraic geometry of special “elliptic curves” to obtain equivalent security to classical cryptographic systems, but with smaller keys. Elliptic curves are a foundational type of geometric curve that sit in the 2D plane, and arise frequently in number theory, algebraic geometry, and physics. They are naturally described using group and ring theory and make a great introduction to algebraic geometry. Although cryptography is just one application, it gives us a framework to learn about elliptic curves and to test our knowledge with hands-on computations, or even computer implementations. After learning the basics of elliptic curve cryptography, there are many possible directions that we could take, depending on your background and interests. For example, we could look at integer factoring, attacks on ECC, or even the basics of quantum computing

Mentor: Aareyan Manzoor

Mentees: Alice Schroeder and Elysia Wang

Description:

Most fields of math are dedicated to studying some kind of theory, e.g. group theory studies the theory of groups. Model theory is the theory of theories. A model is a structure in which the theory is interpreted and true. For example, a group is a model to the theory which consists of the group axioms. When does a theory have a model? How "many" models are there? How does the structure of the theory influence the relationship between its models? These are the types of questions we will attempt to answer.

Model theory can be applied heavily in algebra (especially algebraic geometry), combinatorics, analysis (especially operator algebras) and others. Discussion of applications will be tailored to mentees' interests. Model theory and the metamathematical way of thinking is extremely powerful, and will be beneficial in most things pure math.

Mentor: Jeremy Champagne

Mentees: Jackie Liu and Lydia He

Description:

This project will be centered around the following question: why (and where) does the golden ratio 1.618… show-up in plant patterns? An easy answer would be that plants attempt to optimise some system (e.g. having well spread leaves to catch as much sun as possible), and that the golden ratio happens to be involved in the optimal solution. I have gathered several papers that explore the subject,  which involve a wide array of mathematical theories, such as Diophantine approximation, equidistribution theory and even modular forms. Overall, we will use this simple sunflower question as a stepping stone towards understanding the relevance of pure mathematics in the so-called « real world ».

Mentor: Yiran Wang

Mentees: Keeley Isinghood and Haiyan Zhu

Description:

Bayesian Inference is a fascinating field gaining popularity. We'll kickstart our journey with two books "Bayesian Data Analysis" and "Bayes Rules!". By the end of the first part, you'll grasp the essentials of Bayesian inference: prior and posterier distribution, MCMC, and the Bayesian-Frequentist debate. As you progress, we'll delve into more advanced materials and also the applications in your areas of interest. Get ready to unlock the power of Bayesian thinking!

Mentor: Xianwei Li

Mentees: Yanqi Gao and Micky Liu

Description:

Time-to-event data refers to a specific type of data measuring the time it takes until an event occurs. Survival analysis helps answer questions related to this data, such as predicting how long a patient will survive, estimating how likely an electronic component will fail at a specific time, assessing whether a drug is effective in improving life expectancy, etc. In this project, we will follow the beginner-friendly book: Survival Analysis, A Self-Learning Context to learn basic techniques for analyzing time-to-event data. We will then start reading a few applied papers which were not published in statistical journals but employ survival analyses in their disciplines. Mentees will learn how to interpret the tables and graphs presented in these papers. If time permits, we will skim-read a few papers about the intersection between time-to-event data and other areas in statistics, such as machine learning for survival data, to expose mentees to some cutting-edge research questions.

Mentor: Jingyue Huang

Mentees: Tianai Liao and Ziyi He

Description:

Have you ever wondered how we can determine the true impact of one event on another? Are you intrigued by understanding the impact of policies or uncovering the effects of interventions? Join our project on "A Journey into Causal Inference" to unravel the mysteries of causal effects in the world around us. In this captivating journey, we'll navigate the fundamental concepts of causal inference, learn to distinguish association from causation, and acquire essential tools to compute causal effects. Just bring your curiosity and enthusiasm to delve into the fascinating realm of causal inference. Let's embark on a learning adventure together!

Mentor: Shiyu He

Mentees: Anne Zheng and Andie Cao

Description:

The questions that motivate most studies in the health, social and behavioral sciences are not associational but causal in nature. For example, researchers might be interested in studying: What is the effect of smoking on the incidence of respiratory diseases? What fraction of past crimes could have been avoided by a given policy? Conventional statistical tools, such as linear regression models analyzing relationships between variables, primarily focus on examining associations. In this project, we will explore the shift from merely finding associations to uncovering causal relationships. This exploration is guided by a general theory of causation based on the Structural Causal Model, a framework developed by Judea Pearl, a Turing award recipient. Throughout the project, we will learn the assumptions underlying causal inferences, the languages used in formulating those assumptions, and the nature of causal and counterfactual claims.