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�ѱ�Գٴǰ�:��Saranya Varakunan

Mentees:��Erica Im and��Asia Celeste Mitchell

Description:��

Cancer progresses through complex interactions between tumour cells and the immune system. In this project, we will explore mathematical models that describe these interactions using ordinary differential equations (ODEs). Starting with basic tumour growth models, we will examine how ODE models can explain how immune cells attempt to control cancer and why tumours often evade immune attacks. We will also use stability analysis, equilibrium points, and phase plane analysis to understand long-term tumour-immune dynamics. No prior biology knowledge is required - just curiosity and an interest in mathematical modeling!

�ѱ�Գٴǰ�:��Mohsen Rezaeian

Mentees:��Yidan Wang,��Daria Jiang and��MingMing Zhang

Description:��

Cancer remains one of the leading causes of death, and developing effective treatments is a major challenge. One promising approach in cancer research is using cancer-on-a-chip systems—small, lab-grown 3D tumor models that closely mimic real tumors in the human body. These models offer a better alternative to traditional lab methods, such as 2D cell cultures or animal models like mice, which often fail to fully capture the complexity of human tumors. However, testing drugs directly on these chip models can still be expensive, time-consuming, and labor-intensive. In this project, we will explore how mathematical and machine learning (ML) models can be used to simulate the behaviour of tumors and predict how drugs will work in these chip models. By using these models, researchers can test various treatment scenarios in a cost- and time-effective manner before conducting real experiments.This approach can help improve drug testing and eventually to contribute better cancer treatments.

�ѱ�Գٴǰ�:��Chenxin Lyu

Mentees:��Grace Yin and��Irene Chung

Description:��

This project introduces students to stochastic differential equations (SDEs) and their applications in finance. SDEs help model the random behavior 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. No prior knowledge of stochastic processes is required, but familiarity with calculus and probability will be helpful..

�ѱ�Գٴǰ�:��Martin Liu

Mentees:��Taiqian Zhang and Bandana Bajaj

Description:����

Given a sequence of graphs where the number of vertices goes to infinity, can we establish a sensible notion of "limit"? This question is motivated by the need to study large networks such as the internet, for which some of the most basic questions in graph theory becomes inapplicable (e.g., does the graph have an even or odd number of vertices, is the graph connected, etc.) In a nutshell, the theory of graph limit develops a rigorous system of mathematics to study graphs with a huge number of vertices, using tools such as homomorphisms, random graphs, and regularity lemmas.

�ѱ�Գٴǰ�:��Fernanda Rivera Omana

Mentees:��Sahana Mia Nirmalan and Shahbano Yousafzai

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 Erd's, Simonovits and S's. 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:��Yuheng Wen

Mentees:��Shu Cong

Description:

If you've taken an introductory cryptography course, you've probably learned how to prove the security of basic cryptographic schemes. In this reading project, my goal is to guide you(or learn together)as we progress from writing security proofs for simple cryptographic primitives to analyzing more complex, real-world protocols. Specifically, we will focus on Continuous Group Key Agreement (CGKA), a scheme widely used in secure group messaging. Major companies like Cisco, Facebook, and Google are actively investing in its deployment.

Mentor:��Rian Neogi

Mentees:��Angela Skie Xu,��Lillian Mo,��Fiona Li and��Maria Gharabaghi

Description:

Consider the following tasks: (1) Design a road network in a city that minimizes traffic congestion, (2) Design a procedure to allocate a set of goods from potential sellers to potential buyers, (3) Design an algorithm to match DRP students to DRP mentors. All these tasks have one thing in common: They involve dealing with strategic agents whose goals differ from one another. In (1), people choose a route to their destination that is the shortest, even if it increases congestion. In (2), sellers and buyers may choose to over-report or under-report the value of items, if they find that it improves their net gain. In (3), it is desirable that a student and a mentor do not prefer each other over their assigned candidate. Algorithmic Game Theory involves design procedures in settings with self-interested strategic agents. We will first cover the basics of the field and then dive deeper into a subfield of your interest, potentially even designing our own procedure for an AGT setting.

Mentor: I. Vecna

Mentee:��Paige Schneider and Emilia Dragoev

Description:��

When we talk to our friends using technologies such as SMS or Internet-based messaging apps, third parties such as corporations and governments can learn a lot about us. They can almost always learn who our friends are, and they can frequently even read our private messages! We have technologies such as encryption and onion routing that can help protect our messages and associations with our friends from these third parties. I propose exploring different technologies for private messaging. Depending on the mentee's interests, we could focus on how these technologies are (or are not) implemented in existing messaging apps and/or theoretical improvements that have not (yet) been implemented in existing apps.

Mentor: Ruizhe Wang

Mentee:��Erin Walshaw and��Daria Novikova

Description:

From Design to Adoption: Privacy-Enhancing Techniques in Health Data Sharing description | Consider the following scenario: The Medical School at the University of ��ݮ��Ƶ asks you to donate your medical records to advance diabetes research. Would you be willing to contribute your data? What if the university assures you that your data will remain anonymous, restricted to authorized researchers, not shared externally, and deleted after one year would this change your decision? These privacy guarantees are provided by Privacy Enhancing Techniques (PETs). While designing a PET might be complicated, persuading people to trust and adopt them is also challenging. In the above scenario, adopting PETs might not be a priority to the Med School if it cannot attract people to donate. In this project, we aim to explore both challenges: how to design a good PET and how to persuade people to use it.

Mentor:��Qianqiu Zhang

Mentees:��Alexandra Roszczenko and��Sampoorna Prakash

Description:��

Monoclonal antibodies (mAbs) are versatile tools in both medicine and research. They can be used to treat diseases like cancer, autoimmune disorders, and infections, as well as to diagnose specific molecules and study biological processes. mAb therapy, hoding the key to a new era of innovative, life-changing treatments valued $237.64 billion in 2023 and expected to surge past $679 billion by 2033. Their functionality comes from the verstality of its structure which allows them to lock onto unique targets. Recent AI breakthroughs, including Nobel Prize winning AlphaFold, help researchers predict complex protein structures. In this project, we'll focus on antibody affinity prediction using deep learning. Binding is an essential process that evaluates how tightly an antibody binds its target, boosting the development of antibody design and ensuring more precise and effective therapies for patients.

Mentor:��Shufan Zhang

Mentees:��Ashley Ge, Dana Yuan and��Myra Gupta

Description:

Hybrid search is a cornerstone of today's large language model systems. It allows searching the (approximate) nearest neighbours (ANN) in the vector data space with constraints on structural data attributes. E.g., you may want to search for the treatment of a disease that is similar to an x-ray image you send to the search engine, but you want the results constrained to patients younger than 30. Some recent research [ZSF25, ICLR] shows the possibility of designing private ANN algorithms for sensitive data (such as medical x-rays), while it is still open questions for hybrid searches. This project aims to design efficient algorithms, that potentially explore secure primitives such as SGX or FHE and privacy-enhancing techniques such as differential privacy, for secure and private hybrid search systems.

Mentor:��Xiao Zhong

Mentees:��Alyia Hussein and Bowen Dai

Description:

Arithmetic dynamics asks a thrilling question: How do simple mathematical rules evolve over time when constrained by arithmetic? Imagine iterating a polynomial like f(x) = x² + c. Some orbits explode to infinity, others cycle forever, and a rare few balance on the edge of chaos. If we put some arithmetic restrictions on c, can we control the chaos of the orbits better? This field merges the precision of number theory with the wild beauty of dynamical systems, revealing hidden patterns in seemingly random processes. In this project, we'll explore these questions, starting with foundational concepts and culminating in the study of post-critically finite (PCF) maps, a special class of functions whose elegance bridges arithmetic and geometry. We will learn the dynamical and arithmetical speciality of PCF maps in terms of their Julia sets and height functions.

Mentor:��Faisal Al-Faisal

Mentees:��Isabela Souza Cefrin da Silva and Ellie Hamer

Description:

Elliptic curves lie at the intersection of different areas of mathematics. In popular (mathematical) culture, they're famous for being involved in the proof of Fermat's Last Theorem and for playing a role in mathematical cryptography. In this project, you will learn what elliptic curves are, where they came from, where they're going, and---most importantly---why they're so interesting!

Mentor: Jiahui Huang

Mentees:��Sara Nayar and Abdul Haseeb

Description:

Study the connection between calculus and topology with Morse theory! This project guides you through foundational concepts such as: critical points, gradient flows, and how they reveal the topology of manifolds. Here the word "flow" can literally be thought of as the flow of a river on a mountain, or your tear rolling down your cheek as you study mathematics. How does the behavior of the river reveal topological (geometric) information about the entire mountain?

Mentor:��Roberto Albesiano

Mentees:��Fatma Jadoon and��Olivia Larssen

Description:

Have you ever wondered how music would sound on a Sierpinski gasket? ��Armed with little more than Calculus, we will embark in a journey to the world of fractals, trying to understand how their rough but regular structure shapes the way differential equations--such as the wave equation describing sounds--behave on them. ��This project aims to be a playful introduction to fractal analysis: no prior knowledge is required, but we will have the chance to encounter some extremely fruitful ideas that reverberate in many fields of advanced mathematics.

Mentor: Xiaoya Wang

Mentees:��Jenny Ran and��Lillian Li

Description:

In research and data analysis, measurements are rarely perfect; errors can occur due to instrument limitations, human mistakes, or other factors. These errors can lead to biased conclusions and misleading results if not properly accounted for. This project introduces the concept of measurement error, its impact on statistical analysis, and methods to address it. Students will explore real-world examples, learn fundamental statistical techniques to mitigate errors, and gain insights into how measurement uncertainty affects research findings. No advanced mathematical background is required just curiosity and an interest in understanding how data quality influences conclusions.

�ѱ�Գٴǰ�:��Rhoda Dadzie-Dennis

Mentees: Alex Palmer,��Tanay Kashyap��and Kris Zhang

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 modern portfolio theory and 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:��Aelita Huang

Mentees:��Lana Gacevic and��Jia Sheng Hu

Description:

Have you ever wondered how long a patient might survive after receiving a new treatment? Or how long a product lasts before breaking down? These are the kinds of real-world questions that survival analysis helps answer. Survival analysis is a branch of statistics that deals with time-to-event data, that is, how long it takes for something to happen. Unlike regular data, this kind of data often includes individuals who have not yet experienced the event by the time the study ends (for example, a patient who is still alive at the end of a clinical trial). In this project, we will explore the fundamental ideas behind survival analysis using the textbook Survival Analysis: A Self-Learning Text by Kleinbaum and Klein.

�ѱ�Գٴǰ�:��Yixuan Zeng

Mentees:��Rohan Raghavan

Description:

How do we know if one thing causes another? This project explores causal inference, a statistical approach that helps distinguish correlation from true cause-and-effect relationships. We will also touch on survival analysis, which focuses on analyzing time-to-event data, such as how long a patient survives after treatment. Using real-world examples from public health and medicine, mentees will learn how to critically evaluate research studies and apply statistical methods using Python or R.

Mentor:��Yan Yu

Mentees: Drishti Handa, Kim Shin Tyler Ah Von, and Jill Shen

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!

�ѱ�Գٴǰ�:��Amrita Punnavajhala

Mentees:��Yushi Liu and��Shiqi Cai

Description:

How we, as humans, perceive and react to the climate change can affect the future of climate change. The goal of this project is for you to represent and understand interactions between human and climate systems (and how those could affect future climate change!) by setting up and analyzing a differential equation model. You can choose to work on whatever aspect of these interactions interests you - this is a relatively new area of study, and there's a whole lot of unexplored territory!

�ѱ�Գٴǰ�:��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:��Ronak Pradeep

Mentees: Nahal Habibizadeh, Kshama Patel, Cassidy Li and Roselyn Zhang

Description:

Ever wondered how search engines decide which results to show you? Typically, they rely on keyword matching, but that approach often falls short of identifying the most relevant answers. Our project aims to improve search engines by enabling them to reason more like humans. We employ a method called "test-time compute", which allocates additional processing time for the search engine to evaluate why a document aligns with your query. This approach enhances search effectiveness and provides explanations for why specific results are selected. In this project, you'll investigate how to develop these new paradigm of search models and gain practical experience with cutting-edge techniques in natural language processing and machine learning to improve information retrieval.

Mentor:��Alex Cowan

Mentees:��Sally Ann Hui,��AJ Carson,��Peter Yang and Suhao Hu

Description:

Many mysterious sequences of arithmetic interest, such as the number of divisors of integers or the number of solutions to polynomial equations over finite fields, magically arise in the Fourier coefficients of certain very structured analytic objects called automorphic forms. This project is about answering questions in arithmetic statistics by extracting information from automorphic forms using methods from analytic number theory. Learning these methods unguided can be daunting, as papers in the field are often long, technical, and dense with notation. This project's guided research project will impart practical working knowledge of the rather intuitive underlying techniques, so that one can both more easily understand the literature as well as calculate for one's self, which is quite satisfying and maybe even a little fun.

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