## F24 DRP-Reading Projects

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## AM-Reading-1 Optimal control for ordinary differential equations

**Mentor: **Ala' Alalabi

**Mentees:** Haiyan Zhu and Neha Munje

**Description:**

How can a business maximize profit while using minimal energy? What is the best way to transfer a space vehicle between orbits with the least time or fuel? How can we maximize the yield of a distillation process while ensuring the product's purity? These questions highlight optimal control theory, which seeks the best way to achieve a goal with minimal cost. Our project will develop optimal control strategies for specific equations, focusing on linear quadratic control.

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## AM-Reading-2 An Introduction to Mathematical Climate Modelling

**Mentor: **Yusuf Aydogdu

**Mentees: **Adya Bhardwaj, Neva Wilson, and Sampoorna Prakash

**Description: **

In the era of climate change, climate modeling 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.

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## AM-Reading-3 Numerical solutions of common differential equations in mathematical physics

**Mentor: **Milad Moshayedi

**Mentees: **Aaisha Pathan and Simran Matharu

**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.

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## AM-Reading-4 How Math Unveils the Secrets of Infectious Diseases Like COVID-19

**Mentors: **Zoya Abbasi

**Mentees: **Gaurika Gupta and Yashila Barnwal

**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!

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## AM-Reading-5 Optimization Methods for Image Processing

**Mentor:** Phuong Dong Le

**Mentees:** Ha Dang Vu, Khoi Hoang, and Monica Trinh

**Description:**

In the last decades, image processing have caught attention, since in many real applications such as medical imaging, collecting amount of training examples is not always feasible. In this project, we would focus on image denoising and combine with total variation regularizer. We solve the arising minimization problem via different methods of optimization. The performances of approach would be addressed in several experiments on images.

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## CO-Reading-1 Mechanism Design and Algorithmic Game Theory

**Mentor: **Rian Neogi

**Mentees: **Daisy Li and Shu Cong

**Description:**

Many scenarios are governed by what we call mechanisms. For example, an auction is a mechanism by which a good is sold to potential buyers. The agents in an auction must follow a certain protocol: buyers send their bids to an auctioneer, who sells the good to the highest bid. The protocol by which these agents interact is called the mechanism. Mechanisms arise in elections, economic markets, and network routing. In this project, we'll study the design of mechanisms, and even design our own!

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## CO-Reading-2 Numerical Semigroups via Chicken McNuggets

**Mentor: **Harper Niergarth

**Mentees: **Heer Mehta and Margaret Puzio

**Description:**

In Europe, Chicken McNuggets are sold in packs of 6,9, and 20. What number of Chicken McNuggets can be ordered using packs of 6, 9 and 20? The set of all such numbers, called McNugget numbers, make up the McNugget semigroup, an example of a numerical semigroup. How many numbers are not McNugget numbers? How many different ways can we buy, say, 231 McNuggets? In this project, we will answer all these questions and more by exploring the combinatorics of sets like this.

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## CS-Reading-1 Software Foundations

**Mentor:** Edward Lee

**Mentees:** Feng Xiang Ming and Solara Lin

**Description:**

Ensuring that software behaves correctly is difficult! Checking that a program will behave correctly can require producing a mathematical model of that program and proofs concerning that model as well. In this project, we will study some popular tools that are used to formally check program behaviour -- the Coq proof assistant -- and work through the volumes of Software Foundations -- in particular, Volumes 1 & 2, for verifying properties of the programming languages that we write programs in.

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## CS-Reading-2 Simulating Fluids with Physics-based Animation

**Mentor: **Anchit Mishra

**Mentee:** Khanjan Soni and Yifei Wang

**Description:**

Understanding how fluids (i.e, liquids and gases) behave and being able to simulate them is an important problem. From designing F1 cars to predicting weather patterns, to media applications such as CGI in movies like Avatar, fluid simulation plays an important role in different areas of applied math and computer science. In this project, students will read (and potentially implement) significant papers that, over time, have shaped the state of the art techniques used in different industries.

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## CS-Reading-3 Exploring Explainable AI Methodologies

**Mentor:** Julie Wojtiw-Quo

**Mentee:** Deepika Sree Ramkumar and Harbin Dhillon

**Description:**

This project will explore what “explainable artificial intelligence” is as a domain and examine the different approaches and methods currently being used to facilitate explainable AI. We will briefly touch on what “AI” and “Machine Learning” are before diving deeper into why explanations can be valuable for these systems and models. We will also look at SHAP, one of the popular techniques used for explaining AI models, and time-permitting, examine others to compare and contrast.

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## CS-Reading-4 Evaluating the Efficiency and Security of Blockchain Consensus Models

**Mentor:** Shashank Joshi

**Mentees:** Radha Kotra, Kamakshi Sarvananthan, Lily Wang, and Rena Yang

**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.

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## PM-Reading-1 Singularities of curves

**Mentor:** Jiahui Huang

**Mentees:** Emily Wang and Nayantara Gummalla

**Description:**

The topic is geometry, the tool is algebra, and the result invokes some graph theory. Begin your study in geometry with the most fundamental object: curves (the only thing simpler is a point, which is trivial). We delve deep into their structure using methods developed in algebraic geometry over the past century. Our focus is on the process of resolving singularities, and we explore how graph-theoretic properties of these resolutions can help classify curves.

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## PM-Reading-2 Introduction to Lattices and Order

**Mentor:** Alec Gow

**Mentees:** Debra Rose DeFazio and Shichen Gou

**Description:**

Order theory provides a formal mathematical framework for making statements like "x is less than y". We will study the basics of order theory, with special focus on the theory of a particular structure called a lattice: a set with a (partial) order in which any two elements have an infimum and supremum. There are many possible avenues to explore after the basics are covered, depending on mentees' interests, ranging from combinatorics to analysis (my own area of research) and beyond.

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## PM-Reading-3 Free Groups and the Word Problem

**Mentor:** Nicolas Banks

**Mentees: **Micky Liu and Brenda Li

**Description:**

Groups are the basic objects in abstract algebra. They can be viewed algorithmically as words - strings of formal symbols - that are reduced, i.e. redundant symbols are removed according to some concrete, pre-specified rules. The "word problem for groups" asks whether there is always an algorithm to check if two words are equivalent given some rules. The answer, surprisingly, is "no", which is the content of the Novikov–Boone Theorem. This project is aimed at understanding this theorem.

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## PM-Reading-4 Word Hyperbolic Group

**Mentor:** Aareyan Manzoor

**Mentees:** Kenneth Xiao and Noah Shaw

**Description:**

A theme in math is to study an object through groups acting on it. The point of geometric group theory is to do the opposite, study groups by acting them on some geometric objects. One can define hyperbolic (negatively curved) groups. This is a very well behaved class: free groups and many arithmetic groups are hyperbolic. The undecidable word problem on general groups can be solved in linear time in hyperbolic groups! Only knowledge on groups needed, no geometry required!

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## PM-Reading-5 Primality Testing and Factoring Algorithms

**Mentor:** Owen Sharpe

**Mentees:** Manal Wasif and Sri Meghana Yarlagadda

**Description:**

We will start with a quick review of trial factoring and the sieve of Eratosthenes, then move on to the probabilistic Miller-Rabin primality test, the special-use Lucas-Lehmer test, and finally the theoretically-best AKS test. We will then investigate the Pollard rho and p - 1 factoring algorithms, and conclude with Shor's quantum algorithm. I will assign programming problems throughout. Students will present to each other once or twice a week.

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## PM-Reading-6 The Abel-Ruffini Theorem: or why there is no general formula for the roots of polynomials

**Mentor:** Yash Singh

**Mentees:** AJ Carson and Nicola Ablett

**Description:**

We have seen the quadratic formula which gives us the roots of a two-degree polynomial equation in terms of its coefficients. A beautiful theorem proved by Abel and Ruffini states that such a general formula is not possible for polynomials of degree 5 or higher. In this project, we will understand this theorem using Galois' beautiful ideas and weave across many areas of mathematics to prove this remarkable result.

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## SAS-Reading-1 When the Numbers Lie: An Analysis into the Misuse of Statistics in Research

**Mentor:** Jessica Xu

**Mentees:** Eike Zhao and Meerra Vasuthevan

**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!

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## SAS-Reading-2 Exploring Hotel Operations and Customer Behavior through Data Analytics

**Mentor:** Yan Yu

**Mentees:** Arwen Mao, Ellen He, Julia Wrona, and Rachel Tania

**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!

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## SAS-Reading-3 Deciphering Causality: A Primer on Causal Inference

**Mentor:** Xiaoya Wang

**Mentees:** Catherine McCulloch, Ivena Yeung, and Ridhika Madan

**Description:**

In "Deciphering Causality: A Primer on Causal Inference," we explore how to identify cause-and-effect relationships in data. For example, we’ll examine if studying more hours leads to better grades, or if it's just a coincidence. This project will teach you the basics of causal inference, a powerful tool for understanding why things happen. You'll learn to analyze real-world data, uncover hidden patterns, and make informed decisions based on statistical evidence.

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## SAS-Reading-4 An Introduction to Survival Analysis

**Mentor:** Xianwei Li

**Mentees:** Alina Hu, Calista Kurniawan, Maya Le, and Tina Yang

**Description:**

Survival data refers to a specific type of data measuring the time it takes until an event occurs. Survival analysis can help predict how long a patient will survive, estimate how likely an electronic components will fail, assess whether a drug is effective in improving life expectancy, and so on. We will follow a beginner-friendly book to learn basic techniques for analyzing time to event data and read a few applied papers together.

## F24 DRP-Research Projects

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## AM-Research-1 Modelling epidemic spread of plasmids in microbial communities

**Mentor: **Aaron Yip

**Mentees: **Chenyi Xu and Tong Liang

**Description:**

The rise of multi-drug resistant pathogens poses a significant threat to modern healthcare. One of the primary ways that antibiotic resistance spreads is through mobile genetic elements, such as bacteriophages and plasmids. To counter the spread of antibiotic resistance, it is important to understand factors that promote and inhibit spread of mobile genetic elements. It has been shown that different social interactions between microbes influence spatial patterns in their growth, which could affect gene transfer efficiency between a donor and recipient population. In this project, the student will investigate how different social interactions between microbes influence the spread of antibiotic resistance genes in microbial communities. They will use spatiotemporal models to simulate spread of mobile genetic elements and quantitatively answer hypotheses about gene spread and microbial interactions.

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## CO-Research-1 3-Uniform Hypergraph Gridwalking via The Tutte Polynomial

**Mentor: **Josephine Reynes

**Mentees: **Bandana Bajaj and Paige Stone

**Description:**

The Tutte polynomial is one of the most famous graph polynomials. Much of its invention is attributed to W. Tutte, one of the founders of the C&O department. Evaluations of the polynomial count many graph properties. The terms of the polynomial can be derived in several ways, including deletion and contraction of edges. These operations create moves on a grid. In this project we will investigate a class of graphs for which the Tutte polynomial is not defined called hypergraphs, specifically, 3-uniform hypergraphs by using their bipartite representations. Examining the grid walks of the bipartite representations will inform how the Tutte Polynomial could be extended to hypergraphs. While it is not known what a Tutte Polynomial on hypergraphs would count, this project would provide information possibly leading to a polynomial extension. This project has a lot of visual examples which make it a fun and approachable topic on an interesting new use for a famous polynomial.

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## CO-Research-2 Finding Combinatorial Specifications for Chord Diagrams

**Mentors: **Kimia Shaban and Tiadora Ruza

**Mentees: **Sherry Feng, Hanwu Zhou, Hope Appelmans, and Purev Batdelger

**Description:**

A key goal in enumerative combinatorics is to encode a family of objects into a generating function and count the number of objects which satisfy certain properties. To create the generating function, we often attempt to recursively decompose combinatorial classes to find a structural description. This process involves creativity and expertise, and is generally quite difficult. Recently, a framework, Combinatorial Exploration (CE), has been introduced to automate this challenging aspect.

In this project we will study CE from theoretical and computational perspectives and apply it to open problems. First, we will apply CE on chord diagrams, which have applications in knot theory and physics, specifically in parameterizing solutions of certain Dyson-Schwinger equations. In particular, we will consider chord diagrams which avoid certain patterns, determine decomposition strategies and implement them in Python. Time permitting, we will implement CE on other types of combinatorial objects.

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## CO-Research-3 Cutting-planes for Vehicle Routing Problems

**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.

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## CO-Research-4 The interplay of Tyshkevich decomposition and other graph operations

**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)?

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## CO-Research-5 Pick's theorem and its generalizations

**Mentor:** Jerónimo Valencia Porras

**Mentees:** Fillion Ding and Melody Tian

**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.

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## CS-Research-1 AI in Software Engineering: Evaluating LLMs in SE

**Mentor**: Noble Saji Mathews

**Mentees**: Anusheh Atif, Kriti Sodhi, and Nina Zhang

**Description**:

Are you intrigued by the transformative potential of Artificial Intelligence in Software Engineering? This project focuses on evaluating Large Language Models (LLMs) within the context of software engineering, exploring their capabilities, merits, and challenges. You’ll delve into the evaluation datasets used for LLMs, experiment with building LLM applications tailored for software development, and critically assess their performance. This project is an exciting opportunity to understand the impact of LLMs on software engineering, identify pitfalls, and discover innovative solutions to enhance their utility in the field. The project will involve practical experimentation with different LLMs / Agentic Systems, understanding their integration into software development workflows, and assessing their performance and understanding the limitations of present techniques and benchmarks to create a more rigorous framework to build and improve these systems.

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## CS-Research-2 Fully Encrypted Protocols

**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!