Spring 2025

Projects are expected to be announced during the first week of January 2025.

Fall 2024

Experiments on the random walk hypothesis by Stathis Chrontsios

The “random walk hypothesis” is a financial theory that has motivated several lines of research within different areas of finance and mathematics. It states that stock market price movements follow similar principles to those of a random walk. Consequently, according to this theory, there are no strategies to consistently make a profit out of trading or investing in the stock market. On the other hand, there are researchers and investors who believe stock price movements can be predicted to some extent based on the past trends of the graph and/or certain characteristics of the company that the stock represents. This project aims to experimentally compare certain stock graphs to those of random walks by employing probabilistic and data analytic tools.

Difficulty: Intermediate
Team Meetings: Twice a week
Prerequisites: Coding experience is essential. Specific knowledge of Python and/or Mathematica (Wolfram) would be helpful. Previous exposure to data analysis techniques would be helpful, but not expected.

Growing multi-slits with the complex-driven Loewner equation by Joan Lind

Given one or more functions, the Loewner differential equation provides a way to generate growing families of sets in the complex plane.  In the nicest (and most well-studied) situation, there is a single real-valued function (called a driving function) which generates a growing curve.  There are two different ways to generalize this situation:  first, one could consider multiple driving functions, and secondly, one could consider complex-valued driving functions.  We wish to combine both of these to study multiple complex-valued driving functions.  There are two goals of the project: (1) To create a method to generate simulations in Matlab, building off a program written by a former undergraduate research student.  (2) To analyze some key examples.  

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Coding experience is preferred. Having taken Math 443 (Complex Analysis) would be helpful, but is not required. 

Existence and Isolation Results for Complex Hadamard Matrices by Remus Nicoara

Complex Hadamard matrices are square matrices with entries of absolute value 1 and mutually orthogonal rows. They have important applications in many fields, including cryptography, quantum information theory, functional analysis, and harmonic analysis. A general classification of n x n complex Hadamard matrices is unknown, even for n as small as 6. The purpose of this project is to further the classification by finding new examples, by classifying Hadamard matrices with certain symmetries (such as certain entries being equal), and by proving isolation results. This will be accomplished through a variety of methods: Software will be used to generate approximate examples, which will inspire formulas to be proven for actual new examples. Analysis and number theory methods will be used to generate new examples (for instance based on complex roots of unity), and to study which matrices are isolated among all complex Hadamard matrices.

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Mastery of Math 251 (Matrix Algebra) material and Math 300 (Introduction to Abstract Mathematics) material. Strong proof-writing skills. Some coding knowledge, or experience with Mathematica/Matlab. Experience with more advanced coursework in Analysis and Algebra is not required, but it is useful.

AI for accelerating advances in super-resolution microscopy by Ioannis Sgouralis

Super-resolution microscopy stands at the forefront of biochemical and biological discovery, allowing scientists to visualize molecular processes with unprecedented clarity. However, this intricate technique faces challenges such as complex sample preparation, substantial computational requirements, and potential for phototoxicity during prolonged imaging sessions. Addressing these challenges requires extensive planning prior to an experiment and fine-tuning of the involved devices, such as microscopes, lasers, and cameras, that are time and resource consuming. Remarkably, AI is poised to revolutionize super-resolution microscopy by allowing automation of the fine-tuning process leading to fast, cheap, and reliable setup of a scheduled experiment. In this project, we will use a highly sophisticated mathematical model of fluorescence microscopy to obtain synthetic data on different microscopy configurations and apply state-of-the-art machine learning algorithms to come up with data-driven approaches to experimental design and optimization. This way, we will develop new methods to speed up the preparation of experiments and increase the quality of the data acquired in them.  

Difficulty: Advanced
Team Meetings: Once per week
Prerequisites: Coding experience and/or numerical analysis, and related courses 

Noise representations in microbial time series data by Ioannis Sgouralis

Microbial growth curves are essential for understanding the dynamics of microbial populations, which is critical in fields like biotechnology, medicine, ecology and environmental science. Growth curves depict the stages of microbial development over time, providing insights into replication rates, carrying capacity, and the effects of various conditions on enhancing or suppressing growth. Mathematical modeling translates these biological processes into quantitative descriptions, allowing for precise predictions, control, and principled data analysis. This fusion of biology and mathematics enables researchers to simulate complex scenarios, optimize cultivation methods, devise treatment strategies for infections, and understand ecological impacts, making mathematical models a vital tool in microbiology research and its applications. In this project, we will develop a mathematical model of microbial growth under limited resources to investigate how small microbial populations and random events interfere giving rise to apparent noise patterns. This way, we will develop new methods to represent biological information and facilitate the assimilation of experimentally obtained data through parameter estimation techniques.  

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Coding experience and mathematical biology or differential equations

A traveling salesman algorithm for analysts by Vyron Vellis

The Traveling Salesman Problem (TSP) asks to find the shortest path through a given number of points. The TSP is one of the most famous problems in computer science due to its vast applications in itinerary design, its influence on operations research, and polyhedral theory, and its immense computational complexity. There are many algorithms for TSP that give a “nearly optimal” path in polynomial time. One of them developed in the 90s is called the Analyst’s Traveling Salesman (ATS) algorithm and it has been pivotal in modern analysis. The objective of this project is to write a computer program that visualizes the ATS algorithm. The program should receive a number of points (in Cartesian coordinates) and will return the order in which the points will be visited.

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Very good knowledge of a computer language such as Python, for example.