Spring 2025

Symmetry and Geometric Variational Problems by Letian Chen

Geometric variational problems seek to find special geometric structures that minimize a certain quantity. The most important examples are minimal surfaces, which minimize the surface area functional. In general, such a special structure will satisfy a partial differential equation (PDE). Solutions to PDEs are complicated and often times not explicit. However, under suitable symmetry assumptions, the PDE can be reduced to an ordinary differential equation (ODE) that is much simpler to solve. This project looks at problems arising from minimal surface theory and mean curvature flow, and studies the existence and (non)uniqueness of the symmetric solutions via ODEs.

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Ordinary differential equations (MATH 231) and multivariable calculus (MATH 241) are required. Some familiarity with numerical methods in any programming language is recommended. Differential geometry is not needed but helpful if previously seen or taking MATH 462 concurrently.

Microswimmers: the bi-level set representation and numerical integration by Vladimir Yushutin

Actuated swimming and propulsion through the environment of small shell-like structures is a fascinating phenomenon. Computational modeling of a microswimmer involves accurate numerical integration of functions over its surface. Clearly, this task becomes challenging as a microswimmer evolves and deforms, and one of the approaches is based on the level set description of surfaces. In this prominent method, a closed surface, e.g. a sphere, is represented by the so-called level set function which is positive outside, vanishes on, and is negative inside the surface. However, we are often interested in surfaces that have boundaries, e.g. a spherical patch, and a single level set function is not enough to describe them! To this end, we introduce the bi-level set method and employ a second surface with its own level set function, e.g. a flat plane for the spherical patch, which marks out the boundary on the first surface. The objective of the project is to create, implement and analyze a novel algorithm for the numerical integration over an evolving surface with boundary based on the bi-level set representation. The project outcomes are immensely relevant to the broad family of unfitted finite element methods such as CutFEM and will facilitate the mathematical modeling and simulation of microswimmers. Along the way, we will also learn modern programming techniques and will contribute the code to an open-source library deal.II which is used by thousands of researchers around the world.

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: Calculus III and Computational Methods/Numerical Analysis courses; good programming skills preferably using C++.

Modeling Cell-Cell Adhesion in Two Interacting Populations by Xinyue Zhao

Cell-cell adhesion is one of the most important interaction forces in tissues and organs. Cells communicate with their nearby cells by protrusions known as filopodia. The long fibers of filopodia extend a distance, known as the sensing radius, beyond the cell mass. The communication of neighboring cells by filopodia induces cell-cell adhesion and cell-cell repulsion. Cell-cell adhesion is a key element of organism development, cell aggregation, wound healing, and cancer invasion. A good understanding of this basic cell mechanism is crucial. 

In this project, we will consider a non-local PDE (Partial Differential Equation) model of two populations which interact through adhesion.  For each population, we assume there exist two types of cell-cell adhesive forces: self-population adhesion (adhesive force between cells of the same type) and cross-population adhesion (adhesive force between cells of different types). We plan to employ linear stability analysis and bifurcation analysis and conduct numerical simulations to explore the dynamics of these interactions. By varying the adhesion coefficients, we expect to observe a range of behaviors, such as cell mixing, cell engulfment, and cell sorting.

Difficulty: Intermediate
Team Meetings: Once per week
Prerequisites: MATLAB coding experience is essential. Courses related to Models in Biology, such as Math 405 and Math 411.

Mathematically Modeling Investment Methods by Stathis Chrontsios

Mathematical finance emerged as a discipline in the 1970s, and since then there has been a variety of investment strategies that experts claim to exceed the returns of savings accounts in the long-term. The most notable ones involve the ownership of US stocks, and in particular companies listed in the S&P 500 Index, especially after the introduction of exchange-traded funds in the 1990s. This project aims to build a mathematical model that can be used to theoretically and experimentally compare the returns of different investment methods, if they were to be implemented in different time intervals. Statistical and data analytic tools are to be employed for experiments on data collected on various stock market indexes.

Difficulty: Intermediate
Team Meetings: Once per 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: Intermediate
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.