Fall 2026

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++.

Grationality, with a Spoon

by Jeneva Clark

Interested in doing math using very simple tools? The analogy: It’s like digging a swimming pool with a spoon. At first this might sound foolish, but simplicity both pleases aesthetic senses and ignites learning. Plus, you can find really cool stuff when you are digging in the dirt! 

We will not use high-powered mathematics, but rely on geometric constructions, proportional reasoning, tiling, dissection, the Carpets Theorem, and proof by descent. One reason for this intentional simplicity is to challenge a popular belief that the “Spiral of Theodorus” is described in the works of Plato.

The word Grationality was introduced at a 2025 sectional meeting of the Mathematical Association of America. It’s a concept akin to rationality of numbers, but in a geometric context. A nice n-gon was defined to be a regular n-gon with side lengths that are natural numbers, and a number n was defined to be grational if and only if there exists a nice n-gon such that its area equals the sum of areas of n congruent nice n-gons. You can read the full article on arXiv, Cornell University’s open-access archive.

Difficulty: Easy

Team Meetings: Once a week

Prerequisites: None.

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.

A neural-based model of spatial decision-making and animal behavior

by Christopher Strickland

Recent work by an interdisciplinary team of experimental biologists, physicists, and mathematicians has resulted in an explicit mathematical model of neural decision-making that reproduces spatial patterns observed in the lab. Given two similarly attractive targets, organisms are observed to walk toward both until their angular distance reaches a critical threshold (a bifurcation point), at which point the organisms makes a decision to approach either the left or the right target. A mathematical model built on basic ideas of neural architecture both reproduces this behavior and predicts a bifurcation curve where organisms that will make a decision between two targets. In recent work, Dr. Strickland has expanded on this modeling framework in a way that allows for many avenues of exploration.

In this project, we will more fully explore the implications of this model across various scenarios and environmental conditions. Much of the work will be computational in nature, based on a Python-language code base that Dr. Strickland has developed, but the mathematics of this project are squarely within the realm of dynamical systems and differential equations. There may also be an opportunity to fit the model to real data from the lab. The goal is to better understand this link between neural architecture and behavior, with potential implications for real-time AI decision-making algorithms and our understanding of collective behavior in ecology as well as psychology in general.

Difficulty: Intermediate

Prerequisites: Good programming skills in Python are key for this project, including NumPy and Matplotlib libraries, knowledge of class structures for object-oriented programming, and some experience with debugging. Math 231 and Matrix Algebra are required. Other desirable classes include Math 411, Math 323, Math 371, and Math 431.

Meetings: Once per week

Proofs that Compile: Learning Lean and the Future of Mathematics

by Yulan Qing (undergraduate participants will be paid)

What if mathematical proofs were something you could build, debug, and play with—like LEGO structures or computer programs? In this project, we treat mathematics not just as something to write on paper, but as something we can construct piece by piece using the Lean proof assistant, a powerful programming language for doing mathematics.

We will start by learning Lean as a new language—much like learning Python or C++—but instead of writing programs that manipulate numbers or data, we will write programs that are proofs. Each theorem becomes a structure built from smaller components, and each step must fit perfectly, just like snapping LEGO bricks together. If something doesn’t fit, Lean tells us exactly where it breaks, and we fix it.

As the project progresses, we will move beyond basic constructions and begin formalizing results from upper-level mathematics (such as algebra, topology, or analysis). Along the way, we will learn how to reuse existing pieces (libraries), design our own tools (tactics), and collaborate on a large shared codebase called mathlib, which is used by researchers around the world.

A particularly exciting part of the project is exploring autoformalization: how to teach computers to help us turn human-written mathematics into formal proofs more quickly. We will experiment with strategies, tools, and even AI-assisted workflows that speed up the process—turning proof-building into something increasingly interactive and dynamic.

By the end of the project, students will have built a collection of fully verified proofs, contributed to an open-source mathematical library, and gained a new perspective on mathematics as something precise, programmable, and surprisingly playful.

Difficulty: Intermediate
Team Meetings: Once per week.
Prerequisites: At least one course in proof-based mathematics (e.g., abstract algebra, real analysis, or topology). No prior experience with programming is required—just curiosity and a willingness to build!

GenAI methods for applications in plant morphology

by Ioannis Sgouralis

Plant morphology is the study of plant structure and form which is essential in Biology and Agriculture. By analyzing the physical traits of plants, such as shape and size of roots, stems, leaves, and flowers, scientists can classify species, track evolutionary relationships, and identify adaptations to specific environments. Nevertheless, plant geometries are complex and difficult to study without specialized methods. Shape reconstruction is the process of creating a digital representation of an object’s geometry, from discrete data such as images, point clouds, or various sensor measurements and plays a crucial role in modern plant morphology studies that use discrete data to digitally recreate continuous plant structures and quantify their morphology.

Generative AI (GenAI) is an emerging family of artificial intelligence models that apply advanced mathematics and machine learning algorithms to produce abstract representations of geometrical shapes based on patterns learned from data. GenAI can enhance the accuracy of shape reconstruction, especially for applications in plant morphology, by filling gaps caused by missing data or simulating realistic structures for virtual geometries.

In this project we will apply GenAI and develop novel methods to study plant morphology. The project consists of two parts: 1) we will apply image processing methods to acquire our own data by discretizing visual representation of plants and their botanical elements; 2) we will apply new GenAI methods and machine learning to reconstruct their morphology and physical characteristics. This way we will provide new methods of data analysis that allow for the study of plants. 

Difficulty: Intermediate

Team Meetings: Once per week

Prerequisites: Multivariate Calculus at M241 level or equivalent and programming in MATLAB are necessary; familiarity with numerical methods at the level of M371 or M471 or equivalent is preferable, but not necessary.

Modeling Periodic Tumor Growth in Three Dimensions

by Xinyue Zhao

Tumor growth is influenced by many time-dependent biological factors, including nutrient availability, blood vessel supply, and cell death. In this project, we will study a mathematical model for tumor growth in which the tumor boundary moves over time. The model is a free boundary problem, meaning that the shape and size of the tumor are not known in advance but are part of the solution.

We will focus on the three-dimensional radially symmetric case, where the tumor is assumed to remain spherical and its size is described by a single radius R(t). This simplification reduces a complicated moving-boundary PDE model to an ODE for the tumor radius with time periodic coefficients. The project will investigate how periodic nutrient supply, such as daily cycles in food intake or vascular delivery, affects long-term tumor behavior. Some of main questions will be: When does the tumor vanish? When does it persist? How do the period, amplitude, and average values of the nutrient functions affect the long-term outcome?

Difficulty: Intermediate

Team Meetings: Once per week

Prerequisites: Calculus III and differential equations. Some familiarity with MATLAB is helpful. Prior exposure to PDEs or mathematical biology is useful but not required.