Spring 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.

Lodato proximity structures in coarse geometry

by Jeremy Siegert

Coarse geometry studies objects at the “large scale” in which one is concerned primarily with the properties (called “coarse properties”) of spaces that stay consistent across “backing up” from an object. Within this field notions of “nearness” or “closeness” at a large scale have proven to be useful in the investigation of some coarse properties. One well studied small scale structure related to nearness that has not yet been considered in a coarse context is the Lodato (or Leader-Lodato) proximity. In the small scale, Lodato proximities prove to be useful in the study of hypertopologies, function spaces, and compactifications. There is reason to expect that a coarse analog of Lodato proximities would be useful in studying coarse geometry as well. The goal of this project will be to formulate a coarse analog of Lodato proximities and explore their properties.

Goals:

• Formulate a coarse analog of Lodato proximities.

• Once a formalization has been made, investigate the relationships between the new coarse analog and existing structures in coarse geometry (coarse proximities, coarse structures, asymptotic resemblances, etc.).

• Compare the above relationships to the relationships between the various small scale analogs.

• Determine which interesting results about Lodato proximities in the small scale translate (in an interesting way) to a large scale context.

Difficulty: Intermediate

Team Meetings: Once a week

Prerequisites: Mathematical maturity at a basic level. Ideally Math 300 (Introduction to Abstract Mathematics) and Math 467 (Topology), but we can manage without topology experience.

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.

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.

Small Organism Collective Behavior in Fluid Environments

by Christopher Strickland

The movement and behavior of small organism collectives often play a key role in ecosystem function. Examples include marine larval plankton that are critical for the health of coral reefs, aerial plankton that are used as agricultural biocontrol agents, and locust swarms which can devastate crops. However, holistic modeling of scenarios like these can be a difficult multiscale problem involving individual locomotion dynamics within larger-scale flows. To address this problem, Dr. Strickland has developed an open-source, agent-based modeling library in the Python programming language called Planktos. It is targeted at collective behavior in 2D and 3D fluid environments with immersed structures and readily interacts with computational fluid dynamics data generated externally.

Last semester, KML students created a simulation and data analysis pipeline in Python aimed at studying the predation of Brownian particles (plankton) around different jellyfish swimmers. In Spring 2026, students will compare these results to models of reactive plankton behavior in which plankton act as a group and/or can react to changes in the local fluid velocity field to try and avoid predation. This is the first time anyone has studied the effects of reactive behavior around moving jellyfish models. Additional challenges may be available to students as well, depending on time and interest, including development of numerical libraries and sensitivity analysis for different modeling assumptions. Almost all of the work for this project will be done via coding in Python and using related software – this is not a pencil and paper math project!!

Difficulty: Intermediate

Prerequisites: Comfortable working with Python: NumPy and Matplotlib libraries and knowledge of class structures for object-oriented programming, some experience with debugging. Experience with the Pandas library is a plus. Also, courses in statistics, Calc III, ODE, and matrix algebra. Other desirable math courses would include MATH 323, 371, 411, and 431. Any CS experience is also a plus – particularly an algorithms, data structures class, and experience visualizing data in plots and charts.

Meetings: Once per week