Math Club Research and Internship Opportunities

Román Aranda

FYRE

AY 25-26

Computing the genus of a matrix

The goal of this project is to build examples of multicurves in a surface with prescribed intersection numbers. The student would spend the first part of the semester learning about the topology of surfaces and combinatorics of filling curves.  During this project, the student can explore low-dimensional topology and introduce themselves to the world of mapping class groups.

MATH 106

Amy Bennett

UCARE, Volunteer

AY 25-26

Math Modeling with Future Elementary Teachers

Elementary teachers complete mathematical modeling tasks during preparation courses to learn about the modeling process and how they can help their future students develop early notions of proportional reasoning.  In this project, you will get an introduction to qualitative research methods and analyze artifact and survey data from future teachers in 3 semesters of coursework.  There may be opportunities to collect additional data, if interested.

Interest in teaching, strong writing skills

Lauren Cranton Helleter

UCARE

Summer 25

Monomial ideals from toric fans

Toric fans are collections of vectors and cones which define geometric spaces. They can also be used to define monomial ideals. I would like to investigate the generators of these ideals for different types of fans.

Knowledge of polynomial rings and monomial ideals required; knowledge of graphs, fans, and polytopes or more ring theory helpful

Huijing Du

UCARE, FYRE, Grant funded, Volunteer

AY 25-26

Mathematical model of cancer growth and treatment

This project focuses on developing and analyzing mathematical models of cancer growth and treatment using differential equations based multiscale modeling framework. The work involves studying the dynamics of cancer and healthy cells, extending models to include competitive interactions, and performing numerical analyses. Additionally, we aim to explore the implications and optimization of cutting-edge treatments such as FLASH Radiotherapy.

MATH 208 & 221; familiarity with mathematical modeling or interest in applying math to biological systems; basic knowledge of numerical methods and experience with programming (e.g., Python, MATLAB, or similar); and an interest in biological applications and cancer treatment dynamics

Yvonne Lai

Volunteer, UCARE

Summer 25, AY 25-26

Math Wars: What are they?

In the 1990s-2000s, there was a lot of public fighting about math education ... in a way similar (and also different) from the discourse that is happening right now about education. This project will interview professionals who lived through the math wars of the 1990s-2000s to gain clarity on what the cost of events like Math Wars are, and potentially how to mitigate them.
Willing to give feedback on your writing of a UCARE proposal.

Curiosity about the purpose of education

Glenn Ledder

NEMO Project

Spring 25

Exploring the Exciting World of Dynamical Systems: Stability, Chaos, and Real-World Applications

Have you ever wondered how mathematical equations can predict the behavior of complex systems like the spread of diseases or population dynamics? Dynamical systems, described by equations of the form y′=f(y), provide a fascinating way to explore how such systems behave over time. These systems have special solutions called equilibria, where nothing changes over time.
In this project, you'll dive into the long-term behavior of these systems. Are they stable, meaning nearby solutions settle down to equilibrium? Or are they unstable, leading to fascinating phenomena like periodic behavior (limit cycles) or unpredictable chaos? We'll focus on systems with 3-4 components, exploring whether their solutions resolve into beautiful, recurring patterns or chaotic, unpredictable behavior.
Using a mix of analytical and numerical methods, you'll investigate cutting-edge models in mathematical epidemiology—a field that uses math to understand how diseases spread. This hands-on project is perfect for curious minds eager to combine theory and computation while working on problems with real-world impact.
Apply at https://forms.gle/6ZFDj2EYV7fST4387 ASAP!

MATH 221

Shiying Li

Volunteer

Summer 25, Fall 25

Optimal transport-based learning for single cell analysis 

High-throughput single-cell technologies offer unique opportunities to identify cellular heterogeneity and gain insights into biological processes. Optimal transport (OT) provides a natural framework for comparing single-cell data at an ensemble level, allowing us to treat all measurements of a single sample or patient as an entity without discarding valuable information. This project involves integrating machine learning techniques with OT for single-cell analysis such as uncovering cell-types, discovering cell populations, and differentiating specific phenotypes. Key outcomes will include a survey of these very recent OT-based methods for single-cell analysis, alongside the implementation and comparison of these methods against state-of-the-art unsupervised learning techniques on benchmark datasets. Given the nonlinearity and high-dimensional nature of single-cell data, we will also explore various preprocessing methods and variants of OT-based metrics to enhance computational efficiency. 

MATH 314 or 315. We welcome motivated students interested in mathematical data analysis and biological data interpretation to apply. This project includes both theoretical and computational components. Students who may not have programming experience but are eager to explore data geometry, nonlinear dimension reduction methods, and Optimal Transport (OT) will find the theoretical aspects interesting. While experience with any programming language is a plus and strongly encouraged, it is not a strict requirement.

Petronela Radu

UCARE, FYRE, Grant funded

Summer 25, AY 25-26

Nonlocal models

Nonlocal models basically allow us to approximate derivatives with integrals. There is a lot to discover, theoretically, numerically, and from a modeling point of view.

MATH 107, MATH 221

Wendy Smith

UCARE, FYRE, Grant funded

Summer 25, AY 25-26

STEM education research (exact opportunities vary)

A variety of opportunities to engage in STEM education research, which could involve collecting data through interviews or observation, analyzing data (qualitative and/or quantitative), helping write grant proposals, supporting outreach events, helping write manuscripts. Students working on these projects will also receive instruction on how to do whichever tasks are entailed; work is usually completed with collaborative teams.

Interest in teaching and learning and a willingness to learn about education research methods

Kazuo Yamazaki

Volunteer

AY 25-26

Theoretical Analysis on Deterministic and Stochastic Partial Differential Equations of Fluid Mechanics and Mathematical Physics

A system of partial differential equations is useful for modeling and analyzing turbulence. Stochastic analogue has been utilized in mathematical physics, e.g. quantum field theory and Yang-Mills theory. The purpose of this project is to study recent research developments in such areas, deterministic or stochastic accommodating to students' preferences. Other optional fields are biology and finance.

Experience in rigorous analysis is the most fundamental prerequisite in research of Analysis & PDEs. Experience in proofs using epsilon and delta is needed. The more analysis the student has taken the better (e.g., Topology, Measure Theory, Geometry, Probability Theory, Theoretical Physics). Experience in PDEs is preferred but not required.

Alex Zupan

UCARE, FYRE

Summer 25, AY 25-26

Fusion numbers of ribbon knots

A "knot" is a simple closed curve in 3-dimensional space, considered up to smooth deformations that can bend and stretch, but don't break, the curve.  Distinguishing knots is, in general, a very difficult problem, and so we often pass to "knot invariants", tools to help us distinguish different knots.  In this project, we will study and catalogue a particular geometric invariant, the "fusion number" of a ribbon knot.  

MATH 208, MATH 221, or MATH 314