Summer Undergraduate Research Institute in Experimental Mathematics (SURIEM)
Overview
Lyman Briggs College at Michigan State University will host the Summer Undergraduate Research Institute in Experimental Mathematics (SURIEM) in 2026. This eight-week Research Experience for Undergraduates (REU) is supported by Michigan State University (MSU) and the National Science Foundation (NSF).
Program Highlights
Participants will collaborate with faculty from these MSU units:
- Lyman Briggs College
- Department of Mathematics
- Department of Statistics and Probability
- Department of Computational Mathematics, Science, and Engineering
The research experience will include the following:
- Reading and analyzing research articles
- Formulating conjectures and asking many questions
- Constructing examples, testing ideas using coding, and writing proofs
- Working as part of a small team (2-3 participants, plus mentors)
- Documenting and presenting research discoveries
Research Projects
Projects for 2026 are listed below.
Some computational problems in optimal transport (and related areas)
Mentors: Jun Kitagawa and Farhan Abedin
Description: The optimal transport (Monge-Kantorovich) problem, in its simplest form, asks: If you have a number of factories and warehouses with equal total capacities, how can you transport your goods from the factories to the warehouses in the cheapest way possible? The problem turns out to be related to many areas in pure and applied mathematics. This project will deal with some computational approaches to solving the problem in certain cases. Possible project ideas are: implement a solver for a parabolic partial differential equation related to the problem, in special cases in one or two spatial dimensions, find a computational approach to the principal-agent problem (a related problem from economics) in one dimension, or implement a solver for a near-field optics problem (a related problem which looks for a lens or mirror which creates some pattern of light from a given light source).
References:
- "Computational Optimal Transport" by Gabriel Peyré and Marco Cuturi.
- "Numerical Analysis of the 1-D Parabolic Optimal Transport Problem” by Abby Brauer, Megan Krawick, and Manuel Santana.
- "When is multidimensional screening a convex program?” by Alessio Figalli, Young-Heon Kim, and Robert J. McCann.
- "Optimal Transport and Applications to Geometric Optics” by Cristian E. Gutiérrez
Modeling Tissue Growth
Mentors: Olga Turanova and Christian Parkinson
Description: The goal of this project is to better understand how a tissue's environment affects its growth. Think of, for instance, a cancer tumor growing in an organism. Some of the existing tissue may be favorable to the tumor, and some may be unfavorable. Can we characterize "how much" unfavorable tissue would prevent the tumor's spread?
The model we will be focused on is formulated in terms of a partial differential equation (PDE). It's no problem if you have never worked with PDEs before! We will tackle a simplified version for which you already have many tools --- namely, what you learned about ordinary differential equations (ODEs) in calculus class. And, we will explore the full model using numerical simulation; part of the project will be learning how to do this.
References:
- The following work from 2003 answers some of these questions for the case of organisms, rather than tissues, invading a fragmented environment. Since the publication of this work, there has been a LOT more research on this topic! https://pubmed.ncbi.nlm.nih.gov/14522170
- Dr. Turanova's preprint (with her former PhD student), which studies a model of tissue invasion in a heterogeneous environment: https://arxiv.org/abs/2503.19849
Coefficients of Quantum Knot Invariants
Mentor: Matthew Harper
Description: If you tie a piece of string and glue the ends together, you have created a mathematical knot. When thinking about knots, it is natural to ask whether a given knot can be untangled into a simple loop (an unknot), or whether two seemingly different knots are actually the same. Oftentimes, we will present knots using two-dimensional diagrams, such as with the trefoil knot.
One way to understand knots is to assign an algebraic object to them, such as a number or a polynomial, in a way that is independent of how the knot is represented by a diagram. Such an assignment is called a knot invariant. One class of such invariants, inspired by quantum field theory, is called quantum invariants. They are defined by assigning matrices to each “elementary diagram” in a knot diagram, but in some cases they may be computed entirely diagrammatically. These invariants are often simple to describe but provide powerful insights into the structure and properties of knots. The most famous of these is the Jones polynomial.
In this research project, we will investigate properties of a relative of the Jones polynomial, the quantum sl3 invariant. The plan for the project is to give an expression for the coefficients of the polynomial in terms of quantities which can be read directly from a knot diagram. Specifically, we aim to extend the results of Harper and Kalfagianni on the set of positive alternating knots by computing the fourth and fifth coefficients of the sl3 polynomial. This project can be understood from a purely combinatorial perspective; no background in topology is required.
References:
- Matthew Harper and Efstratia Kalfagianni. On the quantum sl3 invariant of positive links. Preprint 2025. https://arxiv.org/abs/math/2508.15153
- G. Kuperberg. Spiders for rank 2 Lie algebras. Comm. Math. Phys., 180(1):109–151, 1996.
- T. Ohtsuki. Quantum invariants, volume 29 of Series on Knots and Everything. World Scientific Publishing Co., Inc., River Edge, NJ, 2002. A study of knots, 3-manifolds, and their sets.
Program Dates
The tentative dates for the 2026 program will be May 18 – July 10, 2026.
Stipend and Support
Each participant receives a summer stipend. Additional funding is available for conference travel, supported by MSU and the NSF.
Eligibility
Applicants must:
- Be U.S. citizens, nationals, or permanent residents
- Have completed at least one year as a full-time undergraduate
- Be enrolled full-time in an undergraduate program in Fall 2026
How to Apply
Students apply using the NSF's Education and Training Application. Visit https://etap.nsf.gov/search, and in the Search Opportunities field, input the words "SURIEM REU 2026" then submit. Within the program description, there is a link to the SURIEM application.
Applications opened January 5, 2026. Applications received by February 28 will receive full consideration. Applications received after this date will be reviewed so long as positions remain.
Selection Criteria
Preference will be given to students early in their mathematical studies (e.g., rising sophomores or juniors). Competitive applicants will have completed:
- At least two semesters of calculus
- One proof-based mathematics course
Experience with or interest in programming is a plus. Applications will be reviewed starting in March and continue until all positions are filled.
Contact
For more information, please contact Dr. Robert Bell via email.
