Fall 2026

The goal of this project is to develop materials, methods and activities to introduce basic natural approaches of teaching mathematics into the communities of parents, children and teachers with a focus on elementary school (K-5).

The approach will focus on building and developing the following:

1. Games to introduce intuition.
2. Motivation for mathematical ideas.
3. Visualizations of mathematical concepts.
4. Introduction of mathematical language through formal definitions.
5. Introduction of mathematical models – rules for the games.
6. Formulation of mathematical problems – how you “win” a game.
7. Instruction on algorithmic systematic solutions for the mathematical problems.

Project Mentors: Shamgar Gurevich and Ohhoon Kwon

Graduate Team Members: Junyi Cheng, Rahul Panda, and Taiwo Felicia Taiwo

Undergraduate Team Members: Avery Heinrich, Cassie Luo, Jiaxin Man, and Julian Shapiro

Gerrymandering is the phenomenon by which an electoral district is manipulated to benefit a particular political party or demographic of people. Perhaps you’ve seen examples bizarrely drawn precincts, or the high-profile Supreme Court case LULAC v. Abbott which challenged the recent redistricting of congressional districts in the state of Texas. There is an impressive amount of mathematics behind quantifying the “fairness” of a given district; the tools come from analysis, combinatorics, dynamics, geometry and probability. Our goal is to understand both the mathematics underlying the issue and its political consequences. We will mainly follow the Duchin—Walch text “Political Geometry: Rethinking Redistricting in the US with Math, Law, and Everything In Between”, supplementing it with current articles and mathematical papers.

So, what makes a subset of the plane an ideal district? It is agreed upon that districts should be topologically connected (i.e. contiguous) and by some measure “compact”. There are dozens of numerical formulas introduced by researchers to measure the “compactness” of a set. The easiest to state use isoperimetric inequalities to essentially measure the ratio between the perimeter of a set and its area. Duchin–Tenner argue that this unnecessarily penalizes regions with more natural geometry and can lead to the prioritization of land over people. Rather than focusing on the boundary of the set, the cut score and spanning tree score are determined by combinatorial properties of a dual graph associated to the districting plan.

Our goal for the semester is to study these notions of compactness and learn about the probabilitistic methods (Markov chain Monte Carlo, random walks) which determine whether a given district plan is drawn reasonably.

Project Mentor: Brandis Whitfield

Graduate Student Mentor: Kendra Ebke

Undergraduate Team Members: Samuel Goldstein, Neeraj Madeti, Lilliana Ruggini, and Eleanor SomsakHein

We are very accustomed to working in Euclidean geometries and adhering to its rules, but what changes when we break these rules? In this very exploratory project we will learn the fundamentals of hyperbolic geometry and build physical models to help visualize the properties. The depth to which and directions in which we explore hyperbolic geometry will be guided by the interests, questions, and discoveries of the students. Throughout the course of the semester, students will be expected to explore applications, projects, and other avenues and bring these ideas to the group for discussion.

Project Mentor: Grace Work

Graduate Student Mentor: Ren Watson

Undergraduate Team Members: Addie Aird, Samin Ashraf, Aidan Perry, Chloe Song, and Alexander Tierney

The discovery of hyperbolic 3-manifolds that fiber over the circle by Jorgensen around 50 years ago surprised many geometers. These objects have a strange, paradoxical nature — for example, as shown by Cannon and Thurston, they give rise to highly symmetric continuous surjective maps of the circle onto the 2-sphere. To complete this shocking story, Agol, Wise, and others showed that up to a finite cover, every closed hyperbolic 3-manifold fibers over the circle.

A natural question is then, given a closed hyperbolic 3-manifold $M$, how many surfaces in $M$ are, up to a finite cover, fibers in a fibration of $M$ over the circle. (In fancier language, we want to count virtually fibered surfaces in $M$, up to homotopy and finite covers.) Agol’s machinery already says that the number of such surfaces that have genus at most $g$ grows faster than any polynomial in $g$. But there is some reason to think that this growth rate might be superexponential in $g$, as is the case for the essential surfaces in $M$ that are not fibers in a fibration over the circle.

The goal of this project is to try to produce experimental evidence on whether this growth rate is superexponential or not. The idea is we can start with two virtually fibered surfaces, take some finite cover of each of them, then cut and paste them along some curve. There are super exponentially many ways making such cut-and-paste surfaces, but one big problem is we do not know if they are still virtually fibered or not. Our tool to verify this will be work of Cooper, Long, and Reid, who show that the cut-and-paste surface being virtually fibered is equivalent to some (similarity-) interval exchange information not having a fixed point, which is something a computer can check. So our work will be to build lots of these cut-and-paste surfaces, and write code to see if their corresponding interval exchange has a fixed point or not.

Project Mentor: Fernando Al Assal

Graduate Student Mentor: Jia Wan

Undergraduate Team Members: Alex Howen, Kevin Kauflin, Youngyeon Kim, and Xinyi Shi

If $G$ is a group, the braid group on 4 strands has a natural action on the set of 4-tuples $(g_1, g_2, g_3, g_4)$ of elements of the group generating $G$ and satisfying $g_1 g_2 g_3 g_4 = 1$. This action is combinatorially explicit and is surprisingly interesting in group theory, algebraic geometry (Hurwitz spaces), number theory (Galois action on Belyi covers), and topology (branched covers of the sphere with covering group $G$.) And yet it is surprisingly hard to understand in general what the orbits of the braid group are on this set of 4-tuples, though of course it is a finite computation for any particular finite $G$.

I have a few specific questions about “unexpectedly small braid orbits,” which I expect will be hard to answer in general; but I would like us to work out a lot of examples which I hope will lead us to ideas for proofs for certain classes of $G$. For example, one might ask: under what circumstances can one guarantee that $(g’_1, g’_2, g’_3, g’_4)$ is in the same braid orbit as $(g_1, g_2, g_3, g_4)$ whenever $g’_i$ is conjugate to $g_i$ for all $i$? This is often true but not always.

Project Mentor: Jordan Ellenberg

Graduate Student Mentor: Eshaan Bhansali

Undergraduate Team Members: Emily V Ostrowski, Nitiwit Sirimalaisuwan, Pengrui Song, Jack Vandervaart, and Karis Zhuang

An alternating sign matrix (ASM) is a generalization of a permutation matrix, in which some entries are allowed to be -1. ASMs have shown up in many areas of combinatorics; see the survey article James Propp: The many faces of alternating-sign matrices, and also the paper
Paul Terwilliger: A poset $\Phi_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices.

Matrix multiplication gives a natural way to combine two permutation matrices to get another permutation matrix. In this project, we seek a natural way to combine two ASMs to get another ASM.

Project Mentor: Paul Terwilliger

Graduate Student Mentor: Dhruv Kulshreshtha

Undergraduate Team Members: Elkhan Aday, Justin Hu, Tianyi Wang, and Daniel Youngberg

This is a very geometric project, where desirable properties of “reaction systems” are translated into purely geometric properties, and these geometric properties are analyzed in depth. Students will learn about a major conjecture in the theory of reaction systems, called the “global attractor conjecture” and a specific approach for proving it, based on the geometric properties mentioned above.

You will like this project (and its extensions in higher dimensions) if you like classical geometry and/or if you like thinking geometrically about problems from outside geometry.

Among other things, students will be encouraged to create images (for example in MATLAB, because some of these images are 3D, and MATLAB is very good at 3D images) that help us visualize the geometric constructions required by the proof of the global attractor conjecture. Instead of programming in MATLAB, students will be strongly encouraged to take advantage of (and will be shown many successful examples of) the use of AI to create computer code for this project.

Project Mentor: Gheorghe Craciun

Graduate Student Mentor: Hieu Nguyen

Undergraduate Team Members: Chenning Cai, Andrew Kulik, Donghao Liu, Mark Miller, and Honghan (Duncan) Shen