For Summer 2026 we will recruit nine students to work on the three projects described below.
Graphs and groups ()
There are several graphs naturally associated with groups. In this project, we will focus on the commuting graph and the closely related centralizer graph of a group G.
The commuting graph has as its vertex set the noncentral elements of G, with an edge between two vertices g and h whenever gh=hg. The centralizer graph has as vertices the centralizers C_G(g) of noncentral elements g in G. Two vertices C_G(g) and C_G(h) are connected by an edge whenever Z(C_G(h)) is contained C_G(g).
Recall that the diameter of a connected component of a graph is the largest distance between any two vertices in that component, where the distance between two vertices is the length of the shortest path connecting them.
It is known that there is a bijection between the connected components of these two graphs that, in most cases, preserves their diameters. The goal of this project is to investigate whether the diameters of these connected components can be bounded in terms of the index [G:Z(G)].
Prerequisites: At least one semester of undergraduate Abstract Algebra.
References:
- Peter J. Cameron, , Lond. Math. Soc. Newsl. No. 513 (2024), 22–25.
- Peter J. Cameron, , Int. J. Group Theory 11 (2022), no. 2, 53–107.
- Rachel Carleton and Mark L. Lewis, , J. Group Theory 28 (2025), no. 1, 165–178.
- Nicholas F. Bieke, Rachel Carleton, David G. Costanzo, Colin Heath, Mark L. Lewis, Kaiwen Lu, and Jamie D. Peare, , Bull. Aust. Math. Soc. 105 (2022), no. 1, 92–100.
Lattice Point Geometry ()
We will explore questions in lattice point geometry motivated by problems in coding theory and algebraic geometry. No prior background in these areas is expected, and students will be introduced to the necessary concepts as the project progresses. Familiarity with abstract algebra at the undergraduate level may be helpful.
The main object of this project is the lattice size of a lattice polygon. A lattice polygon is a convex polygon P in the plane whose vertices have integer coordinates. The lattice size of P is the smallest number l ≥ 0 such that, after applying an affine unimodular transformation A (that is, a change of coordinates that preserves the integer lattice), the polygon A(P) fits inside the dilated simplex l∆. Here ∆ is the unit simplex, namely the triangle with vertices (0,0), (1,0), and (0,1). See Example 1.2 in the first paper below for an illustration of this definition.
A possible goal for this summer is to describe lattice polygons P of a given lattice size l ≥ 0 with the property that any lattice polygon Q properly contained in P has strictly smaller lattice size.
The papers listed below are results of previous REU projects on this topic. They contain a broad range of results on lattice size. While much of the material goes beyond the scope of this project, you can take a look at them for the precise definitions and for some of the first results.
References:
- Abdulrahman Alajmi, Sayok Chakravarty, Zachary Kaplan, and Jenya Soprunova, , Involve, a Journal of Mathematics 17-1 (2024), 153--162.
- Anthony Harrison, Jenya Soprunova, Patrick Tierney, , SIAM J. Discrete Math. 36, No 1 (2022), 92-102.
Differential Equations and Applications (Fedor Nazarov and )
The project will be devoted to questions concerning the qualitative behavior of the solutions of differential equations that have completely elementary formulations yet are only partially analyzed. These equations originate from various applications ranging from physics to convex geometry and most likely require a fresh look rather than more advanced technical skills. Both the analytical and the numerical approaches may be explored.
A typical convex geometry example is as follows. Let K be a bounded convex body. The body of flotation of K is a convex body obtained by cutting off K all caps of fixed volume by hyperplanes. Assume that the body of flotation F of K is a ball. Does it mean that K is a ball itself? In case when K is a body of revolution and, in addition, the centers of mass of caps lie on a sphere with the same center as F, the problem can be reduced to a system of differential equations in even dimensions. The problem has been fully analyzed in dimensions 2 and 4, but still remains open even in dimension 6. See for more details.
An example from physically motivated problems is an analysis of a model of thermal explosion in reactive jets. The model was derived in and studied in . A possible project may involve the derivation of asymptotically sharp uniform bounds on the solutions of a certain second order differential equation.
References:
- M.A. Alfonseca, D. Ryabogin, A. Stancu, V. Yaskin, , Pure and Appl. Func. Analysis (to appear).
- P.V. Gordon, U.G. Hegde, M.C. Hicks, , SIAM J. Appl. Math., 78(2) (2018), 705-718.
- P.V. Gordon, V. Moroz and F. Nazarov, , Journal of Differential Equations, 269(7), (2020), 5959-5996.
Prerequisites: Undergraduate Analysis, Elementary ODE’s.