For Summer 2027 we will recruit nine students to work on the three projects described below.
Graphs and groups (Mark Lewis)
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.
In this project, we consider the commuting graphs of solvable groups. Parker has proved that if G is a solvable group whose center is trivial then the commuting graph of G is connected if and only if G a Frobenius group or a 2-Frobenius group. And in this case, when the commutating graph is connected, then its diameter is at most 8.
On the other hand, when G is not solvable and the center of G is not trivial, we know we obtain disconnected graphs when G has a maximal centralizer that is abelian or a centralizer that either contains or intersects trivially all other centralizers. One goal is to determine if there are any other sufficient conditions on the group structure that yield that the centralizer graph will be disconnected and ideally lead to a necessary sufficient condition if possible.
Regarding the diameter, Guidici and Parker have shown that there exists a family of two groups with arbitrarily large diameters on their commuting graphs. We would like to explore if we can bound the diameter of connected commuting graphs of solvable groups that are not nilpotent either by a universal constant or by a function in terms of the number of terms in the upper central series. As a first step, can we bound the diameter of commuting graphs of solvable groups where the second center equals the center and the center is nontrival?
Prerequisites: At least one semester of undergraduate Abstract Algebra.
References:
- Peter J. Cameron, Graphs on Groups: An Indian Adventure, Lond. Math. Soc. Newsl. No. 513 (2024), 22–25.
- Peter J. Cameron, Graphs defined on Groups, Int. J. Group Theory 11 (2022), no. 2, 53–107.
- Rachel Carleton and Mark L. Lewis, The Commuting graph of a solvable A-group, 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, Extending results of Morgan and Parker about commuting graphs, Bull. Aust. Math. Soc. 105 (2022), no. 1, 92–100
Differential Equations and Applications (Fedor Nazarov and Peter Gordon)
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 several applications from physics, biology, and 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.
Possible project related to biological applications will involve analysis of large solutions for a certain ordinary differential equation which appears in the analysis of the dynamics of morphogen gradient in the developing cell [1].
An example of a physically motivated problem is the analysis of a model of thermal explosion in porous reactive media. The model was derived and partially analyzed in [2] and consists of a boundary value problem for a system of two parabolic equations. A possible project may involve the study of various parametric regimes the system can exhibit.
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 [3] for more details.
- [1] P.V. Gordon, C.B. Muratov, Eventual self-similarity of solutions for the diffusion equation with nonlinear absorption and a point source, SIAM J. Math. Anal., 47(4) (2015), 2903-2916
- [2] P. Gordon, On thermal explosion in porous media, Nonlinearity 23 (2010), 1433-1447
- [3] M.A. Alfonseca, D. Ryabogin, A. Stancu, V. Yaskin, On questions of uniqueness of Croft, Falconer and Guy for bodies of revolution, Pure and Appl. Func. Analysis, 10(5) (2025),1133–1159.
Directions of equilibrium of floating bodies (Dmitry Ryabogin).
The project is about finding the directions of equilibrium positions of solid convex bodies of uniform density submerged into water.
The body K made of uniform density D∈(0,1) submerged into water of density 1 is said to float in equilibrium in a given orientation if the line passing through its center of mass and the center of mass of its underwater part is orthogonal to the water surface. In other words, if the Gravity force acting down on the center of mass of the body balances off the Archimedes force acting up on the center of mass of its underwater part.
If the submerged bodies are orthogonal prisms with sections being a regular plane n-gon, with the generator parallel to the water surface, this essentially 2-dimensional problem was solved in the 19th century. It was proved that the 4n-gonal prism has at most 4n directions of
equilibrium. In particular, one can show (try to do it! this is a very hard elementary geometry high school problem, see [1] below) that a regular triangle has 12 or 6 such directions depending on the density of the triangle (the prism).
A general polytope might have infinitely many directions of equilibrium, so one has to count them properly, up to the symmetry.
In this project, we will try to find and make a description of the equilibrium positions of simplices, cubes, octahedra and other bodies
(see [2] for some information).
References:
- [1] "On flotation, stability and related questions: A survey", (with M.
Alfonseca, A. Stancu and Vlad Yaskin), to appear in New Probes into
Discrete and Convex Geometry (J. Pach and G. Toth, eds.), Bolyai Society
Mathematical Studies, Vol. 32, Springer-Verlag, Berlin. - [2] E. N. Gilbert, How Things Float, American Mathematical Monthly, 98:3
(1991), 201-216. - [3] P. Erdös, G. Schibler and R. Herndon, Floating equilibrium of
symmetrical objects and the breaking of symmetry. Part 1: Prisms, Am. J. Phys.60 (1992), 335—345.P. Erdös, G. Schibler and R. Herndon, Floating equilibrium of symmetrical objects and the breaking of symmetry. Part 2: Prisms, Am. J. Phys. 60 (1992), 345—356.
