2025 RTG REU

This project will focus broadly on the connections between group theory, geometry and dynamical systems. Groups are ubiquitous throughout much of mathematics and, though seemingly purely algebraic objects, much can be understood about a group by studying how it acts on certain spaces. In particular, the geometry of the space and the dynamics of the action can often yield information about the algebraic structure of a group and vice versa.

Prerequisites: Linear algebra, group theory, and topology.

By analyzing a particular action on the hyperbolic plane, one may show that for |t| at least 2, the vertical and horizontal shearing-matrices of parameter t generate a free subgroup of SL(2,R). On the other hand, when |t| is less than 2, much less is known in general. Analyzing the actions of such matrix groups on natural spaces as part of the more general study of discrete subgroups of Lie groups provides a rich connection between algebra, geometry and dynamics with a number of open problems students may explore. Beyond the world of linear groups, this project may also branch out into a number of different directions, depending on student interests. For instance, directions include the Wiegold conjecture and its connection to the product replacement graphs of finite simple groups, as well as selfless groups and their connection to growth properties of word maps.

A square-tiled surface (a.k.a. origami) is a real 2-dimensional surface built out of squares. These surfaces play a significant role in dynamics and geometric topology. We shall say that such a surface is treelike if the graph depicting the arrangement of the squares is a tree. Treelike surfaces have recently been shown to have interesting triangulations in an ongoing joint project of Freedman and Zykoski, and our question is this: how many of them are there? One can ask this question in a number of ways. Can one obtain a precise formula for the number of such surfaces when the number of squares is fixed? How about the number of such surfaces in a fixed stratum, i.e. when the surface has prescribed singularities? If not precise formulas, can one obtain asymptotics for these counts?

Prerequisites: Linear algebra, group theory, and topology.

The field of smooth dynamics studies points moving in a space according to some deterministic rule. Within this field, there are various notions of equivalence of smooth dynamical systems, along with numerical invariants such as Lyapunov exponents, entropy, and pressure. Generally, dynamicists have come to believe that these invariants can take any value within any flexible enough class of dynamical systems. In this project, we will restrict this study of flexibility to smooth dynamical systems on the circle, with the aim of explicitly constructing systems whose invariants have prescribed values. A natural question also arises: how many inequivalent systems can we construct with the same invariants?

Prerequisites: linear algebra and real analysis.

The granularity of the quantum world is measured by Planck’s constant, h, a tiny number that in the usual units is of the order 10^{-34}. Niels Bohr’s celebrated “correspondence principle” says that if we let h vary as a parameter, toward zero, then quantum mechanics should resemble classical mechanics. This limit, known as the “semiclassical limit,” is still a subject of ongoing mathematical research, as the relationship between quantum systems and their classical analogues can be quite subtle. These connections are explored in two projects, and students working in either of these projects will be exposed to fundamental concepts in the mathematics of quantum mechanics.

A quantum trajectory is a sequence of measured values, using quantum instruments, of the location of a quantum particle. In certain settings, it has been proven that the probabilistic quantum trajectory converges to the evolution of a corresponding classical measure. This project will build numerical models to simulate these trajectories with the goal of generalizing the convergence result to quantum particles in other geometric settings. Prerequisites: linear algebra, real analysis, and basic understanding of probability and writing code.

The eigenvalues of a Schrodinger operators with real-valued potential describe the possible energy levels of a quantum particle. We study instead the eigenvalues of a Schrodinger operator with imaginary potential; such operators are employed in the the complex absorption method used in quantum chemistry, and are of mathematical interest in their own right.

Prerequisites: linear algebra and real analysis. Familiarity with complex analysis and the theory of ODEs will also be useful.

Selection is based on mathematical background, interest in the research topics, and research potential. This program is made possible by the National Science Foundation, and is funded by the RTG Dynamics: Classical, Modern, and Quantum. Mentors for the summer include: Redmond McNamara, Michael Zshornack, Caleb Dilsavor, Izak Oltman, Yuzhou Zou, Bradley Zykoski, Aaron Peterson. Applications received by February 28, 2025 will receive full consideration.

Required application materials:

• Cover Letter: Please include your status (year in school), major, program you are applying for, and any other information you would like to include. This should be short.

• Transcripts (an unofficial one suffices)

• Personal Statement: Explain your interest in participating in this REU. If you wish, you may include any of the following: background on your preparation for the program, on your interest in the subject, on your plans upon graduation, on a math topic that is of interest to you, your interest in any of the research topics. Note that there is no requirement that you be familiar with the precise topics or have any background in dynamics.

• Two reference letters: We recommend select letter writers who are familiar with your academic abilities, particularly your success in courses that are relevant for the topics in this REU.