The Northwestern Department of Mathematics is organizing a Research Experience for Undergraduates (REU) for students interested in dynamics in summer 2023. We seek a diverse group of 6-8 participants. We encourage applications from across the country and will match students with faculty by interest. Participants must be US citizens or permanent residents who are presently enrolled in a US undergraduate institution. 

The eight-week program includes mini-courses, working in small groups with a research mentor, talks on current research in dynamics, and professional development. Participants will receive a living stipend and be provided with housing and meals on campus and are expected to be in residence June 20-August 11. 

Potential research topics include:

  1. Dynamics and representation theory: There are many dynamical systems on spaces of matrices, and, in order to understand them, we need to understand matrices better. In this project, we investigate dynamics on representation varieties, trying to classify cosets inside conjugacy classes of matrices.

     Prequisites: linear algebra and group theory.   

       2. Symmetry in Dynamics: Many dynamical systems, such as the classical n-body problem, posses certain symmetries. Depending on the system, these can be large, such as when the system exhibits rotational invariance, or small, such as when the system consists of a small number of discrete bodies moving in a symmetric orbit. We explore the consequences of symmetry on the dynamical behavior of the system, ranging from the large class of quasi-periodic solutions in the rotational invariant case to insights into stability analysis in the discrete case. This project explores some general theory and its applications to some specific systems, including the Newtonian n-body problem.

          Prequisites: linear algebra and ODEs.

       3. Dynamics on the infinite symmetric group: we study the infinite symmetric group, the group consisting of all permutations of the natural numbers that move only finitely many elements. We investigate Vershik's classification of random subgroups in the infinite symmetric group.

          Prerequisites: probability and group theory. 

       4. Constructing Anosov actions on nilmanifolds: starting with the case of tori, we explore ways to classify Anosov automorphisms and groups acting by Anosov automorphisms. Starting with the construction of examples, we study how all of these symmetries of a given algebraic structure, a nilmanifold, can be classified.

          Prerequisites: linear algebra and group theory. 

For more information see: https://sites.northwestern.edu/dynamicsrtg/reu-summer-2023/ 

Selection is based on mathematical background, interest in the research topics, and research potential. We encourage applications from students who have not previously participated in a summer research program and from students from historically underrepresented backgrounds.

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: Nir Avni, Aaron Brown, Keith Burns, Solly Coles, Tsachik Gelander, Aaron Peterson, Jeff Xia.

Applications received by February 10, 2023 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.