- Schedule, titles, and abstracts
- Reference and Further reading for minicourses
- Problems from Problem sessions

## 2023 RTG Summer School and REU

**Summer School 2023 on Dynamical Systems**took place June 20–June 30, 2023 on the Evanston campus of Northwestern University. It featured preparatory courses (including courses appropriate for advanced undergraduates) as well as advanced courses. Mini-courses were presented by Alena Erchenko, Mikolaj Fraczyk, Nicholas Miller, Amir Mohammadi, Kurt Vinhage, and Daren Wei. The Summer School 2023 organizers were Nir Avni, Aaron Brown, Osama Khalil, Bryna Kra, and Homin Lee. It was funded primarily through the following NSF grants: DMS-2136217 (RTG: Dynamics: Classical, Modern, and Quantum) and DMS-2020013 (CAREER: Rigidity of Group Actions on Manifolds). Additionally funding was provided by Northwestern University’s Department of Mathematics.**REU Summer 2023**took place June 20–Aug 11, 2023 on the Northwestern University’s Evanston campus. Mentors for the summer 2023 REU included NU Mathematics faculty members Nir Avni, Aaron Brown, Keith Burns, Solly Coles, Tsachik Gelander, Aaron Peterson, Jeff Xia. This program was made possible by the National Science Foundation and was funded by the RTG Dynamics: Classical, Modern, and Quantum.

### Research projects for the 2023 REU

**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.Prerequisites: linear algebra and group theory.**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.Prerequisites: linear algebra and ODEs.**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.**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.

### REU 2023 Reports

Dynamics of Aut(Fn): Ergodicity and /Compact, Connected, Semi-simple Groups

- Authors: Mason Cai and Robin Truax

Symmetries of the N-Body Problem

- Authors: Josh Fleckner, Evan Huang, Brennan Jackson, Jen Tang

Smooth Anosov Actions on T3 and Other Nilmanifolds by Surface Groups

- Authors: Tanner Leonard, Nathan Louie, Paul Shin, and Katherine Tung