- 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