Representation Theory, Dynamics and Geometry
| These will be a course on dynamics, geometry and representation theory. These three topics appear rather separate yet have plenty of interactions. The goal for this course is to illustrate some of them, with some emphasis on representation theory/harmonic analysis. We will discuss some of the following (subject to time and general interest):Basic ideas about unitary representations of Lie groups, and how they are used in applications; Explicit description of the unitary dual of SL(2, R); Applications of the latter to dynamics, e.g. work by Flaminio and Forni on horocycle flowsSpectral gap properties and Kazhdan’s property (T); Applications to random walks on groups and graphs; Applications to the Zimmer conjecture that SL(10^44 +2, Z) for example cannot act on compact manifolds of dimension 10^44 or less (except via finite groups); Applications to exponential mixing, higher Teichmueller theory and dynamics; Applications to homogeneous dynamics and Diophantine analysis; Other topics relating analysis with dynamics and geometry e.g. quantum ergodicity phenomenon i.e. whether eigenfunctions of the Laplacian on Riemannian manifold equidistribute to the Riemannian volumeNo textbook will be required. Some will be recommended. | ||
Background: I will assume some basic knowledge of Lie groups, real analysis, differentiable manifolds and Riemannian geometry. Other items will be either explained or neatly put in black boxes. |
ACCOMODATIONS: Please see the Departmetn’s policies here: https://lsa.umich.edu/content/dam/math-assets/math-document/Policies/Mathematics_Department_Policy_on_Testing_Accommodations%20(2).pdfLinks to an external site.
