MATH 636 – TOPICS IN DIFFERENTIAL GEOMETRY – FALL 2024
AN INTRODUCTION TO DYNAMICAL SYSTEMS
Time and Place: M, W, F 3:00 PM – 3:50PM in EH 3096
Instructor: Ralf Spatzier, 5850 East Hall, 763-2192, spatzier@umich.edu
PREREQUISITES: Basic point set topology/manifold theory, real analysis
GOALS: We will discuss various rigidity results in geometry and dynamics. A famous classical problem is: Can you hear the shape of the drum? This is still largely unresolved. By this, we mean whether the frequencies, i.e. the eigenvalues of the Laplace operator, determine the manifold up to isometry. This is known to be false in general but may well hold true under additional hypotheses.
Here is a similar problem: Can you see the shape of the shape of the drum! Well of course. Really though, this means whether the lengths of the closed geodesics in a compact Riemannian manifold determine the manifold up to isometry? For surfaces this turns out to be true (for lengths as a function on the fundamental group). It is conjectured to be true in any dimension in negative sectional curvature, and has been proved if one of the spaces is highly symmetric, e.g. if it has constant sectional curvature -1.
Other highlights are:
Rigidity in representation theory: Mostow, Margulis, Prasad, etc.;
Rank rigidity in nonpositivecurvature: Ballman, Brin and Burns. Hamenstadt, Spatzier.;
Hopf Conjecture – metrics without conjugate points on tori are flat: Burago-Ivanov;
Entropy rigidity: Besson,Courtois, Gallot;
Zimmer Program on Rigidity in homeomorphism and diffeomorphism groups;
Rigidity in homogeneous dynamics: Measure rigidity, Ratner’s Theorem and the Littlewood Conjecture; Rigidity in positive curvature. Sphere theorems.
These have been highlights of research during the last five decades. Remarkably, both geometric and dynamical ideas play a crucial role, and are intricately woven together. I will discuss some of these topics, and develop the necessary background materials from geometry and dynamics
Audience: This course is aimed at math PhD students. Undergraduate students are most welcome, but they should be aware that this is a fast paced course.
Prerequisites: basic point set topology, manifold theory, real analysis and basic theory of groups and Lie groups.
Additional Meetings/Office hours:. I will be available at additional times every week, and reserve a room. The goal is to discuss questions about the course, and howeworks.
Homework: I will give suggested homework problems which we can discuss in the additional meetings.
Grading:. I will expect regular attendance to the class, and possibly a talk on a topic in the class during the term.
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.