#### MATH 636 – DIFFERENTIAL GEOMETRY – WINTER 2023

**AN INTRODUCTION TO DYNAMICAL SYSTEMS**

**Time and Place**: , Th 10:00 AM – 11:30 AM in EH 3096

**Instructor**: Ralf Spatzier, 5850 East Hall, 763-2192, spatzier@umich.edu

**PREREQUISITES**: Basic point set topology/manifold theory, real analysis

**GOALS**: Dynamical systems is the study of the long-time behavior of diffeomorphims and flows. We will discuss ergodicity and mixing, and natural invariants such as entropy. This theory is particularly successful and interesting in the presence of a lot of hyperbolicity, i.e., when the flow or diffeomorphism expands and contracts tangent vectors exponentially fast. For these systems, we will then develop some of the crucial tools in Pesin theory such as unstable manifolds and Lyapunov exponents. We will illustrate the general theory by particular examples from geometry and dynamical systems on homogeneous spaces. I will apply these ideas to study some more general group actions, e.g., actions of higher rank abelian and semi simple groups. As was discovered in the last decade, these actions show remarkable rigidity properties, and sometimes can even be classified. These results have had important applications in other areas, e.g., geometry and number theory. While the material in the first part of the course is fundamental for many investigations in dynamics, geometry, several complex variables. the second half of the semester will bring us right to the forefront of current research.

**BOOKS**: No text is required but the following are recommended:

*Introduction to the Modern Theory of Dynamical Systems*: Hasselblatt and Katok, Revised Edition, ISBN: 978-0521575577

*Ergodic Theory*: Einsiedler, Ward, Graduate Texts in Mathematics, ISBN: 978-0- 85729-021-2

*An Introduction to Ergodic Theory*, Walters, ISBN 978-0-387-95152-2

**HOMEWORK**: I will suggest problems to be discussed in the discussion sessions.

**TALKS**: You are expected to give a talk during the semester.

**DISCUSSIONS SESSION**: One office hour will be a discussion session

**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.