***The class on Aug. 16 is cancel due to Prof. Kao's unwellness.

1. Course Background & Purposes

Ergodic geometrylies at the intersection of dynamical systems and geometry. It employs tools in dynamical systems, especially the ergodic theory, to examine the global geometry of the ambient space. For example, in some geometric dynamical systems, the dynamical system quantity entropy matches the geometric quantity curvature.

This course is an introductory class to ergodic geometry. We will start with the basics of dynamics and geometry. Then move to study the most concrete example in ergodic geometry: geodesic flows over hyperbolic surfaces. Along the way, we will explore many fundamental ideas and results of dynamics (in particular, ergodic theory) and geometry (in particular, differential geometry).

The class will be self-contained and geared toward students (including undergraduate students). Everyone is welcome to attend the course especially those who want to know more about dynamical systems, ergodic theory, and their applications to geometry and other areas. For example, the contributions of Field medalists Grigory Margulis, Bill Thurston, Curtis McMullen, Elon Lindenstrauss, Artur Avila, Maryam Mirzakhani, etc., belong to this area in a broader sense.

2. Course Outline & Descriptions

There is no required textbook for this course. Lectures may based

on the first four chapters of the following book:

1.     Françoise Dal'bo, Geodesic and Horocylic Trajectoies, Springer

2.     William Parry and Mark Pollicott,Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Ast'erisque 188-187

3.     Tim Bedford, Michael Keane, and Caroline Series (editors), Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, Oxford University Press

The detailed course information is as follows:

Canceled- 7/26 Survey on dynamics, geometry, and their interplay

8/2 Complex analysis and hyperbolic geometry

8/9 Fuchsian groups and symbolic dynamics

Canceled- 8/16 Interesting consequences 

3. Grading

Homework assigned in class

4. Registration

https://forms.gle/zp3KSmmDyqu1mWpD8