Erogic Theory, Geometry and Dynamics
Math 275 - Tu Th 12-1:15 - Virtual Reality
Harvard University - Fall 2020
Instructor:
Curtis T McMullen
Description:
A survey of fundamental results and current research.
Topics may include:
- Dynamics on the circle and the torus
- Lie groups and ergodic theory
- Hyperbolic surfaces and SL2(R)
- Kazhdan's property T and SL3(R)
- Amenability and expanding graphs
- Martingales and Furstenberg's theorem
- Hyperbolic 3-manifolds and Mostow rigidity
- Ratner's theorem
- Conjectures of Oppenheim and Littlewood
- Planes in hyperbolic 3-manifolds
- Dynamics on moduli space Mg
Main Course Notes
- C. McMullen,
Notes on Ergodic Theory
These notes will evolve during the course; be sure to use the current version, linked above. - See also: C. McMullen, Notes on Teichmüller Theory and Complex Dynamics
Suggested Texts
- M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, Cambridge University Press, 2000.
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), 2007.
- Benedetti and Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, 1992.
- E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Sem. Bourbaki 1991/92; Asterisque 206.
- M. Gromov, Volume and bounded cohomology
- R. Mañé, Ergodic Theory and Differentiable Dynamics
- D. Witte Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Math. Series, 2005.
- J. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd Edition. Springer, 2006.
- W. P. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997.
Hardware requirements.
Students will be required to have iPads or similar devices to facilitate
online collaboration and to participate in online discussions.
Prerequisites.
Intended for advanced graduate students. Familiarity with measure theory, functional analysis, Lie groups, and
hyperbolic geometry in dimension 2 and 3 will be useful, but
relevant material will be reviewed in class. Background material on some of these topics
will be covered in section, and can also be found in the suggested texts and the course notes.
Readings and Lectures.
Assigned material should be read before class.
Lectures may go beyond the reading, and not every topic in the reading will be
covered in class.
Homework.
Collaboration is encouraged on homework.
Late homework is not accepted, but the lowest homework grade
will be dropped.
Slack.
This class has a
Slack channel.
Please participate!
Expectations.
Enrolled students should attend lectures regularly and work on
the homework assignments, which are an important part of the course.
Students should share their video Zoom lectures whenever possible.
Grades.
Letter grades, for those requiring them, will be based on homework and perhaps 1-2 additional assigments.