· MATH 648K (Fall 2011)

· The principle of minimal action in geometry and dynamics

· Instructor: Vadim Kaloshin

MATH 3111, vadim.kaloshin@gmail.com, (301) 405-5132.

Course Information

The course will be devoted to analysis of various minimization procedures in geometry and Hamiltonian dynamical systems and their deep relation. For example, dynamics of planar billiards in convex domains is closely related to the famous Kac’s question: Can you hear the shape of a drum?

1. Convex billiards. Laplace spectrum, Length spectrum.

2. Maximization of perimeter, Mather beta-function and alpha-functions.

3. Aubry-Mather theory for twist (2-dimensional) maps.

4. Weak KAM and Mather theory for multidimensional Hamiltonian systems

5. Hofer geometry

6. Symplectic geometry

We focus the first part of the class on the book of Siburg ``Principle of minimal action in

geometry and dynamics’’, Lecture Notes in Mathematics 1844

The second part of the class is focused on the book of Polterovich ``The Geometry of

the group of symplectic diffeomorphisms’’.

Lectures: TTh, 11:00--12:15 p.m., MTH B0427

Office hours: appointment.