Course Content:
In life as well as in mathematics, it is not important if you come back where you started, but it is the journey you followed that makes all the difference. Thus, this 20-hour course is dedicated to the study of the group G of (volume-preserving/symplectic) diffeomorphisms of a manifold M. The milestone of the theory, and our goal together, is the Epstein-Thurston-Mather-Banyaga Theorem: the group G is, up to the natural Flux Homomorphisms, simple, that is, it has no non-trivial normal subgroups. As a corollary, two diffeomorphism groups G_1 and G_2 are isomorphic if and only if the associated manifolds M_1 and M_2 are isomorphic. The proof relies on many beautiful ideas in analysis and dynamics such as Herman's theorem on the conjugacy of torus diffeomorphisms and the fragmentation of diffeomorphisms. Our main source for the course is the book The Structure of Classical Diffeomorphism Groups by Augustin Banyaga.
Lectures:
All lectures are in room 133 except for the ones of 15/04 and 22/04 which are in room 136.
- Wed Apr 8 11:00-13:00. Introduction. Diffeomorphism groups and vector fields
- Fri Apr 10 9:00-11:00. Volume-preserving and symplectic diffeomorphisms
- Wed Apr 15 11:00-13:00. Hamiltonian diffeomorphisms.
- Fri Apr 17 9:00-11:00. The linearized Epstein-Thurston-Mather-Banyaga Theorem.
- Wed Apr 22 11:00-13:00. From perfectness to simplicity.
- Fri Apr 24 9:00-11:00. The fragmentation property of diffeomorphisms.
- Wed May 20 11:00-13:00. Herman's theorem on the conjugacy of torus diffeomorphisms.
- Fri May 22 9:00-11:00. The Epstein-Thurston-Mather-Banyaga Theorem.
- Wed May 27 11:00-13:00. The groups are isomorphic if and only if the manifolds are.
- Fri May 29 9:00-11:00. Outlook: regularity issues, quasi-morphisms, open problems.