I. ( Colloquium):
Title: Symplectic Dynamics
Abstract:
The modern theories of dynamical systems and symplectic geometry start
with Poincare. Concerning Hamiltonian dynamics, Poincare had a
very integrated viewpoint combining geometric and dynamical ideas. Over
the time the two fields developed independently. Given the highly developed
states of both fields and the background of some promising results the time
seems ripe to bring them together around the core of Hamiltonian
mechanics, resulting (may be) in a field which one could call
"Symplectic Dynamics". It should have a body of theory and techniques
which combine in a nontrivial way dynamical and symplectic ideas, with
the aim to tackle problems which are neither accessible by dynamical
ideas nor by purely symplectic ideas. The talk will give some ideas
what this field would be about.
II. (Geometry and Dynamics Seminar)
Title: Generalizations of Fredholm Theory
Abstract:
A meanwhile standard idea for producing geometric invariants
(f.e Donaldson Theory, Gromov-Witten Theory, Symplectic Field Theory)
consists of counting solutions of nonlinear elliptic systems associated
to the geometric data. Although the idea is easy, the implementation
can be very difficult and involved, due to a usually large number of technical
issues, which in more classical approaches to such type of problems
are more than "painful". If there weren't these inherent compactness
and transversality problems the solution sets of the elliptic problems
would be nice manifolds or orbifolds and the invariants would be
achieved by integration of suitable differential forms over them. As
it turns the arising difficulties can be overcome by a drastic
generalization of nonlinear Fredholm theory and new methods for
implementing it in concrete problems.