Abstracts of Blumenthal-2011

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.