## New geometric and functional analytic ideas arising from problems in symplectic geometry

- Date: 10/23/2006

Helmut Hofer (New York University)

University of British Columbia

The study of moduli spaces of holomorphic curves in symplectic geometry

is the key ingredient for the construction of symplectic invariants.

These moduli spaces are suitable compactifications of solution spaces

of a first order nonlinear Cauchy-Riemann type operator. The solution

spaces are usually not compact due to bubbling-off phenomena and other

analytical difficulties. Moreover, there are serious transversality

issues. As it turns out one can develop an abstract nonlinear Fredholm

theory (applicable to a variety of other problems as well) for which

the solution sets would be the compactified spaces. Such a general

Fredholm theory takes place in spaces of varying dimensions.

However, in the case of transversality the solution spaces would still

be smooth (finite-dimensional) manifolds, orbifolds or branched

manifolds depending on the generality of the situation. In fact one has

the usual Fredholm package of transversality and perturbation theory.

In order to develop such a theory one needs to have a fresh look at the

ways how finite-dimensional calculus should be generalized to infinite

dimensions. Using a somewhat more exotic generalization one even can

build a differential geometry containing new kind of spaces which can

be used to build the ambient spaces for the generalized Fredholm

theories. These spaces are called polyfolds. In this talk we describe

some of the ideas and goals of the theory.

10th Anniversary Speaker Series 2006