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