arXiv subjects
Publications and source records attributed to Eduard Zehnder.
I outline the history and the original proof of the Arnold conjecture on fixed points of Hamiltonian maps for the special case of the torus, leading to a sketch of the proof for general symplectic manifolds and to Floer homology. This is the written version of my talk at the Geometric Dynamics Days 2017 (February 3-4) at the RUB in Bochum.
In this paper we start with the applications of polyfold theory to symplectic field theory.
The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to deal with compactness and transversality issues as they occur in moduli problems of symplectic geometry. Applications of the theory cover Floer-type theories as they occur in symplectic geometry. M-polyfolds and the more general polyfolds are smooth spaces which can be finite-dimensional as well as infinite-dimensional. In applications of interest they in general have locally varying dimensions. Despite the fact that the spaces are much more general than Banach manifolds a nonlinear Fredholm theory with the usual features is possible (Sard-Smale type perturbation theory). This generalized Fredholm theory can be applied to classes of nonlinear elliptic problems which show bubbling-off phenomena but allow for certain kind of compactifications.
We survey a (nonlinear) Fredholm theory for a new class of ambient spaces called polyfolds, and develop the analytical foundations for some of the applications of the theory. The basic feature of these new spaces, which can be finite and infinite dimensional, is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior like bubbling-off. The theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.
We describe a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. The basic feature of these new spaces is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.
This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory. Some proofs have been improved and a glossary added.
In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.
This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.