Symplectic Topology of Weinstein Manifolds and Related Topics

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Symplectic and contact topology emerged in attempt to answer qualitative problems of Classical Mechanics and Optics. Since its inception in 1980s there were discovered many new connections of symplectic topology with other areas of Mathematics and Physics. The primary techniques used in the subject go back to Gromov's theory of holomorphic curves in symplectic manifolds and its many ramifications, such as Floer homology, Fukaya categories and Symplectic Field Theory. However, there remains a large class of open problems where holomorphic curve techniques seems not to be sufficient for proving expected results. The current project attempts to develop alternative techniques, or in case they do not exist to develop new methods of construction which could show that everything which is not prohibited by holomorphic curve techniques can indeed happen. In particular, it provides a reformulation of symplectic topology of affine symplectic manifolds as differential topology of singular spaces with a certain well defined list of singularities. This approach can bring new tools from differential to symplectic topology.Weinstein symplectic manifolds recently moved to the forefront of the development of symplectic and contact topology. Building on recent developments, both on the rigid and flexible side of symplectic topology of Weinstein manifolds, the project is designed to advance several central problems of the theory, such as symplectic topology of Lagrangian submanifolds. Among the main objectives of the project are: exploration of techniques for simplification of singularities of Lagrangian skeleta of Weinstein manifolds, and in particular for proving exploration of methods for attacking the regularity conjecture for exact Lagrangian submanifolds of Weinstein manifolds; exploration of a new approach to singularity theory: h-principle without pre-conditions; further exploration of flexibility phenomena for Weinstein manifolds and their Lagrangian submanifolds; and further development of Symplectic Field Theory. The project is designed to bridge the gap between the negative results establishing limits for possible symplectic constructions, and positive results involving the advanced symplectic constructions on the frontier of possibilities. The new methods developed for the arborealization project may find applications elsewhere in singularity theory. The work on the project will involve several graduate students and postdocs and there will be written a graduate student level book devoted to new advances in symplectic flexibility. The obtained results and developed methods may find applications in other areas of mathematics and theoretical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.