Conformal Symplectic Structures, Contact Structures, Foliations, and Their Interactions

Stanford University, Stanford CA

Investigators

Abstract

This project explores the new recently emerged links between various areas of mathematics, such as symplectic and contact topology and the theory of foliations. It builds on recent progress and results of prior research in an attempt to advance some of the long-standing problems in these and related areas. A related goal is the development of new and alternative tools beyond the currently used techniques that, while they proved to be quite effective for some applications, fail for a large class of open problems in the subject. The work will involve several graduate students and postdoctoral researchers. A graduate student workshop devoted to dissemination of new ideas, methods and results will be organized. The principal investigator will also write a graduate student level book devoted to new advances in symplectic flexibility, including main findings of the proposed research. Rich links between symplectic and contact topology were known since the inception of these subjects in 1980s. A decade later there were discovered connections of contact topology with the theory of foliations. In recent years it was understood that it is important to add to the above mix the theory of conformal symplectic structures. This project builds on recent progress and results of prior research in an attempt to explore developments in each of these areas for advancing some of the long-standing problems in the others. A related goal is the development of new tools beyond the currently used techniques, such as Gromov's theory of holomorphic curves and its ramifications, e.g., Floer homology, Fukaya categories and Symplectic Field Theory. While these techniques proved to be very effective for some applications, they fail for a large class of open problems in the subject. This project aims to develop alternative tools, or in case they do not exist to prove h-principle type results asserting that whatever is not prohibited by holomorphic curve method is, in fact, possible. The main objectives of the project are: 1) Completion of the arborealization program of simplification of singularities of Lagrangian skeleta of Weinstein manifolds, and, in particular, establishing the combinatorial notion of arboreal homotopy, i.e., finding the minimal set of Reidemeister type moves necessary and sufficient to connect two arboreal skeleta of homotopic Weinstein structures; 2) Finding an appropriate notion of overtwistedness for conformal symplectic structures and proving the corresponding parametric h-principle; 3) Finding a generalization of the notion of arboreal singularities applicable to general Weinstein manifolds beyond the polarized case; 4) Finding conditions on a codimension 1 foliation to admit a leafwise conformal symplectic structure; explorations of this condition for the problem of deformation of foliations into contact structures; and 5) Developing effective invariants for open contact manifolds and proving surgery type formulas for their computations. 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.