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Symplectic and contact topology : interactions and perspectives

The papers presented in this volume are written by participants of the 'Symplectic and Contact Topology, Quantum Cohomology, and Symplectic Field Theory' symposium. The workshop was part of a semester-long joint venture of The Fields Institute in Toronto and the Centre de Recherches Mathematiques in Montreal. The twelve papers cover the following topics: Symplectic Topology, the interaction between symplectic and other geometric structures, and Differential Geometry and Topology. The Proceeding concludes with two papers that have a more algebraic character. One is related to the program of Homological Mirror Symmetry: the author defines a category of extended complex manifolds and studies its properties. The subject of the final paper is Non-commutative Symplectic Geometry, in particular the structure of the symplectomorphism group of a non-commutative complex plane. The in-depth articles make this book a useful reference for graduate students as well as research mathematicians.

Online American Mathematical Society

Flexible Weinstein structures and applications to symplectic and contact topology [electronic resource]

This thesis has three parts. In the first part, we introduce the notions of regular and flexible Lagrangian manifolds with Legendrian boundary in Weinstein domains. We show that flexible Lagrangians satisfy an existence and uniqueness h-principle (up to ambient symplectomorphism) and give many examples of flexible Lagrangians in the standard symplectic ball. In the second part, we show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology, generalizing similar results in the subcritical Weinstein case. We also prove relative analogs of our results for flexible Lagrangian fillings of Legendrians. As an application, we show that any closed exact, Maslov zero Lagrangian in a cotangent bundle that intersects a cotangent fiber exactly once has the same cohomology as the zero-section. In the third part, we construct many new exotic symplectic and contact structures. For instance, we show that many closed n-manifolds of dimension at least three can be realized as exact Lagrangian submanifolds of the cotangent bundle of the n-sphere with possibly an exotic symplectic structure. We also show that in dimensions at least five any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic.

An introduction to contact topology

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.