Contact and Symplectic Topology : Lectures from the Nantes Trimester 2011 and the Budapest Summer School 2012

Symplectic and contact geometry naturally emerged from the mathematical description of classical physics. The discovery of new rigidity phenomena and properties satisfied by these geometric structures launched a new research field worldwide. The intense activity of many European research groups in this field is reflected by the ESF Research Networking Programme "Contact And Symplectic Topology" (CAST). The lectures of the Summer School in Nantes (June 2011) and of the CAST Summer School in Budapest (July 2012) provide a nice panorama of many aspects of the present status of contact and symplectic topology. The notes of the minicourses offer a gentle introduction to topics which have developed in an amazing speed in the recent past. These topics include 3-dimensional and higher dimensional contact topology, Fukaya categories, asymptotically holomorphic methods in contact topology, bordered Floer homology, embedded contact homology, and flexibility results for Stein manifolds.-- Front Matter....Pages I-XIII Vladimir Igorevich Arnold and the Invention of Symplectic Topology....Pages 1-25 Topological Methods in 3-Dimensional Contact Geometry....Pages 27-83 A Beginner’s Introduction to Fukaya Categories....Pages 85-136 Geometric Decompositions of Almost Contact Manifolds....Pages 137-172 Higher Dimensional Contact Topology via Holomorphic Disks....Pages 173-244 Contact Invariants in Floer Homology....Pages 245-273 Notes on Bordered Floer Homology....Pages 275-355 Stein Structures: Existence and Flexibility....Pages 357-388 Lecture Notes on Embedded Contact Homology....Pages 389-484 A Topological Introduction to Knot Contact Homology....Pages 485-530 Vladimir Igorevich Arnold and the invention of symplectic topology / M. Audin Topological methods in 3-dimensional contact geometry / P. Massot Beginner's introduction to Fukaya categories / D. Auroux Geometric decompositions of almost contact manifolds / F. Presas Higher dimensional contact topology via holomorphic disks / K. Niederkrüger Contact invariants in Floer homology / G. Matić Notes on bordered Floer homology / R. Lipshitz, P. Ozsváth and D.P. Thurston Stein structures: existence and flexibility / K. Cieliebak and Y. Eliashberg Lecture notes on embedded contact homology / M. Hutchings A topological introduction to knot contact homology / L. Ng. 8.1 Sections of a Cotangent Bundle and Fixed Points8.2 Graphs of Symplectic Diffeomorphisms; 9 Generating Functions; 9.1 Caustics and Wave Fronts; 9.2 Crossbows ... ; 9.3 ... and Tanks; 10 Generating Functions (Continuation); 11 Twenty Years After ... First Steps Again; 12 Epilogue (2012); References; Topological Methods in 3-Dimensional Contact Geometry; 1 Introduction; 2 Local Theory; 2.1 Contact Structures as Rotating Plane Fields; 2.2 Examples; 2.3 Isotopies; 3 The Tight vs Overtwisted Dichotomy; 4 Convex Surfaces; 4.1 Characteristic Foliations of Surfaces; 4.2 Neighborhoods of Surfaces 1.6 The Floer Cohomology HF(L, L)2 Product Operations; 2.1 The Product; 2.2 Higher Operations; 2.3 The Fukaya Category; 3 Exact Triangles and Generators; 3.1 Exact Triangles and Mapping Cones; 3.2 Twisted Complexes; 3.3 Exact Triangles in the Fukaya Category; 3.3.1 Dehn Twists; 3.3.2 Lagrangian Connected Sums; 3.4 Generation and Yoneda Embedding; 3.4.1 Generators and Split-Generators; 3.4.2 Yoneda Embedding; 4 The Wrapped Fukaya Category, Examples and Applications; 4.1 The Wrapped Fukaya Category; 4.2 An Example; 4.3 Cotangent Bundles; 4.4 Homological Mirror Symmetry Frederic Bourgeois, Vincent Colin, Andras Stipsicz (eds.). Includes Bibliographical References. 2 Lobs & bLobs: Legendrian Open Books and Bordered Legendrian Open Books