By Frédéric Bourgeois, Vincent Colin, András Stipsicz
Symplectic and speak to geometry obviously emerged from the
mathematical description of classical physics. the invention of new
rigidity phenomena and houses chuffed through those geometric
structures introduced a brand new study box world wide. The intense
activity of many eu study teams during this box is reflected
by the ESF study Networking Programme "Contact And Symplectic Topology" (CAST). The lectures of the summer time university in Nantes (June 2011) and of the forged summer time college in Budapest (July 2012) offer a pleasant landscape of many elements of the current prestige of touch and symplectic topology. The notes of the minicourses supply a steady creation to themes that have constructed in an grand velocity within the fresh prior. those subject matters contain third-dimensional and better dimensional touch topology, Fukaya different types, asymptotically holomorphic tools involved topology, bordered Floer homology, embedded touch homology, and suppleness effects for Stein manifolds.
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Additional resources for Contact and Symplectic Topology
I. Arnold. Invent. Math. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology. Funkc. Anal. Prilozh. M. Eliashberg, The structure of 1-dimensional wave fronts, nonstandard Legendrian loops and Bennequin’s theorem, in Topology and Geometry—Rohlin Seminar. , vol. 1346 (Springer, Berlin, 1988), pp. 7–12  Y. Eliashberg, An estimate of the number of ﬁxed points of transformations preserving area, Preprint, Syktyvkar (1978)  Y. Eliashberg, Cobordisme des solutions de relations diﬀ´erentielles, in S´eminaire SudRhodanien de g´eom´etrie, I, Lyon, 1983.
This is the geometric fact underlying the existence of so-called contact Hamiltonians. Theorem 5 (Libermann ). On a contact manifold (V, ξ) the map which sends a contact vector ﬁeld to its reduction modulo ξ is an isomorphism from the space of contact vector ﬁelds to the space of sections of the normal bundle T V /ξ. 39 Topological Methods in 3-Dimensional Contact Geometry If we single out a contact form α then we get a trivialization T V /ξ → V × R given by (x, [u]) → (x, α(u)). Sections of T V /ξ can then be seen as functions on V and the contact vector ﬁeld Xf associated to a function f using the preceding theorem is called the Hamiltonian vector ﬁeld coming from α and f .
A plane ﬁeld ξ on a 3-manifold V is a (smooth) map associating to each point p of V a 2-dimensional subspace ξ(p) of Tp V . All plane ﬁelds considered here will be coorientable, it means one can continuously choose one of the half spaces cut out by ξ(p) in Tp V . In this situation, ξ can be deﬁned as the kernel of some nowhere vanishing 1-form α: ξ(p) = ker α(p). The coorientation is given by the sign of α. We will always assume that V is oriented. In this situation a coorientation of ξ combines with the ambient orientation to give an orientation on ξ.