The Symplectic Topology of Stein Manifolds

Mohammed Abouzaid

"The familiar classification of Riemann surfaces by the genus and the number punctures essentially states that each smooth surface admits a complex analytic structure which is unique up to deformation. Moreover, every open Riemann surface admits a proper analytic embedding in affine space. In higher dimension, the property of admitting such an embedding distinguishes a class of complex manifolds which are called Stein; they are the analytic analogue of affine varieties. In the early 90's, Eliashberg proved an existence theorem for Stein structures: he showed that the condition that a manifold admit such a structure is entirely determined by its smooth topology. The question of uniqueness (up deformation), remained open. By making full use of the modern machinery of symplectic topology (i.e. Floer theory and the

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