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Dietmar Salamon: Lectures

January 11, 2017
TITLE: An open-closed isomorphism in instanton Floer homology

ABSTRACT: The talk explains a proof that the SU(2) instanton Floer homology of a 3-manifold with boundary, equipped with a Lagrangian boundary condition associated to a handlebody, is naturally isomorphic to the instanton Floer homology of the corresponding closed 3-manifold, whenever the latter is an integral homology 3-sphere. This is joint work with Katrin Wehrheim.

January 11, 2017
TITLE: Adiabatic limits in gauge theory

ABSTRACT: Adiabatic limits can appear in gauge theory when one examines a one-parameter family of Riemannian metrics on a product of two Riemann surfaces S and \Sigma, where the metric on \Sigma degenerates to zero. In this limit ASD instantons on the product degenerate formally to holomorphic curves from S into the moduli space of flat connections on \Sigma. The analysis used to make this correspondence precise has two parts: the first is a variant of gluing, which asserts the existence of ASD instantons near a holomorphic curve for sufficiently small parameters; the second is a variant of compactness, which asserts under suitable assumptions that there cannot be any other solutions. This analysis was developed in the early 1990s in a joint work with Stamatis Dostoglou for the proof of the Atiyah-Floer conjecture for mapping tori. The goal of this talk is to give an outline of the main analytic ideas in this story and, if time permits, to discuss some of the geometric settings in which similar adiabatic limit problems arise.