Symplectic topology of Euclidean space - EMS Meeting

Professor I. Smith (Cambridge)

Friday 13th December, 2013 16:00-17:30 Maths 417

Abstract

Symplectic manifolds (which include oriented surfaces and smooth algebraic varieties) are the phase spaces of classical dynamics. Symplectic structures arise naturally in low-dimensional topology, in representation theory and quantisation, in algebraic geometry and elsewhere. A basic result in the subject says that every symplectic manifold locally looks like complex Euclidean space. Whilst this means there are no local invariants (no analogue of curvature), giving the subject a global topological feel, it also means that every symplectic manifold is at least as complicated as Euclidean space. We will illustrate some of that complexity by (re-)considering the classical question of which manifolds admit ``Lagrangian'' immersions into flat space, i.e. ones whose tangent spaces are everywhere orthogonal to their image by multiplication by i.

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