Séminaires : Séminaire d'Analyse et Géométrie

Equipe(s) :
Responsables :O. Biquard, A. Deruelle, E. Di Nezza, I. Itenberg, X. Ma
Email des responsables : {olivier.biquard, alix.deruelle, eleonora.dinezza, ilia.itenberg, xiaonan.ma}@imj-prg.fr
Salle : 15–25.502
Adresse :Jussieu

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Orateur(s) Joel Fine - Université Libre de Bruxelles,
Titre Knots, minimal surfaces and J-holomorphic curves
Horaire14:00 à 15:00

Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space, H. The main theme of my talk is that it should be possible to count minimal surfaces in H which fill K and obtain a link invariant. In other words, the count doesn’t change under isotopies of K. When one counts minimal disks, this is a theorem. Unfortunately there is currently a gap in the proof for more complicated surfaces. I will explain “morally” why the result should be true and how I intend to fill the gap. In fact, this (currently conjectural) invariant is a kind of Gromov—Witten invariant, counting J-holomorphic curves in a certain symplectic 6-manifold diffeomorphic to SxH. The symplectic structure becomes singular at infinity, in directions transverse to the S fibres. These singularities mean that both the Fredholm and compactness theories have fundamentally new features, which I will describe. Finally, there is a whole class of infinite-volume symplectic 6-manifolds which have singularities modelled on the above situation. If there is time, I will explain how it should be possible to count J-holomorphic curves in these manifolds too, and obtain invariants for links in other 3-manifolds.