J. Fine - Knots, minimal surfaces and J-holomorphic curves

Réalisation : 2 juillet 2021 Mise en ligne : 2 juillet 2021
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I will describe work in progress, parts of which are joint with Marcelo Alves. Let L be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to L, and in this way obtain a knot invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT.“Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how this “minimal surface polynomial" can be seen as a Gromov-Witten invariant for the twistor space of hyperbolic 4-space. This leads naturally to a new class of infinite-volume 6-dimensional symplectic manifolds with well behaved counts of J-holomorphic curves. This gives more potential knot invariants, for knots in 3-manifolds other than the 3-sphere. It also enables the counting of minimal surfaces in more general Riemannian 4-manifolds, besides hyperbolic space.

Langue :
Fanny Bastien (Réalisation), Hugo BÉCHET (Réalisation)
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Citer cette ressource:
I_Fourier. (2021, 2 juillet). J. Fine - Knots, minimal surfaces and J-holomorphic curves. [Vidéo]. Canal-U. (Consultée le 21 mai 2022)

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