Séance du 9 Janvier 2015

Lieu : IHP, salle 201

11:00 Alessia Mandini (Pavia)  
Gromov width of polygon spaces
Abstract: After Gromov’s foundational work in1985, problems of symplectic embeddings lie in the heart of symplectic geometry. TheGromov width of a symplectic manifold (M,ω) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in (M,ω). I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in R3 with edges of lengths (r1,...,rn). Under some genericity assumptions on lengths ri, the polygon space is a symplectic manifold. After introducing this family of manifolds, I will concentrate on the spaces of 5-gons and 6-gons and calculate their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.

14:15  Joel Fine (Bruxelles) 
Circle invariant definite connections and symplectic Fano six-manifolds. 
Abstract: A definite connection is an SO(3) connection over a 4-manifold, whose curvature is non-zero on every tangent 2-plane. Given such a connection, the associated 2-sphere bundle is naturally a symplectic manifold. This is a special case of Weinstein’s “fat connections” and they have a particularly rich geometry, which I will briefly describe. I will then focus on definite connections invariant under a circle action, in which case the corresponding symplectic six-manifold is “Fano” (or “monotone”). I will explain how to classify them, the only possibilities being connections over S^4 or CP^2, giving the Fanos CP^3 and the complete flag on C^3 respectively. Finally, I will explain some conjectures and open questions concerning definite connections.

16:00  Damien Gayet (Grenoble)  
Nombres de Betti moyens des ensembles nodaux aléatoires.
Abstract: Dans une variété compacte Riemannienne de dimension n, on prend au hasard une fonction f dans l'espace engendré par les fonctions propres du laplacien avec valeurs propres plus petites que L. J'expliquerai qu'il est possible de donner une majoration, quand L tend vers l'infini, de chacun des nombres de Betti moyens du lieu d'annulation de f (son ensemble nodal). C'est un travail réalisé en commun avec Jean-Yves Welschinger.
Prochaines séances: 6/03.