Séance du 17 avril 2015

Lieu IHP, salle 201

11:00  Dimitri Zvonkine (Jussieu)
Cycles de double ramification. 
Abstract: Soit une courbe complexe  C  avec  n  points marqués  x_1, ... x_n. Étant donné  n  entiers  a_1, ..., a_n  de somme nulle on peut se demander si le diviseur  sum a_i x_i  est principal sur  C.  Le lieu des courbes  (C, x_1,...,x_n)  telles que ce diviseur est principal forme une sous-variété de codimension  g  dans l'espace des modules des courbes de genre  g  avec  n  points marqués. Il s'appelle le "cycle de la double ramification". Le problème consistant à calculer sa classe d'homologie s'appelle le "problème d'Eliashberg" et a des applications potentielles dans la théorie symplectique des champs. Nous présenterons une solution de ce problème.Travail commun avec F. Janda, R. Pandharipande et A. Pixton.

14:15  Yasha Savelyev (Madrid)   
On the Hofer geometry injectivity radius conjecture.   
Abstract: I will discuss some basics of Hofer Finsler geometry originating in the study of symplectic geometry, and closely connected with the theory of Hamiltonian dynamical systems. One may understand this geometry as one attempt to formalize optimal mechanical transformations of a phase space. Then I will describe one basic outstanding conjecture in this field and discuss some partial verifications of this conjecture. The techniques employ Gromov-Witten theory but it should be possible to indicate basic ideas without much analysis.
16:00  Roman Golovko (Budapest)   
On Arnold-type inequality and linear representations of a characteristic algebra.

Abstract: Given a chord-generic horizontally displaceable Legendrian submanifold L of P x R with the property that its characteristic algebra admits a finite-dimensional matrix representation, we show an Arnold-type lower bound for the number of Reeb chords on L. This result is a generalization of the results of Ekholm-Etnyre-Sullivan and Ekholm-Etnyre-Sabloff which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds L of C^n x R, n > 0, whose characteristic algebras admit finite-dimensional matrix representations, but whose Chekanov-Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct examples of Legendrian submanifolds L of C^n x R with the property that the characteristic algebra of L does not satisfy the rank property. This is a joint work with Georgios Dimitroglou Rizell.

Prochaine séance: 22/05 (E. Giroux, K. Ono).