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).
