Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Daniel Rudolf (Ruhr-Universität Bochum): Viterbo‘s conjecture for Lagrangian products in $\mathbb{R}^4$

We show that Viterbo‘s conjecture (for the EHZ-capacity) for convex Lagrangian products in $\mathbb{R}^4$ holds for all Lagrangian products (any trapezoid in $\mathbb{R}^2$) x (any convex body in $\mathbb{R}^2$). Moreover, we classify all equality cases of Viterbo’s conjecture within this configuration and show which of them are symplectomorphic to a Euclidean ball. As by-product, we conclude sharp systolic Minkowski billiard inequalities for geometries which have trapezoids as unit balls. Finally, we show that the flows associated to the above mentioned equality cases (which are polytopes) satisfy a weak Zoll property, namely, that every characteristic that is almost everywhere away from lower-dimensional faces is closed, runs over exactly 8 facets, and minimizes the action.

Miguel Pereira (Augsburg University):  The Lagrangian capacity of toric domains

In this talk, I will state a conjecture giving a formula for the Lagrangian capacity of a convex or concave toric domain. First, I will explain a proof of the conjecture in the case where the toric domain is convex and 4-dimensional, using the Gutt-Hutchings capacities as well as the McDuff-Siegel capacities. Second, I will explain a proof of the conjecture in full generality, but assuming the existence of a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology. This second proof makes use of Siegel's higher symplectic capacities.

Maksim Stokić (Tel Aviv University):  $C^0$ contact geometry of isotropic submanifolds

Homeomorphism is called contact if it can be written as $C^0$-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold $L$ of a contact manifold (Y, ξ) is called isotropic if ξ|TL =0. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic.  In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.