Homogeneous quasimorphism, C^0-topology and Lagrangian intersection

Symplectic Geometry Seminar - 7/17/20

Yusuke Kawamoto, Shira Tanny, Javier Martínez-Aguinaga
École Normale Supérieure, Tel Aviv University, Complutensian University of Madrid
July 17, 2020
Yusuke Kawamoto: Homogeneous quasimorphism, C^0-topology and Lagrangian intersection

Abstract: We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the 2- and 4-dimensional quadric which is continuous with respect to both C0-topology and the Hofer metric. This answers a variant of a question of Entov-Polterovich-Py which is one of the open problems listed in the monograph of McDuff-Salamon. A comparison of spectral invariants for quantum cohomology rings with different coefficient fields plays a crucial role in the proof which might be of independent interest.

Shira Tanny: Floer theory of disjointly supported Hamiltonians

Abstract. We discuss the Floer-theoretic interaction between disjointly supported Hamiltonians, a problem considered earlier by Polterovich, Seyfaddini, Ishikawa and Humilière-Le Roux-Seyfaddini. In aspherical symplectic manifolds, we find new constraints on Floer trajectories, and derive applications to spectral invariants and the boundary depth, as well as to the action selector constructed by Abbondandolo, Haug, and Schlenk. Furthermore, we prove that the spectral invariants, with respect to the fundamental and point classes, of Hamiltonians supported in certain domains, are independent of the ambient manifold. This is a joint work in progress with Yaniv Ganor.

Javier Martínez-Aguinaga: Formal Legendrian and horizontal embeddings

Abstract. In this talk we will discuss some recent results about the space of Formal Legendrian embeddings in contact 3-manifolds and Formal Horizontal embeddings in Engel manifolds. At the π0−level, formal invariants of Legendrian embeddings correspond to the well understood classical invariants. We will introduce analogous invariants at the π1−level, which can be described in a geometrical way. As an application we can construct new examples of non-trivial loops of Legendrian embeddings which are trivial as loops of smooth embeddings. Joint work with Eduardo Fernández (ICMAT-UCM) and Francisco Presas (ICMAT).