Séminaire Symplectique
organisé par l'équipe Géométrie
-
Yuan Yao
Symplectix seminar, by zoom : Symplectic packings in higher dimensions
10 novembre 2023 - 15:45Salle de séminaires IRMA
Abstract: The problem of symplectically packing k symplectic balls into a larger one has been solved in dimension four, i.e. there is now a combinatorial criteria of when this is possible. However, not much is known about symplectic packing problems in higher dimensions. We take a step in this direction in dimension six, by considering a “stabilized” packing problem, i.e. we consider symplectically packing a disjoint union of four dimensional balls times a closed Riemann surface into a bigger ball times the same Riemann surface. We show this is possible if and only if the corresponding four dimensional ball packing is possible. The proof is a mixture of geometric constructions, pseudo-holomorphic curves, and h-principles. This is based on work to appear with Kyler Siegel. -
Ciprian Manolescu
Symplectic Zoominar: A knot Floer stable homotopy type
17 novembre 2023 - 15:15Salle de séminaires IRMA
Given a grid diagram for a knot or link K in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.) -
Matthias Meiwes
Symplectic Zoominar: C^0 stability of topological entropy for 3-dimensional Reeb flows
24 novembre 2023 - 15:15Salle de séminaires IRMA
The C^0 distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms. In this talk, I will explain some recent progress on the stability properties of the topological entropy with respect to this distance obtained in collaboration with M. Alves, L. Dahinden, and A. Pirnapasov. Our main result states that the topological entropy for closed contact 3-manifolds is lower semi-continuous in the C^0 distance for C^infty-generic contact froms. Applying our methods to geodesic flows of surfaces, we obtain that the points of lower-semicontinuity of the topological entropy include non-degenerate metrics. In particular, given a geodesic flow of such a metric with positive topological entropy, the topological entropy does not vanish for sufficiently C^0-small perturbations of the metric.
-
Tba Tba
Zoominaire Symplectix : TBA
1 décembre 2023 - 15:45Salle de séminaires IRMA
TBA -
Three 20min Research Talks
Symplectic Zoominar
8 décembre 2023 - 15:15Salle de séminaires IRMA
Julien Dardennes (Toulouse) Title: The coarse distance from dynamically convex to convex Abstract: Chaidez and Edtmair have recently found the first examples of dynamically convex domains in that are not symplectomorphic to convex domains, answering a long-standing open question. In this talk, we present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We also show that these domains are arbitrarily far from the set of symplectically convex domains in R^4 with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity (joint work with J. Gutt, V. Ramos and J. Zhang). Arnaud Maret (Paris) Title: Complex projective spaces via surface groups representations Abstract: My plan is to explain how complex projective spaces can be identified with components of totally elliptic representations of the fundamental group of a punctured sphere into PSL(2,R). I will explain how this identification realizes the pure mapping class group of the punctured sphere as a subgroup of the group of Hamiltonian diffeomorphisms of the complex projective space. Luya Wang (Stanford) Title: Deformation inequivalent symplectic structures and Donaldson's four-six question Abstract: Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan. -
Yusuf Barış Kartal
Symplectic Zoominar : Equivariant Floer homotopy via Morse-Bott theory
22 décembre 2023 - 15:15Salle de séminaires IRMA
Floer homotopy type refines the Floer homology by associating a (stable) homotopy type to an Hamiltonian, whose homology gives the Hamiltonian Floer homology. In particular, one expects the existing structures on the latter to lift as well, such as the circle actions. On the other hand, constructing a genuine circle action even in the Morse theory is problematic: one usually cannot choose Morse-Smale pairs/Floer data that is invariant under the circle action. In this talk, we show how to extend the framework of Floer homotopy theory to the Morse-Bott setting, in order to tackle this problem. In the remaining time, we explain how to relate the Floer homotopy type to the free loop spaces of exact Lagrangian submanifolds equivariantly, and discuss applications to recovering information about the topology of the underlying manifold from its symplectic cohomology. Joint work with Laurent Cote.