Tag - Symplectic geometry

David White: Symplectic Instanton Homology of Knots and Links in 3-manifolds

10th February, 2023

Powerful homology invariants of knots in 3-manifolds have emerged from both the gauge-theoretic and the symplectic kinds of Floer theory: on the gauge-theoretic side is the instanton knot homology of Kronheimer-Mrowka, and on the symplectic the (Heegaard) knot Floer homology developed independently by Ozsváth-Szabó and by Rasmussen. These theories are conjecturally equivalent, but a precise connection between the gauge-theoretic and symplectic sides here remains to be understood. We describe a construction designed to translate singular instanton knot homology more directly into the symplectic domain, a so-called symplectic instanton knot homology: We define a Lagrangian Floer homology invariant of knots and links which extends a 3-manifold invariant developed by H. Horton. The construction proceeds by using specialized Heegaard diagrams to parametrize an intersection of traceless SU(2) character varieties. The latter is in fact an intersection of Lagrangians in a symplectic manifold, giving rise to a Lagrangian Floer homology. We discuss its relation to singular instanton knot homology, as well as the formal properties which this suggests and methods to prove these properties.

Kai Hugtenburg: Open Gromov-Witten Invariants from the Fukaya Category

10th February, 2023

Enumerative mirror symmetry is a correspondence between closed Gromov-Witten invariants of a space X, and period integrals of a family Y. One of the predictions of Homological Mirror Symmetry is that the closed Gromov-Witten invariants can be obtained from the Fukaya category. For Calabi-Yau varieties this has been demonstrated by Ganatra-Perutz-Sheridan. Recently, enumerative mirror symmetry has been extended, by including open Gromov-Witten invariants and extended period integrals. It is natural to expect that open Gromov-Witten invariants can be obtained from the Fukaya category. In this talk I will outline a construction which will demonstrate this for certain open Gromov-Witten invariants.

Nicole Magill: Symplectic Embeddings and Infinite Staircases

7th February, 2023

The 4-dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. This function can have a property called an infinite staircase, which implies infinitely many obstructions are relevant in determining whether embeddings exist. Based on various work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. The argument relies on a correspondence between constructing embeddings via almost toric fibrations and finding obstructions via exceptional spheres. The talk will focus on explaining this correspondence.

Shira Tanny: Closing Lemmas in Contact Dynamics and Holomorphic Curves

30th January, 2023

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960s, various facets of this question remain largely open. I will review recent advances on this problem in the context of contact flows, which are closely related to Hamiltonian flows from classical mechanics. In particular, I'll discuss a proof of a conjecture of Irie stating that rotations of odd-dimensional ellipsoids admit a surprisingly large class of perturbations creating periodic orbits. The proof involves methods of modern symplectic topology including pseudo-holomorphic curves and contact homology.

Shaoyun Bai: Arnold Conjecture over Integers

20th January, 2023

We show that for any closed symlectic manifold, the number of 1-periodic orbits of any non-degenerate Hamiltonian is bounded from below by a version of total Betti number over ℤ, which takes account of torsions of all characteristics. The proof relies on an abstract perturbation scheme (FOP perturbations) which allows us to produce integral pseudo-cycles from moduli space of J-holomorphic curves, and a geometric regularization scheme for moduli space of Hamiltonian Floer trajectories generalizing the recent work of Abouzaid-McLean-Smith. I will survey these ideas and indicate potential future developments.

Christian Lange: Orbifolds and Systolic Inequalities

13th January, 2023

In this talk, I will first discuss some instances in which orbifolds occur in geometry and dynamics, in particular, in the context of billiards and systolic inequalities. Then I will present topological conditions for an orbifold to be a manifold together with applications to foliations and to Besse geodesic and Reeb flows (joint work with Manuel Amann, Marc Kegel and Marco Radeschi). Here a flow is called Besse if all its orbits are periodic. Such flows are related to systolic inequalities. Namely, I will explain a characterization of contact forms on 3-manifolds whose Reeb flow is Besse as local maximizers of certain 'higher' systolic ratios, and mention other related systolic-like inequalities (joint work with Alberto Abbondandolo, Marco Mazzucchelli and Tobias Soethe).

Marcelo Alves: Hofer’s Geometry and Braid Stability

16th December, 2022

Hofer's metric dH is a remarkable bi-invariant metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. In my talk, I will explain a result, obtained jointly with Matthias Meiwes, which says that the braid type of a set of periodic orbits of a Hamiltonian diffeomorphism on a closed surface is stable under perturbations that are sufficiently small with respect to Hofer's metric. As a consequence of this we obtained that the topological entropy, seen as a function on the space of Hamiltonian diffeomorphisms of a closed surface, is lower semi-continuous with respect to the Hofer metric dH.

If time permits, I will explain related questions for Reeb flows on 3-manifolds and Hamiltonian diffeomorphisms on higher-dimensional symplectic manifolds, and recent progress on these problems obtained by myself, Meiwes, Abror Pirnapasov and Lucas Dahinden.

Robert Cardona: Periodic Orbits and Birkhoff Sections of Stable Hamiltonian Structures

9th December, 2022

In this talk, we start by reviewing recent results on the dynamics of Reeb vector fields defined by contact forms on three-dimensional manifolds, and then introduce Reeb fields defined by stable Hamiltonian structures. These are more general and arise, for instance, in stable regular energy level sets of Hamiltonian systems. We give a characterization of Reeb fields that are aperiodic or that have finitely many periodic orbits (under a certain nondegeneracy assumption). Finally, we give sufficient conditions for the existence of an adapted broken book decomposition or the existence of a Birkhoff section.

Thomas Massoni: Non-Weinstein Liouville Domains and Three-Dimensional Anosov Flows

25th November, 2022

Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years. Little is known about non-Weinstein Liouville domains, whose first instance is due to McDuff. I will describe two key examples of such domains in dimension four, and then explain how they fit into a general construction based on Anosov flows on three-manifolds. The symplectic invariants of these 'Anosov Liouville domains' constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case.