Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more- we show it can detect the framed bordism class of certain Lagrangians and in particular recover a result of Abouzaid-Alvarez-Gavela-Courte-Kragh about smooth structures on Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large's recent construction of a stable-homotopical enrichment of Lagrangian Floer theory.
We discuss the relation between hypersurface singularities (e.g. ADE, Ẽ6, Ẽ7, Ẽ8, etc) and spectral invariants, which are symplectic invariants coming from Floer theory.
Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a nondegeneracy condition called maximal non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will then explain how to construct and compute Floer-theoretic contact invariants. These are a sort of infinite-dimensional version of Morse theory wherein the chain complexes are generated by closed Reeb orbits and the differential counts certain J-holomorphic curves. This talk will feature numerous graphics and anecdotes.
Filtered Lagrangian Floer homology gives rise to a barcode associated to a pair of Lagrangians. It is well-known that the lengths of the finite bars and the spectral distance are lower bounds of the Lagrangian Hofer metric. In this talk we are interested in a reverse inequality. I will explain an upper bound of the Lagrangian Hofer distance between equators in the cylinder in terms of a weighted sum of the lengths of the finite bars and the spectral distance.
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.
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.
In this talk, I want to show that in the planar circular restricted three body problem there are infinitely many symmetric consecutive collision orbits for all energies below the first critical energy value. By using the Levi-Civita regularization we will be able to distinguish between two different orientations of these orbits and prove the above claim for both of them separately. In the first part of the talk, I want to explain the motivation behind this result, especially its connection to powered Flybys. Afterward I will introduce the main technical tools, one needs to prove the above statement, like Lagrangian Rabinowitz Floer Homology and its G-equivariant version. To be able to effectively calculate this G-equivariant Lagrangian RFH, we will relate it to the Tate homology of the group G. With this tool at hand, we will then finally be able to prove that there are infinitely many consecutive collision orbits all facing in a specific direction.
The Hochschild cohomology of the Floer algebra of a Lagrangian L, and the associated closed-open string map, play an important role in the generation criterion for the Fukaya category and in deformation theory approaches to mirror symmetry. I will explain how, in the monotone setting, one can build a map from the Floer cohomology of L with certain local coefficients to (a version of) Hochschild cohomology. This map makes things much more geometric, by transferring the algebraic complexity to the world of matrix factorisations, and is an isomorphism when L is a torus.
In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.
In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces.
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