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.
Given a trivalent plane graph embedded in the Euclidean plane (up to isotopy), Treumann and Zaslow constructed and studied a certain associated Legendrian surface embedded in standard contact ℝ5, nowadays referred to as a Legendrian 2-weave. Using gradient flow trees, Casals and Murphy computed its Legendrian contact dg-algebra with commutative coefficients (i.e. working over the group ring of the first homology group). We extend their computation to the non-commutative setting (i.e. working over the group ring of the fundamental group). For these Legendrian 2-weaves, we further verify the well-known conjecture that the moduli space of representations of this fully non-commutative version of the Legendrian contact dg-algebra are in bijective correspondence with a certain moduli space of sheaves.
I will review two combinatorial constructions of integrable systems: Goncharov-Kenyon construction based on counting perfect matchings in bipartite graphs, and Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on counting paths in networks. After that I will outline my proof of equivalence of those constructions.
I will discuss a recent work constructing quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. The existence of these quasimorphisms answers a question of Entov, Polterovich, and Py. As an immediate corollary, we learn that the commutator length is unbounded, sharply contrasting a result of Tsuboi regarding the group of homeomorphisms that do not preserve area. A key role is played by 'link spectral invariants', constructed using a kind of quantitative variant of the Heegaard Floer cohomology for links.
In a work with Jacques Fejoz, we consider the conformal dynamics on a symplectic manifold, i.e. for which the symplectic form is transformed colinearly to itself. In the non-symplectic case, we study the problem of isotropy and uniqueness of invariant submanifolds. More precisely, in this talk, I will explain a relation between topological entropy and isotropy and some uniqueness results.
If G is a Lie group whose adjoint representation preserves a nondegenerate symmetric bilinear form on its Lie algebra (e.g. a semisimple group) and F is the fundamental group of a closed oriented surface S, then the spaces of equivalence classes of representations F-greater than G (equivalently gauge-equivalence classes of flat G-connections over S) enjoys a rich symplectic geometry. These symplectic manifolds generalize the Kähler structures on the Jacobi variety, moduli of holomorphic vector bundles, and Teichmüller space. This talk will describe this geometry and several open questions about these symplectic manifolds.
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.
I will discuss some recent work showing that a generic area-preserving diffeomorphism of a closed surface has an equidistributed sequence of periodic orbits. The proof uses several properties of spectral invariants from periodic Floer homology, along with a variational argument originating in work of Marques-Neves-Song and Irie regarding equidistribution results for minimal hypersurfaces and Reeb flows, respectively.
I will start by explaining the construction of a formal scheme starting with an integral affine manifold Q equipped with a decomposition into Delzant polytopes. This is a weaker and more elementary version of degenerations of abelian varieties originally constructed by Mumford. Then I will reinterpret this construction using the corresponding Lagrangian torus fibration X→Q and relative Floer theory of its canonical Lagrangian section. Finally, I will discuss a conjectural generalization of the story to decompositions of CY symplectic manifolds into symplectic log CY's whose boundaries are 'opened up'.
I will talk about ongoing work with S. Venugopalan on computing disk potentials (which are counts of holomorphic disks with boundary on a Lagrangian) via multi-directional neck stretching. The focus will be on examples.
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