Tag - Symplectic geometry

We show that, for closed Legendrians in 1-jet bundles, when there is a sheaf with singular support on the Legendrian, then (1) its self Reeb chords are bounded from below by half the sum of Betti numbers, and (2) the Reeb chords between itself and its Hamiltonian push off is bounded from below by Betti numbers when the C⁰-norm of the Hamiltonian is small. I will show how to visualize Reeb chords/Lagrangian intersections in sheaf theory, and then explain the duality exact triangle and the persistence structure used in the proof. If time permits, I will state a conjecture on the relative Calabi-Yau structure that arises from the duality exact triangle.

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (at the 'prime' zero). I will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals), which interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogues of topological localization of Sullivan, Quillen, and Bousfield.

We will discuss the existence of rational (multi)sections and unirulings for projective families f: X→ℂP¹ with at most two singular fibres. In particular, we will discuss two ingredients that are used to construct the above algebraic curves. The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains. The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre.

We describe two extensions, called the virtual Morse-Bott index and circle-equivariant virtual Morse-Bott index, of the classical Morse-Bott index of a Morse-Bott function on a smooth manifold to the setting of (a) suitably defined analytic functions on singular analytic spaces and (b) suitably defined Hamiltonian functions on almost symplectic, singular analytic spaces equipped with circle actions. Almost symplectic, singular analytic spaces with circle actions are pervasive in gauge theory and key examples include the moduli spaces of Higgs pairs over Riemann surfaces, moduli spaces of projective vortices over complex Kaehler manifolds, and moduli spaces of non-Abelian monopoles over smooth Riemannian four-manifolds. We explain how the concept of circle-equivariant virtual Morse-Bott index can potentially be used to answer questions arising in the geography of smooth four-manifolds, such as whether constraints on the topology of compact complex surfaces of general type (the Bogomolov-Miyaoka-Yau inequality) continue to hold for symplectic four-manifolds or even for smooth four-manifolds of Seiberg-Witten simple type.

I will discuss joint work with McLean and Smith, lifting the results of Seidel, Lalonde, McDuff, and Polterovich concerning the topology of Hamiltonian fibrations over the 2-sphere from rational cohomology to complex cobordism. In addition to the use of Morava K-theory (as in the recent work with Blumberg on the Arnold Conjecture), the essential new ingredient is the construction of global Kuranishi charts for genus 0 pseudo-holomorphic curves; i.e. their realisation as quotients of zero loci of sections of equivariant vector bundles on manifolds.