I shall present two billiard-like systems associated with a convex hypersurface in a symplectic space, the outer and an inner ones. The talk will survey the known results and focus on open problems.
Tag - Symplectic geometry
Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space, H4. The main theme of my talk is that it should be possible to count minimal surfaces in H4 which fill K and obtain a link invariant. In other words, the count doesn’t change under isotopies of K. When one counts minimal disks, this is a theorem. Unfortunately there is currently a gap in the proof for more complicated surfaces. I will explain 'morally' why the result should be true and how I intend to fill the gap. In fact, this (currently conjectural) invariant is a kind of Gromov–Witten invariant, counting J-holomorphic curves in a certain symplectic 6-manifold diffeomorphic to S2×H4. The symplectic structure becomes singular at infinity, in directions transverse to the S2 fibres. These singularities mean that both the Fredholm and compactness theories have fundamentally new features, which I will describe. Finally, there is a whole class of infinite-volume symplectic 6-manifolds which have singularities modelled on the above situation. I will explain how it should be possible to count J-holomorphic curves in these manifolds too, and obtain invariants for links in other 3-manifolds.
I will discuss an idea of constructing a category associated with a pair of holomorphic Lagrangians in a hyperkähler manifold, or, more generally, a manifold equipped with a triple of almost complex structures I, J, K satisfying the quaternionic relation IJ = -JI = K. This category can be seen as an infinite-dimensional version of the Fukaya-Seidel category associated with a Lefschetz fibration. While many analytic aspects of this proposal remain unexplored, I will argue that in the case of the cotangent bundle of a Lefschetz fibration, our construction recovers the Fukaya-Seidel category.
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.
Most work on Lagrangian fillings of Legendrian knots to date has concentrated on orientable fillings, but instead I will present some first steps in constructions of and (especially) obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings.
The existence of rational plane curves of a given degree with prescribed singularities is a subtle and active area in algebraic geometry. This problem turns out to be closely related to difficult enumerative problems which arise in symplectic field theory, which in turn play a central role in the theory of high-dimensional symplectic embeddings. In this talk, I will discuss various perspectives on these enumerative problems and present a new closed formula for relevant curve counts as a sum over decorated trees.
Because of the presence of non-trivial automorphisms of stable maps, Gromov-Witten invariants of a general symplectic manifold are usually rational-valued. Realizing a proposal of Fukaya-Ono back in the 1990s, I will explain how to construct integer-valued Gromov-Witten type invariants by virtually counting stable maps with trivial automorphism groups after a suitable abstract perturbation of the Cauchy-Riemann equation. I will also discuss how this idea would fit into a programme initiated by Abouzaid-McLean-Smith and Pardon on finding refinements of Gromov-Witten invariants with values in generalized cohomology theory.
I will present two constructions of Kähler manifolds, endowed with Hamiltonian torus actions of infinite dimension. In the first example, zeroes of the moment map are related to isotropic maps from a surfaces in ℝ2n. In the second example, which is actually a hyperkähler moment map, the zeroes are related to symplectic maps of the torus T4. The corresponding modified moment map flows have short time existence. Polyhedral analogues of these constructions can be used to investigate piecewise linear symplectic geometry.
I will explain the construction of a new class of Liouville domains that live in a complex torus of arbitrary dimension, whose boundary dynamics encodes information about the singularities of a toric compactification. The primary motivation for this work is to find a symplectic interpretation of some curious Laurent polynomials that appear in mirror symmetry for Fano manifolds; it also potentially opens a path to bound symplectic capacities of polarized projective varieties from below.
I will talk about an ongoing project that explores the construction of high-dimensional Legendrian spheres from supporting open books and contact structures. The input is a Lagrangian disk filling of a Legendrian knot in the binding. We try to understand the relationship between different constructions from the same input, and suggest parallels, in the S2n+1 case, to a construction defined by Ekholm for ℝ2n+1.
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