Mathematics

Let Λ, Λ ′ be a pair of closed Legendrian submanifolds in a closed contact manifold (Y,ξ=Ker(α)) related by a Legendrian cobordism W⊂(C×Y, ξ ~ =Ker(−ydx+α)) . In this note, we show that in the hypertight setting, if Λ intersects a closed, weakly exact or monotone pre-Lagrangian P⊂Y for reasons of Floer homology, then so does Λ ′ .

For a convex domain in the standard Euclidean symplectic space which is invariant under a linear anti-symplectic involution τ we show that its Ekeland-Hofer-Zehnder capacity is equal to the τ -symmetrical symplectic capacity of it.

We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some high dimensional overtwisted spheres is not homotopically equivalent to a finite dimensional Lie group.

We construct an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov compactness.

I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. Schlenk.

This paper discusses homological mirror symmetry for the Fargues-Fontaine curve of equal characteristic.

We define an extended field theory in dimensions 1+1+1 , that takes the form of a `quasi 2-functor' with values in a strict 2-category Ham ˆ , defined as the `completion of a partial 2-category' Ham , notions which we define. Our construction extends Wehrheim and Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's construction of symplectic instanton homology. It can be seen, in dimensions 1+1 , as a real analog of a construction by Moore and Tachikawa. Our construction is motivated by instanton gauge theory in dimensions 3 and 4: we expect to promote Ham ˆ to a (sort of) 3-category via equivariant Lagrangian Floer homology, and extend our quasi 2-functor to dimension 4, via equivariant analogues of Donaldson polynomials.

This is the first of a series of two articles where we construct a version of wrapped Fukaya category WF(M∖K; H g 0 ) of the cotangent bundle T ∗ (M∖K) of the knot complement M∖K of a compact 3-manifold M , and do some calculation for the case of hyperbolic knots K⊂M . For the construction, we use the wrapping induced by the kinetic energy Hamiltonian H g 0 associated to the cylindrical adjustment g 0 on M∖K of a smooth metric g defined on M . We then consider the torus T=∂N(K) as an object in this category and its wrapped Floer complex C W ∗ ( ν ∗ T; H g 0 ) where N(K) is a tubular neighborhood of K⊂M . We prove that the quasi-equivalence class of the category and the quasi-isomorphism class of the A ∞ algebra C W ∗ ( ν ∗ T; H g 0 ) are independent of the choice of cylindrical adjustments of such metrics depending only on the isotopy class of the knot K in M . In a sequel [BKO], we give constructions of a wrapped Fukaya category WF(M∖K; H h ) for hyperbolic knot K and of A ∞ algebra C W ∗ ( ν ∗ T; H h ) directly using the hyperbolic metric h on M∖K , and prove a formality result for the asymptotic boundary of (M∖K,h) .

We define a unital A ∞ -category whose objects are exact Lagrangian cobordisms in the symplectization of Y=P×R , with negative cylindrical ends over Legendrians equipped with augmentations. The morphism spaces are given in terms of Floer complexes Ct h + ( Σ 0 , Σ 1 ) which are versions of the Rabinowitz Floer complex defined by Symplectic Field Theory (SFT) techniques.

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=U(n),SO(n) . Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniques of Poisson-Lie theory and the geometric crystals of Berenstein-Kazhdan. They provide a bridge between dual spaces of Lie algebras Lie(K ) ∗ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G= K C . We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K , we construct an exhaustion by symplectic embeddings of toric domains. As a by product we arrive at a conjectured formula for Gromov width of regular coadjoint orbits. We prove similar results for multiplicity free K -spaces.