This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian submanifolds. The examples are constructed using a special class of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer homology. 
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Lagrangian two-spheres can be symplectically knotted

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed arboreal Lagrangian skeleta for Weinstein pairs and Weinstein 4-manifolds associated to max-tb Legendrian representatives of algebraic links. We provide computations of Legendrian and Weinstein invariants, and discuss the symplectic topological nature of the Fomin-Pylyavskyy-Shustin-Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.

Lagrangian Skeleta and Plane Curve Singularities

We prove that the augmentation variety of any positive braid Legendrian link carries a natural cluster $\mathrm{K}_2$ structure. We present an algorithm to calculate the cluster seeds that correspond to the admissible Lagrangian fillings of the positive braid Legendrian links. Utilizing augmentations and cluster algebras, we develop a new framework to distinguish exact Lagrangian fillings.

Augmentations, Fillings, and Clusters

Author(s): Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Simental, Jose | Abstract: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

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Algebraic Weaves and Braid Varieties

We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.

Honghao Gao

Papers

Augmentations, Fillings, and Clusters

Positive Braid Links with Infinitely Many Fillings