Symplectic Geometry Papers of the Week

We obtain new constraints on the topology of Lagrangian submanifolds in the standard symplectic vector space $\mathbb{C}^n$ using Floer theory on the based loop space, generalizing a theorem of Fukaya-Irie.

We introduce a new surgical operation on Anosov flows, which we call bicontact padding surgery.

We prove the nearby Lagrangian conjecture for the Klein bottle: every closed connected embedded exact Lagrangian submanifold of its cotangent bundle is Hamiltonian isotopic to the zero section by a compactly supported isotopy.

Local mirror symmetry, originated from the work of Chiang-Klemm-Yau-Zaslow arXiv:9903053, indicated a potential relation between mirrors to a vector bundle and mirrors to its zero section. In this paper, we constructed a homological mirror to these local varieties and proved the corresponding Homological Mirror Symmetry statement.

Following the knot-quiver correspondence, we investigate quiver counterparts of the HOMFLYPT skein relations by interpreting link invariants in terms of holomorphic worldsheets and their multicover contributions. Extending this framework to more general worldsheet geometries, we conjecture that, up to quiver equivalence, the adjacency matrix of the quiver associated with any link in a skein triple admits a block decomposition whose diagonal blocks are quiver adjacency matrices associated with the other two links.

We show that a decomposable Lagrangian concordance between two Legendrian links must induce a closed immersion between the sheaf moduli of its ends.

We survey the recent development of spinal open books and nearly Lefschetz fibrations.

It is obtained by coupling two angular momenta. We construct a p -adic analog of this system for any prime number p and describe its symplectic normal forms at the critical points.

We consider the category of sheaves on a manifold with microsupport in a singular conic Lagrangian, and more generally microsheaves on the Lagrangian skeleton of a Weinstein manifold with stop, which are equivalent to ind-completions of partially wrapped Fukaya categories.

We prove that there is a nonconstant $J$-holomorphic plane $u:\mathbb{C}\rightarrow M\times \mathbb{T}^2$ for any almost complex structure $ J$ on $M\times \mathbb{T}^2$ tamed by $\omega_M\oplus\omega_{\text{std}}$.

We describe the shifted Lagrangian geometry of curves in Radko surfaces, through their global integrations and their homotopy Poisson algebras.

We study the chiralization of Poisson--Lie theory through graded symplectic geometry.

This paper proves the orientation relations required for worldsheet skein relations to cancel codimension-one boundary strata.

Recently, Pardon introduced a universal curve counting framework, which allows him to show that curve counting on 3-folds is "generated" by local curves (in a precise sense). We implement this universal curve counting program for Calabi-Yau 4-folds, where a significant technical difficulty comes from the dependence of the virtual dimension on the genus, which leads to more complicated generators than just local curves.

Inspired by the rigidity of exact Lagrangian submanifolds, a conjecture by P.~Biran and O.~Cornea asks if a simple exact Lagrangian cobordism $W$ in a tame exact symplectic manifold is always ``symplectically trivial'' (i.e. the Lagrangian suspension of an Hamiltonian isotopy). In this short note, we prove a topological version of this conjecture~: if the cobordism has dimension at least $6$, it is always topologically trivial.

This paper is motivated by the recent discovery, in symplectic topology, of a relation (partly proved, partly conjectural) between quantum connection and quantum Steenrod operations. We introduce an abstract TQFT framework, based on certain features of the definition of deformed symplectic cohomology, in which such a relation holds generally.

Using Floer theories, one can define two different entropies of $f$, called "Floer-theoretic" entropy and "categorical" entropy.

We introduce a relative version of Weinstein's morphism associated to a Lagrangian submanifold \(L\) of a closed symplectic manifold \((M,\omega)\).

Let $Y$ be a smooth cubic 3-fold. We show that $S^2 \times Y$ admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on $(S^2)^2 \times T^2$, but these appear to be the first simply-connected and monotone examples.

In this paper, we study the fiber connectivity of proper almost-toric systems $\Phi$ on connected four-dimensional symplectic manifolds.