Symplectic Geometry
We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type $\mathsf{ADE}$ or affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with…
The spectral diameter of a symplectic ball is shown to be equal to its capacity; this result upgrades the known bound by a factor of two and yields a simple formula for the spectral diameter of a symplectic ellipsoid. We also study the…
We prove the generic existence of spectral networks for a large class of spectral data.
This paper introduces a new algebraic notion - triangulated persistence category (TPC) - that refines that of triangulated category in the same sense that a persistence module is a refinement of the notion of a vector space. The spaces of…
For an oriented manifold $M$ and a compact subanalytic Legendrian $\Lambda \subseteq S^*M$, we construct a canonical strong smooth relative Calabi--Yau structure on the microlocalization at infinity and its left adjoint $m_\Lambda^l:…
We show that each cluster seed in the augmentation variety is inhabited by an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to…
We use time-independent incomplete Hamiltonian flows to excise interesting closed subsets of positive codimension from symplectic manifolds. Examples of such subsets include what we call a "Cantor brush", a "box with a tail", and -- more…
In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the…
In this article we prove that the space of Floer Hessians has infinitely many connected components.
We show that the category of linearly topologized vector spaces over discrete fields constitutes the correct framework for algebraic structures on Floer homologies with field coefficients. Our case in point is the Poincar\'e duality theorem…
We construct a stable infinity category with objects flow categories and morphisms flow bimodules; our construction has many flavors, related to a choice of bordism theory, and we discuss in particular framed bordism and the bordism theory…
For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are…
We resolve the periodic square peg problem using a simple Lagrangian Floer homology argument. Inscribed squares are interpreted as intersections between two non-displaceable Lagrangian sub-manifolds of a symplectic 4-torus.
We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all…
We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC \`a la…
Generalized complex geometry was classically formulated by the language of differential geometry. In this paper, we reformulated a generalized complex manifold as a holomorphic symplectic differentiable formal stack in a homotopical sense.…
We upgrade the natural weakly-filtered structure of Fukaya categories discussed in arXiv:1806.06630 to a genuinely filtered one. The main tools are a Morse-Bott, or 'cluster', model for Fukaya categories and a particular choice of class of…
We construct global Kuranishi charts for the moduli spaces of stable pseudoholomorphic maps to a closed symplectic manifold in all genera. This is used to prove a product formula for symplectic Gromov-Witten invariants. As a consequence we…
We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface…
We study loops of symplectic diffeomorphisms of closed symplectic manifolds. Our main result, which is valid for a large class of symplectic manifolds, shows that the flux of a symplectic loop vanishes whenever its orbits are contractible.…