The study finds infinitely many Lagrangian fillings for most Legendrian torus links.
problem Infinitely many Lagrangian fillings for Legendrian torus links except for a few.
method Constructing infinite order Lagrangian concordances and using actions of modular and mapping class groups.
result There exist infinitely many Lagrangian fillings for most Legendrian torus links.
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X,L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L⊂X which agrees with cylinders over Legendrian links $…
We prove that any Legendrian knot in (S3,ξstd) bounds an exact Lagrangian surface in R4∖B4 after a sufficient number of stabilizations. In order to show this, we construct a family combinatorial moves on knot projections with some additional data that correspond to Lagrangian cobordisms betw…
Holomorphic disks on compact Lagrangian surfaces are shown to exist.
problem Existence of holomorphic disks on compact Lagrangian surfaces.
method Analyzes compact Lagrangian surfaces in Kähler manifolds with specific curvature properties.
result Existence of holomorphic disks whose boundary is freely homotopic to a given class in the fundamental group.
New knots found without Lagrangian fillings.
problem Existence of Legendrian knots without Lagrangian fillings.
method Provided a family of Legendrian knots with augmentations not induced by any exact Lagrangian filling.
result Negative answer to the converse of the original statement.
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
Proves existence of exact lagrangian cobordisms between legendrians.
problem Existence of exact lagrangian cobordisms between closed legendrians.
method Proves existence through mathematical proof.
result Existence of non-compact exact lagrangian cobordisms.
Researchers found unique exact Lagrangian fillings for a specific type of knot.
problem Tackling the uniqueness of exact Lagrangian fillings for Legendrian (2,n) torus knots. method Computed augmentations induced by exact Lagrangian fillings to distinguish them.
result Identified that these exact Lagrangian fillings are pairwise non-isotopic.
The study explores surgeries on Lagrangian skeleta in 4-manifolds, connecting geometric and algebraic structures.
problem Understanding the combinatorics of exact Lagrangian surfaces in 4-manifolds.
method Analyzes surgeries on Lagrangian skeleta and their impact on the Fukaya category and cluster transformations.
result Induces cluster transformations on spaces of local systems and higher rank local systems.
Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.
problem Understanding the geometry and topology of Lagrangian submanifolds with Riemannian constraints.
method Investigation of metric properties and symplectic structures on spaces of Lagrangian submanifolds with uniform Riemannian bounds.
result There are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold.
Cluster structures on moduli spaces are linked to Legendrian knots.
problem Constructing cluster structures on moduli spaces.
method Quantization of exact Lagrangian fillings corresponding to Legendrian knots.
result Cluster structures can be deduced from Legendrian knots.
We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the nega…
Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.
problem Understanding Lagrangian cobordisms and their properties in cotangent bundles.
method Use microlocal theory of sheaves, sheaf quantization, and cone decompositions.
result Interleaving distance of sheaves is bounded by the shadow distance of the cobordism.
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
Paper constructs exact discrete Lagrangian for variational integrators.
problem Error analysis between exact and discrete trajectories.
method Construction of exact discrete Lagrangian in Lie groupoid and algebroid setting.
result Rigorous construction of exact discrete Lagrangian for variational integrators.
Characterizes Legendrian knots with exact Lagrangian fillings.
problem Identifying Legendrian knots with exact Lagrangian fillings.
method Characterization through exact orientable Lagrangian fillings.
result Underlying smooth knot types of fillable Legendrian 4-plats are positive.
New method bounds intersections of exact Lagrangians in cotangent bundles.
problem Counting intersections of exact Lagrangian submanifolds in cotangent bundles.
method Sheaf quantization in Tamarkin's category for clean and degenerate intersections.
result Cardinality of intersections is bounded by sheaf Hom spaces.
The study finds an obstruction for decomposable exact Lagrangian fillings of certain Legendrian knots.
problem Properties of decomposable exact Lagrangian cobordisms between Legendrian links.
method Analysis of normal rulings associated with decomposable exact Lagrangian fillings.
result A Legendrian knot has a decomposable exact Lagrangian filling if and only if it has an even number of normal rulings.
New exact Lagrangian tori found in torus cotangent bundle.
problem Existence of non-Hamiltonian isotopic Lagrangian tori.
method Constructing exact Lagrangian submanifolds in T∗Tn. result Found Lagrangian tori that are symplectically but not Hamiltonian isotopic to the zero section.
The paper finds non-isotopic exact Lagrangians in symplectic manifolds with C∗-actions.
problem Finding non-isotopic exact Lagrangians in symplectic manifolds.
method Using contracting C∗-actions, the paper constructs families of non-isotopic closed exact Lagrangian submanifolds. result The Floer cohomologies of these Lagrangians are topological, recovering ordinary cohomologies of intersections.
Paper studies Lagrangian submanifolds and their homological monodromy.
problem Understanding the homological monodromy of Lagrangian submanifolds.
method Proves triviality of homological Lagrangian monodromy under specific conditions.
result Homological Lagrangian monodromy is trivial if Hofer energy is less than minimum energy of J-holomorphic spheres and discs.
We establish an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly on…
Study symplectic mapping classes and their relations to surface mapping classes.
problem Comparing symplectic mapping classes with classical ones.
method Analogous configurations of Lagrangian submanifolds in Weinstein domains.
result Relations between Dehn twists in symplectic manifolds imply similar relations in surface mapping classes.
We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already ava…
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor …
Ancient solutions and translators identified for Lagrangian flow.
problem Characterizing ancient solutions and translators of Lagrangian mean curvature flow.
method Analyzing almost calibrated, exact, ancient solutions with specific geometric properties.
result All ancient solutions with entropy less than 3 are special Lagrangian, planes, or translators in \(\mathbb{C}^2\).
Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact s…
This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
Flexible Lagrangians in Weinstein domains lead to new constructions and exotic structures.
problem Understanding regularity and flexibility of Lagrangian manifolds.
method Introducing and discussing regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains.
result Many closed n-manifolds can be exact Lagrangian submanifolds of T∗Sn with exotic Weinstein structures. Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. Study corrects previous computation of Lagrangian topology in Calabi-Yau threefold.
problem Correcting a previous computation of the topology of a real Lagrangian in Schoen's Calabi-Yau threefold.
method Used two methods to compute mod 2 cohomology: Mayer-Vietoris calculation and an exact sequence.
result Agreement between the two methods confirms the topology of the real Lagrangian.
Non-orientable Lagrangian cobordisms exist for Legendrian knots, differing from orientable ones.
problem Understanding non-orientable Lagrangian cobordisms for Legendrian knots.
method Analyzing symplectization of standard contact 3-space and using exact, non-orientable Lagrangian cobordisms.
result Existence of non-orientable Lagrangian endocobordisms with crosscap genus being a positive multiple of 4.
We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.
The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.
problem Understanding the relationship between Legendrian links and cluster theory.
method Using exact Lagrangian fillings and cluster theory, the paper establishes connections between Legendrian links and cluster varieties.
result The augmentation variety of certain Legendrian 2-bridge links is isomorphic to a product of cluster varieties.
We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivia…
Constructs BGG resolutions for symplectic case.
problem Exactness of BGG resolutions in singular infinitesimal characters.
method Penrose transform over Lagrangian Grassmannian.
result Exactness of constructed complex over big affine cell.
The paper studies Lagrangian surfaces in a specific Riemannian product space.
problem Characterizing and classifying Lagrangian surfaces in a particular geometric space.
method Analyzes various types of Lagrangian surfaces and their properties.
result Classification of different types of Lagrangian surfaces.
A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional…
Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined …
New method solves constrained optimization problems efficiently.
problem Equality-constrained nonlinear, nonconvex optimization problems.
method Adaptive inexact Newton method with randomized iterative sketching.
result Global almost sure convergence and local linear/superlinear convergence.
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two…
We establish an h-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball B in the standard symplectic R2n,2n≥6, there exists an embedded Lagrangian n-disc transversely attached to B along its Legendrian boundary.
Geometrically interprets exact triangles of projectively flat bundles on complex tori.
problem Understanding exact triangles of projectively flat bundles on complex tori.
method Interprets projectively flat bundles geometrically and focuses on intersections of Lagrangian submanifolds.
result Geometric interpretation of exact triangles of projectively flat bundles.