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.
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.
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 $…
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 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.
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.
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 \(\mathbb{C}^2\).
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…
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…
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.
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.
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…
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.
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.
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.
Study finds many Lagrangian fillings for certain Legendrian links.
problem Understanding Lagrangian fillings for Legendrian links of finite type.
method Use of N-graphs and combinatorics of seed patterns.
result Proves existence of at least seeds many exact embedded Lagrangian fillings for Legendrian links of type ADE.
We prove the existence of Lagrangian fillings for Dn-type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of Dn-type. method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
Torsion found in knot homology, challenging augmentation theories.
problem Torsion in linearized contact homology for Legendrian knots.
method Examples of Legendrian knots with non-trivial homology over Z.
result Augmentations not induced by exact fillings, even mod 2.
We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin…
We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study…
Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold (X,ω). We define a functional $\CC:\OO \to \R$ for each differential form β of middle degree satisfying β∧ω=0 and an exactness condition. If the exactness condition d…
Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence …
LDDNN learns physical dynamics from data without exact solutions.
problem Learning physical dynamics from data without exact solutions.
method LDDNN topology that learns Lagrangian density from data.
result LDDNN can learn physical dynamics from data.
Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.
problem Understanding how Lagrangian cobordisms impact DGAs of Legendrian ends.
method Adapting the map induced by cobordisms on DGAs to linearizations using augmentations, and showing invariance under Lagrangian isotopy.
result The induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses.
Exact discrete mechanics for nonholonomic systems defined.
problem Discrete mechanics for nonholonomic systems.
method Constructing an exponential map and deriving exact discrete nonholonomic integrators.
result Reproduces continuous nonholonomic flow as discrete flow on constraint submanifold.