Nearby pinwheels are isotopic, solving Arnold's conjecture.
problem Proving isotopy of Lagrangian pinwheels in rational homology balls.
method Combining neck-stretching, symplectic blow-up, and computation of isotopy groups.
result Two pinwheels are isotopic, confirming Arnold's conjecture.
Study symplectic forms on manifolds to find Lagrangian pinwheels that can be separated.
problem Determine conditions for symplectic forms to carry disjoint Lagrangian pinwheels.
method Use rational blow-up to analyze Lagrangian pinwheels in symplectic manifolds.
result Conditions for disjunction of Lagrangian pinwheels in specific manifolds.
The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
We study Lagrangian embeddings of a class of two-dimensional cell complexes Lp,q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type p21(pq−1,1) (Wahl singularities). We show that …
We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight -- namely the nonexistence of a polygonal region which we call a virtual pinwheel.
We present a method for finding embedded nullhomologous tori in standard 4-manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth 4-manifolds with b^+ = 1 and b^- = 2,...,7, via surgery on nullhomologous tori embedded in th…
We consider a novel stochastic multi-armed bandit setting, where playing an arm makes it unavailable for a fixed number of time slots thereafter. This models situations where reusing an arm too often is undesirable (e.g. making the same product recommendation repeatedly) or infeasible (e.g. compute job scheduling on ma…
Constructing translating solitons from Lagrangian Grim Reapers.
problem Creating Lagrangian translating solitons from intersections of Grim Reapers.
method Desingularizing intersections with special Lagrangian Lawlor necks.
result Constructing Lagrangian translating solitons with multiple ends and loops.
In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian …
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
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.
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
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 prove that any Lagrangian translating soliton is Lagrangian L-stable.
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 \(\mathbb{C}^2\).
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…
Proves a conjecture about Lagrangian intersections using new theory.
problem Homological Arnol'd conjecture on Lagrangian intersections.
method New Lagrangian Ljusternik-Schnirelman theory and fundamental quantum factorizations.
result Uniform lower bounds on Lagrangian intersection numbers.
We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
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.
Paper proves estimates for Lagrangian flow singularities.
problem Understanding Lagrangian flow singularities.
method Interior a priori estimates and Jacobi inequality.
result Proves estimates for supercritical Lagrangian phase.
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…
In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in certain symplectic manifolds, conditions on low dimensional topological invariants…
We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…
In this paper we show how to lift Lagrangian immersions in CPn−1 to produce Lagrangian cones in Cn, and use this process to produce several families of examples of Lagrangian cones and special Lagrangian cones. Moreover we show how to produce Lagrangian cones, isotopic to the Harvey-Lawson a…
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
problem Homotopy rigidity of nearby Lagrangian cocores in Weinstein sectors.
method Spectral wrapped Donaldson-Fukaya category with orthogonal group coefficients.
result Inclusion followed by retract and quotient is null-homotopic.
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
The paper classifies certain types of Lagrangian translators and self-expanders in complex 2-space.
problem Classifying Lagrangian translators and self-expanders in complex 2-space.
method Using a new Omori-Yau type maximum principle proved by Chen and Qiu.
result Several classification results of 2-dimensional complete Lagrangian translators and self-expanders.
The abstract discusses special Lagrangians and their flow, proving conjectures and observing related phenomena.
problem Existence and long-time existence of special Lagrangian representatives and Lagrangian mean curvature flow.
method Gibbons-Hawking ansatz, circle-invariant hyperkaehler 4-manifolds, Calabi-Yau 2-folds, Thomas conjecture, Thomas-Yau conjecture.
result Proves versions of the Thomas conjecture and Thomas-Yau conjecture.
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.
Generalizes Thomas-Yau theorem for special and minimal Lagrangians.
problem Proving uniqueness of special Lagrangians and minimal Lagrangians.
method Hamiltonian perturbations using Imagi, Joyce, and Oliveira dos Santos method.
result Generalized uniqueness theorem for special and minimal Lagrangians.
Paper solves flat bi-Lagrangian structure problems in ray space.
problem Existence of flat bi-Lagrangian structures in ray space.
method Established geometric conditions for flat canonical connections.
result Complete solutions to two problems regarding flat bi-Lagrangian structures.
New Lagrangian and special Lagrangian examples found in complex space.
problem Constructing exact Lagrangian and special Lagrangian submanifolds with symmetries.
method Using an Ansatz generalizing Castro-Lerma's construction, with admissible compact and non-compact subgroups.
result Explicit examples of Lagrangian translators and special Lagrangians with various symmetries.
Study on deformations of special Lagrangians with boundary in Calabi-Yau manifolds.
problem Deformation problem for special Lagrangians with boundary constraints.
method Identifying tangent vectors with harmonic 1-forms vanishing on the boundary, proving unobstructed deformations.
result Moduli space of special Lagrangians with boundary is a smooth manifold.
Real Lagrangians in toric manifolds are classified by combinatorial data.
problem Classifying real Lagrangian submanifolds in toric symplectic manifolds.
method Established a real analog of the Delzant construction.
result Real Lagrangians in toric del Pezzo surfaces have all possible diffeomorphism types.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
Upper bounds for Lagrangian capacities of Liouville domains
problem Lagrangian capacity of Liouville domains
method Using S1-equivariant techniques result Extremal Lagrangian torus on the boundary of ellipsoid
Variational reduction simplifies Lagrangian systems with scaling symmetries.
problem Simplifying Lagrangian systems with scaling symmetries.
method Defining a variational reduction procedure for homogenous Lagrangian systems.
result Reconstructing trajectories from critical points of reduced variational principle.
In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagran…
Study Lagrangian Floer theory in smooth divisor complements.
problem Intersection Floer theory of Lagrangians in smooth divisor complements.
method Complete construction of Floer homology.
result Floer homology for Lagrangians in smooth divisor complements constructed.
We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Visible Lagrangians in Hitchin systems are studied for pillowcase covers.
problem Visible Lagrangians in Hitchin systems intersecting non-trivially.
method Computation of Fourier-Mukai transforms and study of mirror dual branes.
result Mirror dual branes are closely related to Hausel's toy model.
Study decomposability of Lagrangian classes on K3 surfaces.
problem Decomposability of Lagrangian classes on K3 surfaces.
method Lattice theory and special Lagrangian submanifolds.
result Prove decomposability of Kähler classes on dense subsets of the Kähler cone.