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…
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48 results for “Lagrangian states”
Real analytic solutions found for special Lagrangian equation.
problem Analyzing convex solutions of the special Lagrangian equation.
method Interior regularity established for convex viscosity solutions.
result All convex solutions are real analytic in the interior.
A new Lagrangian method for graph neural networks accelerates state computation.
problem Efficiently computing states in graph neural networks for complex data.
method Lagrangian optimization for state convergence in graph neural networks.
result The proposed method accelerates state computation without iterative phases.
We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl.] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel.
The paper shows how Lagrangian duality improves deep learning for constrained problems.
problem Learning optimization problems with complex constraints in science and engineering.
method Lagrangian duality applied to deep learning models.
result Lagrangian duality brings significant benefits for constrained learning tasks.
We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
Study on Lagrangian phase operator solutions in compact domains.
problem Existence of solutions to the Dirichlet problem for the Lagrangian phase operator.
method Analysis of subsolutions and boundary conditions.
result Existence of solutions depends on the existence of subsolutions.
Alternative approach to regularize time-dependent singular Lagrangian systems.
problem Regularizing time-dependent singular Lagrangian systems.
method Employing the coisotropic embedding theorem and the Tulczyjew isomorphism.
result Uniqueness of the Lagrangian regularization to first order.
Develops geometric quantum mechanics in infinite dimensions.
problem Quantum dynamics in infinite-dimensional spaces.
method Tulczyjew triple concept for Lagrangian formalism.
result Self-adjoint operators as Lagrangian submanifolds.
Formulates a strategy to prove Atiyah-Floer conjecture.
problem Relate instanton Floer homology and Lagrangian Floer homology.
method Develops a strategy to construct the desired isomorphism.
result Formulates a possible approach to address singular space of flat connections.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for .
method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
Proposes a new method for GNNs that avoids iterative node state convergence.
problem Iterative computation of node states in GNNs is inefficient and requires many epochs.
method Constrained optimization in the Lagrangian framework to learn transition function and node states simultaneously.
result The proposed method compares favorably with existing models on various benchmarks.
CLWF improves time series imputation speed and accuracy.
problem Slow convergence in diffusion model-based imputation methods.
method CLWF uses Lagrangian mechanics to learn velocity and integrates a denoising autoencoder to estimate gradient.
result CLWF outperforms state-of-the-art imputation approaches.
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…
Unified DICE estimators as regularized Lagrangians for improved off-policy evaluation.
problem Improving off-policy evaluation from behavior-agnostic data.
method Unified derivation of DICE estimators as regularized Lagrangians of a linear program.
result Dual solutions offer greater flexibility and provide superior estimates in practice.
New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.
problem Categorification of Witten-Reshetikhin-Turaev invariants for 3-manifolds.
method Constructing a topological model from quantum group U_q(sl(2)) using Lagrangian intersections in configuration spaces.
result Witten-Reshetikhin-Turaev invariants are encoded by intersections of Lagrangian submanifolds in a fixed configuration space.
A membrane technique, in which the symplectic and Ricci forms are integrated over surfaces in a complexification of the phase space, as well a ``creation" connection with zero curvature over lagrangian submanifolds, is used to obtain a unified quantization including a noncommutative algebra of functions, its representa…
A distributed semismooth Newton based augmented Lagrangian method for distributed optimizationmath.OC
A new method solves distributed optimization problems over networks.
problem Solving optimization problems over networks with local cost functions and limited communication.
method Distributed semismooth Newton based augmented Lagrangian method.
result The method efficiently solves distributed optimization problems over networks.
Paper proposes method to recover quantized data with missing info.
problem Recovering quantized data with missing information.
method Regularized convex cost function with Bi-factorization and Augmented Lagrangian Method.
result The method finds global minimizer of the cost function.
New approach to symplectic isotopy problem using Lagrangian disks.
problem Symplectic isotopy conjecture in complex projective plane.
method Using nodal symplectic isotopy problem to guide model symplectic isotopies of smooth surfaces.
result Equivalence between smooth symplectic isotopy problem and existence problem of certain embedded Lagrangian disks.
New proof of Witten asymptotic conjecture for Seifert manifolds.
problem Proving Witten asymptotic conjecture for Seifert manifolds.
method Semiclassical analysis on a two-dimensional phase space torus.
result Witten-Reshetikhin-Turaev invariant expressed as scalar product of Lagrangian states.
We propose an efficient algorithm for sparse signal reconstruction problems. The proposed algorithm is an augmented Lagrangian method based on the dual sparse reconstruction problem. It is efficient when the number of unknown variables is much larger than the number of observations because of the dual formulation. More…
Certain solutions of a sextic sigma-model Lagrangian reminiscent of Skyrme model correspond to perfect fluids with stiff matter equation of state. We analyse from a differential geometric perspective this correspondence extended to general barotropic fluids.
Zonal states and improved bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfacesmath.AP
Improved bounds for eigenfunctions on hyperbolic surfaces found.
problem Establishing improved bounds for eigenfunctions of magnetic Laplacians.
method Using explicit eigenstates called magnetic zonal states.
result Explicit eigenstates called magnetic zonal states found.
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex -Toda equation and their geometric implications.
method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of corresponds to a specific surface class with integrable frames.
The study introduces flows for lagrangian varifolds to converge to special lagrangian cycles.
problem Understanding the behavior of lagrangian cycles in Calabi-Yau manifolds.
method Constructing mass-decreasing flows of lagrangian varifolds and cycles with vanishing Maslov index.
result Special lagrangian cycles are generated by the flow of cycles, and they represent a part of the lagrangian homology.
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.
A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in must be greater than one. In this paper we prove this for -smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in , viewed as…
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…
The paper proves stability of translating solitons in Lagrangian geometry.
problem Stability of translating solitons in Lagrangian geometry.
method Proof of Lagrangian -stability.
result Lagrangian translating solitons are Lagrangian -stable.
The paper lifts Lagrangian immersions to cones in complex space.
problem Creating Lagrangian cones from immersions in complex projective space.
method Developing a method to lift immersions to cones, producing examples and analyzing projections.
result Examples of Lagrangian cones and special cones are produced, with projections showing few transverse double points.
Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problemmath.DG
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…
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.
Improved non-squeezing theorem for calibrated geometries proved.
problem Proving an improved non-squeezing theorem for calibrated geometries.
method Two proofs: direct and reduction to classical case.
result Established an improved non-squeezing theorem for calibrated geometries.
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.
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.
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.
Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
A sparse semismooth Newton based augmented Lagrangian method for large-scale support vector machinesmath.OC
Efficiently solves large-scale SVMs with sparse semismooth Newton method.
problem Numerical difficulties in solving large-scale SVMs.
method Sparse semismooth Newton based augmented Lagrangian method.
result Outperforms state-of-the-art solvers for large-scale SVMs.
New Lagrangian torus found in nearly Kaehler S3×S3.
problem Classifying Lagrangian submanifolds in nearly Kaehler .
method Constructing and classifying Lagrangian immersions.
result Complete classification of Lagrangian immersions in nearly Kaehler .
Proves uniqueness of real Lagrangians up to cobordism.
problem Proving uniqueness of real Lagrangians in symplectic manifolds.
method Cobordism theory and Hamiltonian isotopy.
result Real Lagrangians are unique up to cobordism.
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.