Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
Survey and extend work on singular foliations in diffeology.
problem Understanding singular foliations and their properties in diffeological settings.
method Survey Stefan and Sussmann's work, introduce transverse equivalence, and define basic cohomology.
result Transverse equivalence of singular foliations preserves leaf spaces diffeologically but not conversely.
New equivalence for singular foliations preserves transverse geometry.
problem Transverse geometry of singular foliations.
method Introducing a new notion of equivalence for singular foliations and showing compatibility with holonomy groupoids.
result Unified invariants and equivalence of singular foliations.
Simplicial sets deformation retract onto transverse simplices.
problem Deformation retraction of simplicial sets.
method Showed deformation retraction of singular simplicial set onto transverse simplices.
result Singular simplicial set deformation retracts onto transverse simplices.
Study shows boundary measurements can determine transversal singularities in anisotropic geometries.
problem Determining transversal singularities in anisotropic geometries from boundary measurements.
method Geometric condition on transversal manifold and FBI type transform.
result Recovery of transversal singularities in the linearized problem.
Refined theorem on linear perturbations with applications in singularity theory and optimization.
problem Linear perturbations and their implications in singularity theory and optimization.
method New perspective of Hausdorff measures for refined transversality theorem.
result Applications in singularity theory and optimization.
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…
Introduces a new equivalence for singular foliations and their groupoids.
problem Preserving transverse geometry in singular foliations.
method Introduces a new equivalence relation for singular foliations and connects it to holonomy groupoids.
result Establishes a connection between singular foliations and their associated holonomy groupoids.
Quantizes symplectic manifolds with toric singularities using Toeplitz operators.
problem Quantize symplectic manifolds with toric singularities.
method Establishes quantization for compact toric symplectic manifolds with transversal singular real polarizations using Toeplitz operators.
result Toeplitz operators determine a star product on compact toric symplectic manifolds with toric singularities as ℏo0+. Local models and Lie groupoids for Riemannian foliations.
problem Understanding the geometry of Riemannian foliations locally.
method Construction of Lie groupoids and algebroids controlling foliation transverse geometry.
result Local models and Lie groupoids provide control over foliation geometry.
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …
Study on singularities of frontal surfaces, classifying under equivalence.
problem Classifying singularities of frontal surfaces.
method Classification under left-right-equivalence, introduction of frontalisation, definition of cuspidal and transverse double point curves.
result Frontal surfaces have finite codimension if and only if the curves are reduced.
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
The study proves a transverse diameter theorem for Lorentzian foliations.
problem Understanding the geometry of foliations in Lorentzian spacetimes.
method Developed a novel causality structure on leaf spaces via transverse Lorentzian geometry.
result Derived a transverse diameter theorem for Lorentzian foliations and orbifolds.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Classifies foliations with a hypersurface as the singular leaf and open leaves.
problem Classifying foliations with a hypersurface as the singular leaf and open leaves.
method Analyzes the transverse order k foliations, showing that a loop in the singular leaf induces a well-defined holonomy transformation.
result A complete classification of these foliations and concrete descriptions of their associated groupoids and algebras.
Classifies singularities of smooth vector fields on the line.
problem Classifying singularities of smooth vector fields on the line.
method Local classification with respect to C1-conjugacy, including normal forms and unfoldings. result Complete description of the 1-d case achieved.
The study of geometric structures around transversals using deformation spaces.
problem Understanding local behavior of geometric structures around singular foliations.
method Using deformation spaces to study local behavior of geometric structures.
result Obtained normal form theorems around transversals for various geometric structures.
Study on cohomology of singular foliations with localization results.
problem Understanding cohomology of singular Riemannian foliations.
method Introduced equivariant basic cohomology and proved its properties.
result Equivariant basic cohomology localizes to closed leaves.
We develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.
Defines singular grid diagrams for various types of links.
problem Unified description of singular links and related objects.
method Definition of singular grid diagrams and classification of equivalence relations.
result Unified description of singular links and related objects.
Homotopy equivalence found between Milnor-Lê fibers of specific singularities.
problem Analyzing non-isolated singularities and their Milnor-Lê fibers.
method Using transversality property and homotopy equivalence to relate Milnor-Lê fibers of different singularities.
result Homotopy equivalence between negative Milnor-Lê fibers of specific singularities.
This study addresses transitions in conically singular associative submanifolds and their desingularizations.
problem Counting closed associative submanifolds of G2-manifolds and understanding transitions arising from degenerations. method Analysis of moduli spaces, transversality results, and desingularization techniques for conically singular associative submanifolds.
result For generic co-closed G2-structures, there are no CS associative submanifolds with stability-index greater than 0 or 1. We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space.
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler c…
Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.
problem Hamilton-Tian conjecture for specific Sasakian manifolds.
method Sasaki-Ricci flow, compact transverse Fano Sasakian 5-manifolds, klt foliation singularities.
result Confirmed Hamilton-Tian conjecture for compact transverse Fano Sasakian 5-manifolds.
Let F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective out…
In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.
problem Characterizing Sasaki-Ricci solitons on Sasakian manifolds of up to seven dimensions.
method Analysis of the Sasaki-Ricci flow and convergence to solitons.
result Existence and classification of Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.
This paper compares two invariants of foliated manifolds which seem to measure the non-Hausdorffness of the leaf space: the transversal length on the fundamental group and the foliated Gromov norm on the homology. We consider foliations with the property that the set of singular simplices transverse to the foliation sa…
The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as appli…
Survey on Killing foliations with technical advantages.
problem Understanding closures of Riemannian foliations.
method Review of Molino's structural theory and transverse isometry theory.
result Closures of Killing foliations described by transverse Killing vector fields.
The paper shows convergence of Sasaki-Ricci flow on Sasakian 5-manifolds.
problem Analyzing convergence of Sasaki-Ricci flow on Sasakian manifolds.
method Uniform L^4-bound of transverse Ricci curvature, application of normalized Sasaki-Ricci flow.
result Solutions converge to unique singular Sasaki η-Einstein metric.
Holonomy defined for singular leaves in foliations.
problem Defining holonomy for singular structures in foliations.
method Introducing a sequence of group morphisms from π_n(L) to π_{n-1} of the universal Lie ∞-algebroid.
result Holonomy sequence relates to Androulidakis-Skandalis and Brahic-Zhu constructions.
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
Study families of Lie algebroids on complex spaces, introducing unfoldings.
problem Investigate singular holomorphic Lie algebroids on complex analytic spaces.
method Introduce and study unfoldings of Lie algebroids, showing a correspondence with holomorphic flat connections.
result Existence of a one-to-one correspondence between transversal unfoldings and holomorphic flat connections.
We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally comple…
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…
A 2-web in the plane is given by two everywhere transverse 1-foliations. In this paper we introduce the study of singular 2-webs, given by any two foliations, which may be tangent in some points. We show that such two foliations are tangent along a curve, which will be called the polar curve of the 2-web, and we study …
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…
Positive curvature forces foliation leaf spaces to have boundaries.
problem Understanding boundaries in foliated leaf spaces with positive curvature.
method Analyzing singular Riemannian foliations with positive sectional curvature.
result Polar foliations of positively curved manifolds have leaf spaces with nonempty boundaries.
We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds M having non-empty compact singular set $\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if the transverse type of $\fa$ at each singular point is a center and $\fa$ has a compact leaf with fini…
Completes preliminary structures in 3D flows to foliations.
problem Characterizing completability of lamination pairs in 3-manifolds.
method General approach for various types of foliations and flows.
result Characterizes when lamination pairs can be completed to foliations.
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.