Integrates singular subalgebroids using diffeological groupoids.
problem Integration of singular subalgebroids.
method Definition of integration via diffeological groupoids with specific properties.
result Holonomy groupoids correspond to singular subalgebroids with submersive property.
The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
Proves singular set of certain integral hypercurrents has measure zero.
problem Characterizing singular sets of specific integral hypercurrents.
method Proof based on varifold stationarity.
result Singular set has measure zero.
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
The study classifies singularities in curved 3D shapes.
problem Understanding nonnegatively curved polyhedral 3-manifolds.
method Classification of singularities with nonnegative curvature and integral monodromy.
result 32 isometry types of codimension 3 singularities are classified.
Symplectic classification for a specific type of singularity in integrable systems.
problem Symplectic classification of integrable systems near singular points of type An. method Real-analytic symplectic normal forms and classification of Lagrangian foliations.
result All integrable systems are symplectically equivalent near singular points of this type.
We compare the homology groups HnIC(X) of the chain complex of integral currents with compact support of a metric space X with the singular Lipschitz homology HnL(X) and with ordinary singular homology. If X satisfies certain cone inequalities all these homology theories coincide. On the other hand, for…
Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
The paper defines singular evolutoids and uses them to derive an integral equality.
problem Understanding singular points of evolutoids of smooth curves.
method Application of the Gauss-Bonnet Theorem to the extended front of evolutoids.
result Integral equality for smooth periodic curves derived from evolutoids.
The paper explores hidden torus symmetries in integrable systems and their stability.
problem Structural stability of singularities in integrable systems.
method Use of hidden torus actions near singular orbits and integrable perturbations.
result Persistence of toric symmetries and structural stability of Kalashnikov's parabolic orbits.
We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We a…
The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…
Smooth approximation of integral cycles mod 2 in Riemannian manifolds.
problem Approximating mod 2 integral cycles by smooth submanifolds.
method Approximation of mod 2 integral cycles by smooth submanifolds with controlled singularities.
result Every mod 2 integral cycle can be approximated by a smooth submanifold with a controlled singular set.
New action-angle coordinates found for singular symplectic manifolds.
problem Existence of action-angle coordinates for singular symplectic manifolds.
method Action-angle theorem for folded symplectic integrable systems.
result New topological obstructions found for global existence of action-angle coordinates.
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated in…
The paper improves estimates on singular sets in manifolds with integral curvature bounds.
problem Estimating the singular set of manifolds with integral curvature bounds.
method Using Gromov-Hausdorff limits and Cheeger-Naber methods.
result Improved Minkowski dimension estimate for singular sets.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth C∞ symplectic classification of Lagrangian fibrations near singularities. result Action variables form complete C∞ symplectic invariants for parabolic orbits and cuspidal tori. The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite (m−2)-dimensional Hausdorff measure and Minkowski content bounds. result The set of flat singular points has locally finite (m−2)-dimensional Hausdorff measure. In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under a…
Study on homology groups of cDV singularity links, identifying their topology.
problem Identify the topology of links of cDV singularities of types cAn and cDn. method Analyzing the second integral homology group of the links, using results from Smale and Thom-Sebastiani sums.
result The homology groups of the links are determined for cDV singularities of types cAn and cDn. Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
Formula calculates Riemann-Roch number for singular symplectic quotients.
problem Computing Riemann-Roch number for singular symplectic quotients.
method Complete singular stationary phase expansion of Witten integral.
result New explicit local invariant of singularities in symplectic quotients.
Constructs a Lie groupoid integrating singular foliations.
problem Integrating singular foliations into higher Lie groupoids.
method Recursive use of bi-submersions and geometric resolutions.
result Finite-dimensional Lie groupoid integrating singular foliations.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
This work computes integral variation and monodromy maps for plane curve singularities.
problem Computing integral variation and monodromy maps for plane curve singularities.
method Constructing analytic models, vector fields, and gyrographs to compute maps explicitly.
result Effective algorithms and gyrographs for computing integral variation and monodromy maps.
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
We classify linear Nambu structures (which are generalized Poisson structures in Hamiltonian dynamics and which give rise to integrable differential forms and singular foliations), then give a linearization for Nambu structures anf integrable differential forms near a nondegenerate singular point.
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface Σ. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.
Rectifies flat singular points of area-minimizing currents with singularity degree > 1.
problem Rectifying flat singular points of area-minimizing currents with singularity degree > 1.
method Subdividing singular points based on singularity degree and proving rectifiability of points with singularity degree > 1.
result The set of points with singularity degree > 1 is (m-2)-rectifiable.
Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.
We prove a singular Darboux type theorem for homogeneous polynomial closed 2-forms of degree one on Cn. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces.
Generalizes Gauss-Bonnet to metrics with logarithmic singularities.
problem Calculating curvature for metrics with singularities on compact surfaces.
method Proves a generalized Gauss-Bonnet formula under Lebesgue integrability condition.
result Establishes formula for special Kähler metrics with meromorphic cubic differentials.
Study on flat singular points of area-minimizing currents, defining a singularity degree.
problem Understanding the structure of singular points in area-minimizing integral currents.
method Analysis of vanishing sequences of scales around a singular point, defining a singularity degree.
result The singularity degree is independent of the chosen vanishing sequence and has interesting properties.
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singu…
Study optimal control of diffusion processes with infimum or supremum costs.
problem Optimizing control of a diffusion process with costs dependent on its infimum or supremum.
method Introduced novel integral operators to solve two-dimensional singular control problems.
result Explicit solutions for optimal dividend problem with time-dependent preferences.
The paper proves a Hawking-type singularity theorem using worldvolume quantum strong energy inequalities.
problem Improving classical singularity theorems with weakened energy conditions.
method Integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities.
result Past geodesic incompleteness proven in cosmological scenarios.
The paper is devoted to differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. Such distributions are defined as images of the tangent bundle under smooth endomorphisms. We prove the novel divergence theorem with the divergence type operator and deduce the Codazzi equatio…
Paper rigorously defines Feynman graph integrals on Kähler manifolds.
problem Establishing convergence of Feynman graph integrals on Kähler manifolds.
method Using Getzler's rescaling technique, graph integrands are extended to forms with divisorial-type singularities in the compactification of configuration spaces.
result Feynman graph integrals are rigorously defined as Cauchy principal value integrals.
We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the…
We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regard…