Proves Goh conditions for singular curves with specific properties.
problem Finding optimal paths with specific geometric constraints.
method Proof of Goh conditions of order n and open mapping theorem.
result Establishes conditions for strictly singular curves of corank 1.
For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7 singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical A−D−E singularities of planar…
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
The paper examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…
Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
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.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.
Study of singular solutions to a fourth order system in a ball with a singularity.
problem Asymptotic behavior of singular solutions to a conformally invariant fourth order system.
method Spectral analysis and a priori estimates for Jacobi fields.
result Solutions near the singularity behave like Emden--Fowler solutions.
Study on solutions to conformally invariant fourth order equations, classifying their properties.
problem Classify qualitative properties of solutions to conformally invariant fourth order equations.
method Analyze two cases: removable and non-removable singularities, using Pohozaev-type invariant.
result Non-existence of semi-singular solutions, classifying them as multiples of the Emden--Fowler solution.
Study of origamis' singularities for groups of prime-power order.
problem Classifying singularities of origamis for groups of prime-power order.
method Geometric and group-theoretic ideas used to classify strata.
result Many groups of prime-power order have only one stratum, but some do not.
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We furthe…
We study singular monopoles on open subsets in the 3-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. The other is given in terms of the growth order of the norms of the…
Study finds solutions to Yamabe equation with specific behavior near singular points.
problem Existence of solutions with prescribed asymptotic behavior near singular points of the Yamabe equation.
method Analysis of positive solutions with isolated singularities and asymptotic expansions.
result Existence of solutions with arbitrarily high order of approximation near singular points.
We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.
problem Calculating volumes of moduli spaces of flat surfaces with prescribed conical singularities.
method Induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.
result Explicit computation of volumes is possible.
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
We consider a closed Willmore surface properly immersed in Rm (m>2) with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the "conservative" reformulation of the Willmore equation introduced in a previous paper by the second author, we show that, i…
Removes singularity order for Willmore immersions, reducing bubbling scenarios.
problem Understanding the singularity order of weak limits of Willmore immersions.
method Obtains removability result on singularity order, reducing bubbling scenarios.
result Only three out of twelve non-planar minimal surfaces may occur as bubbles of Willmore immersions.
Classifies singular foliations of a specific type and studies their extensions.
problem Classifying singular foliations of a particular type and understanding their extensions.
method Introduces and classifies singular foliations of bk+1-type, proving they are encoded by k-th order foliations. result Singular foliations of bk+1-type are encoded by k-th order foliations, and these groupoids fiber over certain character stacks. Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.
problem Understanding the monodromy of Milnor fibrations of surface singularities.
method Using Seiberg-Witten invariant and monopole Floer homology.
result Infinite order non-triviality results for boundary Dehn twists.
Compactness of metrics with positive sixth order Q-curvature on a sphere with punctures.
problem Compactness of conformally flat singular metrics with constant, positive sixth order Q-curvature.
method Introduced necksize concept, used moving planes and blow-up arguments, proved upper and lower bounds, introduced homological invariant.
result A subsequence of metrics converges with respect to Gromov--Hausdorff metric if punctures remain separated and necksize is bounded away from zero.
The purpose of this paper is to prove the a priori estimates for constant scalar curvature Kaehler metrics with conic singularities along normal crossing divisors. The zero order estimates are proved by a reformulated version of Alexandrov's maximum principle. The higher order estimates follow from Chen-Cheng's frame …
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of …
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical…
Study 3D shapes in 5D space with sharp points.
problem Understanding shapes with sharp points in higher dimensions.
method Define curvature locus using fundamental forms at sharp points.
result Local second order geometrical information captured.
Study of light function singularities on surfaces.
problem Characterizing singularities of the slant function on surfaces.
method Analyzing the differential geometry of the parabolic set and its spherical image under the Gauss map.
result The type of singularities of the slant function is determined by the geometry of the parabolic set and its spherical image.
In this paper, we develop the theory of Perelman's W-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of W-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…
At each point in an immersed surface in R4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in R3, a curvature parabola in the normal plane which codifies all the …
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in ord…
Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.
problem Understanding local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeros and poles, and k-residues at poles.
result For a given pattern of zeros, there exists a primitive holomorphic k-differential with these zeros.
We introduce thermodynamic response functions for singular Bayesian models.
problem Singular Bayesian models violate regular asymptotics due to non-identifiability and degenerate Fisher geometry.
method Posterior tempering induces thermodynamic response functions, linking WAIC, WBIC, and singular fluctuation.
result WAIC, WBIC, and singular fluctuation are unified within a thermodynamic response framework.
We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere, when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached all Darboux transforms converge to the original …
Study shows Kähler-Einstein metric singularities linked to curvature.
problem Understanding singularities of Kähler-Einstein metrics.
method Relates singularities to holomorphic sectional curvature of conical geometry.
result Provides second-order estimates with explicit constants.
This paper constructs metrics with constant fractional higher order curvature on punctured spheres.
problem Constructing complete metrics with constant fractional higher order curvature on punctured spheres.
method The approach involves constructing singular solutions for a conformally invariant integro-differential equation, reducing the problem to solving an infinite-dimensional Toda-type system.
result Unified approach for fractional and higher order cases, proving Fredholm properties for the linearized operator.
Extends results on smoothability of singular Fano and Calabi-Yau varieties.
problem Smoothability of singular Fano and Calabi-Yau varieties under terminal singularities.
method Generalizes deformation theory results for Calabi-Yau and Fano threefolds to higher dimensions, using higher Du Bois and rational singularities.
result Identifies a class of singularities for which smoothing results hold, including generalized Fano and Calabi-Yau varieties.
Study of singular foliations of b^k-type and their geometric properties.
problem Classify and understand singular foliations of b^k-type.
method Introduce and study singular foliations of b^k-type, classify them, and relate them to geometric structures.
result Obstructed by a characteristic class when extending k-th order foliations to (k+1)-th order foliations.
Unified framework for singular statistical models using observable charts.
problem Non-identifiability and breakdown of classical asymptotic theory in singular models.
method Invariant framework based on observable charts to define local coordinate systems in model space.
result Observable order provides a lower bound on KL divergence vanishing rate in singular models.
Lecture notes on singular foliations, smooth and holomorphic.
problem Understanding singular foliations in geometry.
method Review of foundations, recent tools from non-commutative geometry, and homotopic notions.
result Introduction of various homotopic notions and open questions.
Paper studies third order open mapping in sub-Riemannian geometry.
problem Analyzing third order open mapping in sub-Riemannian geometry.
method Third order open mapping results for maps from a Banach space into a finite dimensional manifold. Computing third order term in the Taylor expansion of the end-point map.
result Specialization of abstract theory to study length-minimality of sub-Riemannian strictly singular curves and third order analysis of specific extremal curves.
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…
This paper shows how to construct Abelian differentials with any prescribed singularities.
problem Constructing Abelian differentials with specific orders and residues.
method Flat representation of Abelian differentials.
result Every pattern of orders and residues can be realized in Abelian differentials, except for two families in genus zero.
Localized Kasner-like singularities constructed in spacetime.
problem Constructing localized singular solutions to Einstein vacuum equations.
method First order symmetric hyperbolic formulation, adapted orthonormal frame.
result Localized Kasner-like singularities with refined uniqueness and general asymptotic data.
A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.
Defines axial curvatures for corank 1 singular manifolds in higher dimensions.
problem Characterizing singular n-manifolds in Rn+k with corank 1 singular points. method Using curvature locus and second fundamental form, defining up to l(n−1) axial curvatures. result Umbilic curvatures are absolute values of our axial curvatures.