New technique shows non-orbifold singularities are undetectable by G-invariant spectrum.
problem Detecting non-orbifold singularities using the G-invariant spectrum.
method Generalized Sunada-Pesce-Sutton technique to the G-invariant setting.
result Found an isospectral pair of spaces, one orbifold and one with non-orbifold singularities.
Compact Ricci-flat manifolds with conical singularities are found.
problem Finding compact Ricci-flat manifolds with conical singularities.
method Analyzing flat limits of smooth polarized Calabi-Yau manifolds and using the Calabi ansatz.
result The existence of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.
A new algebraic method extracts symmetry anomalies from 5D SCFTs.
problem Extracting global symmetry anomalies from 5D superconformal field theories.
method Path algebra of branes probing Calabi-Yau cones provides a complementary approach.
result Combinatorial approach to symmetry anomalies in 5D SCFTs.
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
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.
Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
New singularity concept in GR: volume singularities.
problem Understanding spacetime singularities in General Relativity.
method Introducing a new type of singularity: volume singularities.
result Volume singularities are hidden by event horizons.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
Study describes singularities of distance squared functions on singular surfaces.
problem Characterizing singularities of distance squared functions on singular surfaces.
method Using smooth map-germs Sk, Bk, Ck, and F4 singularities, the study describes singularities via blowing-ups. result Characterization of singularities of wave-fronts and caustics of singular surfaces.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a 5/2-cuspidal edge is considered. Study singularities of Weinstein skeleta and arboreal singularities.
problem Understanding singularities in Weinstein skeleta and their relation to arboreal singularities.
method Deforming the skeleton via homotopies, Morse-Bott representative, generic perturbation, localization procedure.
result Singularities of Weinstein skeleta can be classified into arboreal singularities or tangency singularities.
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.
The paper studies singularities of pedal curves of hyperbolic frontals.
problem Investigating singularities of pedal curves of spacelike frontals in hyperbolic 2-space.
method Analyzing singularities of pedal curves based on dual curve germs and pedal point locations.
result The singularities of pedal curves depend on the singularities of the first hyperbolic Legendrian curvature germ and the pedal point for non-singular dual curve germs. For singular dual curve germs, additional dependence on both Legendrian curvature germs is observed.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
Maxfaces can have cuspidal edges near certain singularities.
problem Characterizing singularities on maxfaces.
method Analyzing singular Björling data and proving geometric properties.
result Near a maxface with a specific type of singularity, there exists another maxface with a cuspidal edge.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.
problem Understanding geodesic behavior near singularities in Riemannian manifolds.
method Analytical study of geodesics on Riemannian manifolds near singularities.
result The winding number of geodesics around a singularity depends on the singularity type and approaches infinity as the singularity becomes cuspidal.
This paper studies singular improper affine spheres from Lagrangian submanifolds, classifying stable singularities.
problem Understanding singularities of improper affine spheres from Lagrangian submanifolds.
method Analyzes canonical improper affine spheres and their on-shell singularities from Lagrangian submanifolds in arbitrary even dimensions.
result Classifies stable Lagrangian/Legendrian singularities on shell for improper affine spheres.
Defines algebraic structures from singular knot theory.
problem Distinguishing singular knots and links.
method Generalized Reidemeister moves and colorings.
result Set of colorings is an invariant of singular links.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Criteria for Morin singularities in lower dimensions with applications to bifurcation.
problem Understanding Morin singularities in maps into lower dimensions.
method Providing criteria for Morin singularities and studying bifurcations of Lefschetz singularities.
result Criteria for Morin singularities in maps into lower dimensions.
Generates moves for oriented singular links using algebraic structures.
problem Understanding and distinguishing oriented singular links.
method Introduced algebraic structures to study oriented singular knots and links.
result Colorings of singular knots by new algebraic structures are invariants and distinguish some links.
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.
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 calculates Euler obstruction for determinantal varieties with ICIS singular set.
problem Calculating Euler obstruction for determinantal varieties with ICIS singular set.
method Obtained formulas to calculate Euler obstruction for determinantal varieties with ICIS singular set.
result Formulas to calculate the Euler obstruction for determinantal varieties with ICIS singular set.
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and b-tangent bundles approach to resolve singularities. result Established a Poincaré-Hopf theorem for bm-tangent bundles. Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
Defines generic singularities in planar flat 3-webs and classifies them.
problem Classifying singularities in planar flat 3-webs.
method Proposes a definition of genericity and classifies singularities.
result Classifies generic singularities of planar flat 3-webs.
Curvature criteria for A-simple singularities and their parallel curves identified.
problem Determining singularity types of A-simple singularities and their parallel curves.
method Defined curvature parameters and criteria for A-simple singularities.
result Criteria to determine singularity types of A-simple singularities and their parallel curves.
Algorithm classifies saddle-focus singularities in Hamiltonian systems.
problem Classifying nondegenerate saddle-focus singularities in integrable Hamiltonian systems.
method Developed an algorithm based on semi-local equivalence to represent singularities as almost direct products.
result Obtained complete lists of saddle-focus singularities of complexities 1, 2, and 3.
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.
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Study on estimating curvature and volume in singular Ricci flows.
problem Quantitative analysis of singular Ricci flows.
method Estimates on curvature and volume, set of singular times.
result Established several quantitative results about singular Ricci flows.
Uniqueness of conical flows helps understand singularities in surface flows.
problem Understanding singularities in surface flows.
method Analyzing asymptotically conical tangent flows.
result Uniqueness of multiplicity-one asymptotically conical tangent flows.
Study singularities of CMC 1 surfaces in de Sitter space.
problem Characterize singularities of spacelike CMC 1 surfaces in de Sitter space.
method Proved duality between singular points and introduced invariants.
result Classified non-degenerate singular points on spacelike CMC 1 surfaces.
Study on oriented disingquandles for distinguishing singular links.
problem Distinguishing singular links of different colors.
method Introduced oriented dichromatic singular links and oriented disingquandles to define invariants.
result Found invariants that can distinguish some singular links.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
Proves heat expansion for Laplacian on a singularity.
problem Analytic hypersurface with isolated singularity and Laplacian heat expansion.
method Local parametrization, Newton scheme, quasihomogeneous tangent cone, local models with irregular singularities.
result Existence of small time heat expansion for Laplace operator.
Differential forms and symmetric tensors show contrasting singular behaviors in a specific geometric setting.
problem Exploring differential forms and symmetric tensors on a specific geometric setting.
method Analyzing differential forms and symmetric tensors on the quadrant C2 with subset diffeology. result Symmetric tensors exhibit singularities that accumulate, while differential forms are smooth.
Paper identifies conditions for singularity removal in equations with conical singularities.
problem Removability of singularities in equations with conical singularities.
method Identification of the Pohozaev constant to measure failure of Pohozaev identity.
result Vanishing Pohozaev constant is necessary and sufficient for singularity removal.