Study connects Morin singularities to sphere homotopy groups.
problem Computing stable homotopy groups of spheres.
method Apply Morin singularities to sphere homotopy groups.
result Differentials in spectral sequence linked to sphere homotopy groups.
Study connects Morin singularities to sphere homotopy groups.
problem Understanding stable homotopy groups of spheres.
method Establishes a connection between Morin singularities and sphere homotopy groups.
result Describes behavior of singularity strata images around more complex strata.
Recent work on stable minimal hypersurface singularities.
problem Understanding singularities of stable minimal hypersurfaces.
method Simplifications of technical discussion in previous work.
result Simplified approach to analyzing hypersurface singularities.
Classifies singular fibers of stable maps and computes associated cohomology groups.
problem Understanding singular fibers of stable maps between 3-manifolds with boundary and surfaces.
method Classification of singular fibers, computation of cohomology groups, cobordism invariants.
result Obtained cobordism invariants for Morse functions on compact surfaces with boundary.
Classifies normal stable Horikawa surfaces with smoothable singularities.
problem Characterizing surfaces with specific singularities and smoothability criteria.
method Classification and smoothability criterion based on log canonical singularities.
result Provides a criterion for global Q-Gorenstein smoothability of Horikawa surfaces. Constructs special Lagrangians with stable singularities.
problem Creating compact special Lagrangians with stable singularities.
method Constructs families of compact almost Calabi-Yau manifolds and special Lagrangians.
result Models stable T^2-cones as compact special Lagrangians.
Study on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.
problem Understanding the behavior of stable maps from S3 to R3. method Analysis of codimension-one transitions, singular set behavior, and global invariants.
result Effects of decompositions on global invariants with prescribed branch sets.
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
Sharp bounds on singularities of stable minimal hypersurfaces.
problem Understanding the singularities of stable minimal hypersurfaces.
method Generalized Schoen inequality and branched sheeting theorem.
result Sharp bounds on Hausdorff dimension of singular sets.
Stable harmonic maps into certain Lie groups have singularities with specific codimensions.
problem Understanding the singularities of harmonic maps into compact Lie groups.
method Analyzing stable stationary harmonic maps and their singular sets using Hausdorff codimension.
result The singular set of stable stationary harmonic maps into certain Lie groups has a Hausdorff codimension of at least four.
The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.
problem The behavior of singular minimal hypersurfaces in dimensions 7 and above.
method Analyzes the local behavior of minimal hypersurfaces under perturbations and convergence of families of hypersurfaces.
result Existence and smoothness of nearby minimal hypersurfaces under perturbations, uniqueness of homological minimization, and existence of Jacobi fields.
Characterizes stable singularities of affine equidistants on surfaces in 4-space.
problem Understanding the singularities of affine equidistants on surfaces in 4-dimensional space.
method Characterizes stable singularities of λ-equidistants in terms of bi-local extrinsic geometry. result Characterizes the stable singularities of λ-equidistants as Ak,C2,2±. The paper constructs stable Higgs bundles for hyperbolic metrics with singularities.
problem Existence of conformal hyperbolic metrics with prescribed singularities.
method Stable parabolic Higgs bundles of rank two.
result Alternative proof of Heins' theorem and extension of Hitchin's work.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
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.
Study isotopy of Morin singularities, strengthening A-equivalence.
problem Classify Morin singularities under A-isotopy.
method Define A-isotopy, analyze A-isotopy classes of Morin singularities.
result Determine the number of A-isotopy classes of Morin singularities.
Optimal regularity theory for stable minimal hypersurfaces with small singular set.
problem Optimal regularity of stable minimal hypersurfaces with small singular set.
method Analysis of stable minimal hypersurfaces in a specific domain with small singular set.
result Optimal size assumption on the non-immersed singular set guarantees optimal regularity.
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
The paper shows how stable minimal spheres emerge in certain 3D spaces under Ricci flow.
problem Construction of spherical space forms with no stable minimal surfaces.
method Ricci flow on spherical space forms with positive scalar curvature.
result Stable minimal spheres appear in spherical space forms during Ricci flow.
The paper solves conditions for non-singular extensions of fold maps.
problem Conditions for the existence of non-singular extensions of horizontal stable fold maps.
method Defined a combinatorial object called a pairing map to prove equivalence of existence.
result Existence of non-singular extensions is equivalent to the existence of a pairing map.
Characterizes hyperbolic links with stable maps to the plane.
problem Understanding hyperbolic links through stable maps.
method Characterization of hyperbolic links via stable maps to the plane.
result Complete characterization of hyperbolic links with specific stable maps.
The paper constructs stable minimal hypersurfaces with specific singularities.
problem Creating minimal hypersurfaces with controlled singularities.
method Constructing hypersurfaces with a given singular set in a modified Euclidean space.
result Embedded minimal hypersurfaces with stable properties and specified singularities.
The paper studies stable surfaces in sub-Riemannian 3-space forms.
problem Finding criteria for strong stability of CMC surfaces in sub-Riemannian 3-manifolds.
method Analyzing functional minimization and studying specific examples of surfaces.
result Examples of complete strongly stable non-vertical surfaces in sub-Riemannian hyperbolic 3-space.
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
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.
Characterizes stable sheaves for equality in orbifold BG inequality.
problem Stability of sheaves on compact Kähler varieties with klt singularities.
method Characterization of stable reflexive sheaves for BG equality.
result Characterizes stable reflexive sheaves for equality in BG inequality.
We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.
New potential theory on minimal hypersurfaces shows stable growth of solutions near singularities.
problem Analyzing potential theory on minimal hypersurfaces.
method Introducing Hardy structures to study classical operators and showing stable growth of solutions.
result Minimal growth of positive solutions of Lw = 0 is stable and persists under perturbations or blow-ups.
Study on stability of cylindrical singularities in MCF of finite codimensions.
problem Stability of cylindrical singularities in mean curvature flow.
method Construction of stable manifold, explicit solutions, asymptotic analysis.
result Asymptotic stability of cylindrical singularities under generic perturbations.
Research describes all possible gradient vector fields on a sphere with up to ten singular points.
problem Characterizing gradient vector fields on a sphere with limited singular points.
method Using a graph to represent one-dimensional stable manifolds, specifying singularities and connections.
result Identified all topological structures of codimension one gradient vector fields on a sphere with up to ten singular points.
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…
Notes on Khovanov and knot Floer theories' stable homotopy types.
problem Understanding stable homotopy types in Khovanov and knot Floer theories.
method Introduction to Khovanov and knot Floer theories' stable homotopy types.
result Introduction of stable homotopy types in Khovanov and knot Floer theories.
Proves CM line bundle positivity for K-stable klt Fano varieties.
problem Proving ampleness of the CM line bundle for K-stable klt Fano varieties.
method Algebraic approach, including probability theory for limit computations.
result Proves semi-positivity and positivity statements for K-semi-stable and uniform K-stable cases.
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.
We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
New solutions found for G2 system using K3 orbifolds.
problem Finding smooth solutions to the G2 Hull-Strominger system. method Torus fibrations over K3 orbifolds, adapted Serre construction for singular settings.
result Constructed new smooth solutions to the G2 Hull-Strominger system. Characterizes W-congruences to study their stable umbilical points.
problem Understanding singularities of W-congruences.
method Characterization of W-congruences to study umbilical points.
result Examples of isolated stable umbilical points of type Am. The paper studies singularities in a complex flow related to mean curvature.
problem Investigating singularities in a complex flow related to mean curvature.
method Constructing two distinct examples of singularities using the line bundle mean curvature flow.
result Found a finite time singularity, ruling out long time existence of the flow.
Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.
problem Understanding stationary measures on hyperbolic surfaces with cusps.
method Analyzing exponential decay of cusp excursions and proving quasi-symmetry stability.
result Stationary measures on hyperbolic surfaces with cusps are quasi-symmetrically stable and singular.
Constructs minimal immersions with singularities.
problem Minimal immersions with singularities in metric spaces.
method Constructs minimal immersions with catenoidal necks or floating disks converging to a singular point.
result Constructs minimal immersions with singularities.
Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine λ-equidistants of n-dimensional closed submanifolds of Rq, for q≤2n, whenever (2n,q) is a pair of nice dimensions.
Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of …
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries.