Geometric study of cuspidal S1 singularities using diffeomorphisms and isometries.
problem Understanding geometric properties of cuspidal S1 singularities. method Form representing deformation using diffeomorphisms and isometries, necessary and sufficient condition for frontal maps.
result Investigation of geometric properties and cuspidal cross caps in deformations.
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
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.
We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a 2n-dimensional manifold. In the C-analytic category this set consists of the Martinet hypersurface Σ2, the restriction of the singular symplectic form ω to TΣ2 and the kern…
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
Novel singularity models for 4D harmonic forms and spinors from polytopes.
problem Understanding harmonic forms and spinors in 4D.
method Homogeneous singularity models based on regular 4-polytopes.
result Models describe cones on the 1-skeletal of polytopes.
We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential compo…
A generalised Thurston-Bennequin invariant for a Q-singularity of a real algebraic variety is defined as a linking form on the homologies of the real link of the singularity. The main goal of this paper is to present a method to calculate the linking form in terms of the very good resolution graph of a real normal unib…
Formal Normal Form created for special CR singularities.
problem CR singularities in complex manifolds.
method Formal Normal Form construction for real-smooth submanifolds.
result A new Formal Normal Form established.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
New form of D4−-singularities for fronts in 3D space.
problem Understanding singularities of fronts in 3D space.
method Coordinate transformation on source and isometry on target.
result Computed differential geometric invariants near D4−-singularity. New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
The paper studies metrics on vector bundles with singularities and their associated forms.
problem Analyzing singular Hermitian metrics on vector bundles and their associated forms.
method Defines and analyzes the Segre and Chern forms of singular metrics, proving properties of their Lelong numbers.
result Lelong numbers of the associated forms are integers if singularities are integral.
We call a singularity of a presymplectic form ω removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinor…
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
We simplify D4+-front singularities and apply to geometric invariants.
problem Simplifying D4+-front singularities in R3. method Coordinate transformation on source and isometry on target.
result Gauss-Bonnet type theorem for fronts with D4+-singularity. Proves Riemannian positive mass theorem with singularities.
problem Proves Riemannian positive mass theorem for specific types of singular manifolds.
method Uses initial data sets with a second fundamental form to transfer convexity between different singularity components.
result Proves the theorem for manifolds with some mean-concave components and others mean-convex.
Virtual singular braids embed in a group with normal form.
problem Embedding virtual singular braids into algebraic structures.
method Presented a semi-direct product structure and provided a normal form.
result Virtual singular braid group VSGn is a semi-direct product of VSPGn and Sn. 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.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
Study geometric properties of S1 singularities and their deformations.
problem Understanding differential geometric properties of S1 singularities and deformations.
method Representing deformation using diffeomorphisms and isometries, studying geometric properties.
result Differential geometric properties of S1 singularities and Whitney umbrellas in deformations.
Study of singularities in mean curvature flow with focus on S3imesR.
problem Understanding singularities in mean curvature flow.
method Detailed analysis of singularities modeled on S3imesR, using normal form transformations and rescaled MCF. result Proves mean convexity and singularity isolation in a small neighborhood, conjectures singularity formation in entire neighborhood.
It is constructed a normal form for a class of real-smooth surfaces M\subset\mathbb{C}^{2} defined near a degenerate CR singularity.
A differential 1-form α on a manifold of odd dimension 2n+1, which satisfies the contact condition α∧(dα)n=0 almost everywhere, but which vanishes at a point O, i.e. α(O)=0, is called a \textit{singular contact form} at O. The aim of this paper is to study local normal forms (formal, analytic …
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. 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.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
Study Z2 harmonic functions with singularities on flat space.
problem Understanding Z2 harmonic functions with point singularities.
method Analyzes Z2 harmonic functions on R2 with point singularities. result Characterizes Z2 harmonic functions on R2 with point singularities. Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
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.
We construct a formal normal form for a real 2-codimensional submanifold M⊂CN+1 near a CR singularity approximating the sphere. This result gives a higher dimensional extension of Huang-Yin's normal form in C2.
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.
The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.
problem Finding normal forms near critical points of sub-Riemannian exponential maps.
method Singularity theory applied to sub-Riemannian structures.
result Normal forms for sub-Riemannian exponential maps in specific cases.
Study of singularities in two-dimensional Nijenhuis operators with non-zero trace differential.
problem Characterizing singularities of two-dimensional Nijenhuis operators.
method Analyzing the smoothness of functions related to the determinant of the operator.
result Complete description of singularities for certain function classes.
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. 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.
The paper connects orbifold singularities to higher symmetries in SQFTs.
problem Understanding higher symmetries in supersymmetric quantum field theories.
method Cutting and gluing of orbifold singularities to determine symmetries.
result Local orbifold singularities encode 0-form, 1-form, and 2-group symmetries.
Study magnetic fields on special Lie groups, proving non-existence of certain types.
problem Existence of closed 2-forms with specific properties on non-singular 2-step nilpotent Lie groups.
method Analyzing left-invariant magnetic fields on 2-step nilpotent Lie groups, proving non-existence and existence results.
result Strong obstruction and non-existence of closed 2-forms of type II on non-singular Lie algebras.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable 1-forms on $R^3
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…
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.
Reduces symplectic manifolds with singularities for quantum reduction.
problem Quantization commutes with reduction for singular symplectic manifolds.
method Reduction theory for bm-symplectic manifolds and folded symplectic manifolds under general symmetries. result New constructions of (singular) quasi-Hamiltonian spaces via reduction and fusion product.
Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is co…
New forms generalize Whitney forms with rational coefficients for numerical analysis.
problem Numerical problems with singularities near simplex faces.
method Introduce shadow forms and degrees of freedom for integration over faces of blow-up simplices.
result Obtain isomorphism between shadow forms cohomology and cellular cohomology of blow-up simplices.
We study singular stochastic control of a two dimensional stochastic differential equation, where the first component is linear with random and unbounded coefficients. We derive existence of an optimal relaxed control and necessary conditions for optimality in the form of a mixed relaxed-singular maximum principle in a…