Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
Study on conditions for singular local tube fibrations.
problem Conditions for singular local tube fibrations.
method Analyzes the most general condition for singular local tube fibrations.
result Provides conditions for the existence of singular local tube fibrations.
Study on images and singularities of pseudoholomorphic maps.
problem Characterize images and singularities of pseudoholomorphic maps.
method Analyzes pseudoholomorphic maps in domains and targets of dimension four.
result Proves properties of images and singularities of pseudoholomorphic maps.
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.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…
Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. 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.
Paper constructs fold maps with useful singular value sets.
problem Creating fold maps with specific singular value sets.
method Surgery operations to construct fold maps with crossings.
result Fold maps with singular value sets containing crossings.
Defines renormalised energies for singular harmonic maps into compact manifolds.
problem Analyzing harmonic maps with singularities in planar domains.
method Introduces renormalised energies and synharmony to study singularities and minimising configurations.
result Renormalised energies are coercive and Lipschitz-continuous, and associated with minimising singular harmonic maps.
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.
The paper derives Gauss-Bonnet formulas for mappings between surfaces with boundary.
problem Calculating topological invariants for mappings between surfaces with boundaries.
method Defining singular points, constructing coherent tangent bundles, and applying Gauss-Bonnet formulas.
result Derives two Gauss-Bonnet type formulas for mappings between surfaces with boundaries.
Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
Maps with many singularities found in complex space.
problem Constructing maps with singularities in complex space.
method Created a map with infinitely many Schoen-Wolfson singularities on a disc.
result Found a Ck map with smooth trace in C2. The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.
problem Analyzing the structure of constraint maps with singularities and free boundaries.
method Establish continuity near singularities using a new quantitative unique continuation principle, and investigate the presence of branch points leading to new singularities.
result Topological singularities can only lie in the interior of the contact set in the uniformly convex setting, and the optimality of this result is proven.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
Study symmetry of cross-cap surfaces with folding maps.
problem Reflectional symmetry of cross-cap surfaces.
method Characterization of singularities in folding maps.
result Characterized generic singularities on cross-cap.
The paper studies harmonic maps to the circle with complex singular sets.
problem Finding harmonic maps with prescribed singular sets in higher-dimensional spaces.
method Considered variational relaxations of the problem, showing energy convergence to a renormalised volume plus lower-order interaction energy.
result The energy of minimisers converges, after renormalisation, to the volume of the singular set plus a lower-order interaction energy.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. Note on minimal maps' uniqueness via singular values.
problem Uniqueness of minimal maps into \(\mathbb{R}^n\).
method Using singular values and convexity of area functional, proving local linearity of singular value vectors.
result Improved uniqueness theorem for minimal graphs.
We obtain relations among the characteristic classes of a manifold M admitting corank one maps. Our relations yield strong restrictions on the cobordism class of M and also nonexistence results for singular maps of the projective spaces. We obtain our results through blowing up a manifold along the singular set of a sm…
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.
Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
We give criteria for Morin singularities for germs of maps into lower dimensions. As an application, we study the bifurcation of Lefschetz singularities.
Lips and swallow-tails are generic local moves of singularities of a smooth map to a 2-manifold. We prove that these moves of singularities of the product map of two functions on a 3-manifold can be realized by isotopies of the functions.
Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for n-harmonic maps. result Regularized n-conformal heat flow does not develop finite time singularities. Milnor fibrations have been studied since 1960's. In this paper, we study singular points of differentiable maps, called Milnor fibration product maps, obtained by several Milnor fibrations. We give a characterization of singular points of such product maps, and for the case of certain weighted homogeneous polynomials,…
By using Moser's iteration technique, we show some removable singularity theorem of the tension field for biharmonic maps into manifolds of non-positive curvature, and the bubbling theorem of biharmonic maps and also harmonic maps.
Wave maps (or Lorentzian-harmonic maps) from a 1+1-dimensional Lorentz space into the 2-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructin…
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property…
Study of harmonic maps with extreme Kerr-like singularities.
problem Analyzing harmonic maps with specific singularities.
method Asymptotic analysis of harmonic maps from 3D Euclidean space to hyperbolic plane.
result Existence and classification of tangent harmonic maps at extreme black hole horizons.
Study the singularities of Gauss map components of surfaces in 4D.
problem Characterize singularities of Gauss map components of surfaces in R4. method Analyze the geometric properties and stability of singularities.
result Singularities of Gauss map components are generically stable and related to surface geometry and J-holomorphic curves. We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…
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.
Simple criteria for codimension two surface singularities.
problem Identifying singularities in surfaces of codimension two.
method Provided criteria for singularities in surfaces of codimension less than or equal to two.
result Conditions for codimension two singularities in ruled surfaces and center maps.
A generic smooth map of a closed 2k-manifold into (3k−1)-space has a finite number of cusps (Σ1,1-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities (Σ1,0-singularities). Two fold maps are fold bordant if the…
A method to analyze maps into circles with singularities.
problem Analyzing maps with singularities into circles.
method Renormalization of Dirichlet Lagrangian for S1-harmonic maps. result Applications in Willmore energy and frame energies.
The paper finds a geometric explanation for coinciding Thom polynomials of cusp and corank-2 singularities.
problem Explaining the coincidence of Thom polynomials for cusp and corank-2 singularities.
method Analyzing geometrically the coincidence of Thom polynomials for Morin and corank-2 singularities.
result Found a geometric explanation for the coincidence of Thom polynomials for Morin and corank-2 singularities.
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
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.
Paper presents new cobordism sequences for singular maps.
problem Understanding cobordism groups of singular maps.
method Develops analogous exact sequences to classical ones.
result Positive answer to Szűcs' question on cobordisms of immersions.
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space Xτ giving homotopy representation of cobord…
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.