Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than 2 into the 2--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algori…
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.
Smooth maps show Gromoll filtration for spheres.
problem Understanding Gromoll filtration for special generic maps.
method Analyzing smooth maps with definite folds and Gromoll filtration.
result Gromoll filtration equals p for special generic maps.
Visual construction of maps linking to two-bridge links.
problem Understanding stable maps and their properties for two-bridge links.
method Visual construction and analysis of stable maps from 3-sphere to plane.
result Determination of stable map complexities for some two-bridge link exteriors.
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.
problem Classifying Thurston geometries and understanding their properties.
method Introduces an axiomatic definition for the Kodaira dimension and studies its compatibility with traditional notions.
result Establishes a connection between the simplicial volume and the holomorphic Kodaira dimension, showing implications for smooth Kähler 3-folds.
In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.
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.
A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.
Unified definition of hallucinations in language models.
problem Persistent hallucinations despite mitigation efforts.
method Unified definition of hallucination as inaccurate world modeling.
result Unified framework distinguishes hallucinations from other errors.
This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularit…
A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers p for which a given homotopy sphere admits a special generic map into Rp. By means of t…
The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.
problem Whether rational homology n-spheres admit special generic maps into Rp for p<n. method Stein factorization technique to derive a necessary homological condition.
result New results on the (non-)existence of special generic maps for specific rational homology spheres.
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions …
Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's di…
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…
In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fo…
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative…
The paper proves the existence of a folded annulus with multiple creases.
problem Existence of a folded annulus with multiple creases.
method Analyze developable surfaces, use normal curvature and relative torsion, compute geometric descriptors, prove propagation of folds.
result Proves the existence of a folded annulus with multiple creases.
First steps towards a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of Hilb^2(K3). We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hype…
Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.
problem Estimating the ribbonlength of different types of knots.
method Using Kauffman's model of folded ribbon knots, we derive upper bounds on ribbonlength for specific knot types.
result Upper bounds on ribbonlength are linear in crossing number for some knots and sub-linear for others.
Enhanced coloring invariant distinguishes folded molecular chain topologies.
problem Apparent indistinguishability of folded chain topologies using current coloring invariants.
method Introduced Boltzmann weights to improve the resolving power of quandle colorings.
result Improved resolution in distinguishing folded chain topologies.
Origami can create complex knots, with minimum creases defining a new knot invariant.
problem Creating complex knots using origami folds.
method Developed a new knot invariant called the fold number, defined as the minimum number of creases required to obtain an equivalent knot.
result No proper foldings can produce nontrivial knots, but improper foldings can.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuou…
A smooth map having only fold singularities is called a fold-map. We will give effective conditions for a continuous map to be homotopic to a fold-map from the viewpoint of the homotopy principle.
Paper classifies pillow box isometric deformations preserving crease patterns.
problem Classifying pillow box isometric deformations preserving crease patterns.
method Continuous isometric deformations from pillow boxes to double rectangles, preserving crease patterns.
result Such deformations necessarily change pillow box topology.
This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.
problem Understanding the kinematics and nonlinear phenomena of Spidron units.
method Analysis of single unit cell kinematics and recursive construction of multiple cells.
result Non-periodic folding restricts isotropic folding as the number of unit cells increases.
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…
The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for (2,q)-torus links, twist knots, and pretzel links. Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotic…
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.
New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lag…
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in th…
Sharp lower bound on fold singularities self-intersections.
problem Finding a lower bound on the number of self-intersections of fold singularities.
method Established a sharp lower bound on the number of self-intersections of the boundary of an immersed surface, then applied this to fold singularities.
result Sharp lower bound on the number of self-intersections of fold singularities.
Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …
It is known that for coprime integers p>q≥1, the lens space L(p2,pq−1) bounds a rational ball, Bp,q, arising as the 2-fold branched cover of a (smooth) slice disk in B4 bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each Bp,q. Whereas, Yamada gives an …
We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were in…
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.