New Poisson and near-symplectic structures found on 6D wrinkled fibrations.
problem Finding compatible geometric structures on 6D wrinkled fibrations.
method Extending wrinkled fibrations to 6D and proving the existence of rank-2 Poisson structures and near-symplectic structures.
result 6D wrinkled fibrations can be equipped with Poisson and near-symplectic structures.
Local formulas for Poisson structures on wrinkled fibrations are derived.
problem Understanding Poisson structures on specific geometric fibrations.
method Local formulæ for Poisson bivectors and symplectic forms on wrinkled fibrations.
result Local formulas for Poisson structures on wrinkled fibrations are derived.
We study simple wrinkled fibrations, a variation of the simplified purely wrinkled fibrations introduced by Williams, and their combinatorial description in terms of surface diagrams. We show that simple wrinkled fibrations induce handle decompositions on their total spaces which are very similar to those obtained from…
An R_2-move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations, flip and slip for example. In this paper, we first investigate how monodromies are chang…
Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow …
This paper generalizes wrinkling techniques to Haefliger structures, linking them to foliations.
problem Proving h-principles for partial differential relations with controlled singularities.
method Generalizing wrinkled embeddings to Haefliger structures and interpreting them as holonomic approximations.
result Haefliger structures provide a framework for making general wrinkling statements and imply connectivity results.
Wrinkles form on a thin sheet bonded to a sphere, revealing energy and length scale behaviors.
problem Understanding the wrinkle formation on a thin sheet bonded to a sphere.
method Analyzing the energy of the system with the sheet's thickness as a small parameter, determining leading and next-order behaviors.
result The wrinkling pattern varies with radius, with the number of wrinkles being approximately integer multiples of the sheet thickness.
Paper generalizes wrinkled embedding concept to jet spaces.
problem Approximating homotopies of embeddings with singularities.
method Defines wrinkled embeddings for differential information in jet spaces.
result Holonomic approximation theorem holds for multi-valued sections with simple singularities.
A {\it wrinkled embedding} f:Vn→Wm is a topological embedding which is a smooth embedding everywhere on V except a set of (n−1)-dimensional spheres, where f has cuspidal corners. In this paper we prove that any rotation of the tangent plane field TV⊂TW of a {\it smoothly embedded} submanifold $V\s…
We use the wrinkling theorem proven in Y. Eliashberg and N. Mishachev, "Wrinkling of smooth mappings and its applications - I", Invent. Math., 130(1997), 345-369, to fully describe the homotopy type of the space of S-immersions, i.e. equidimensional folded maps with prescribed folds.
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate tha…
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…
According to Kiyoshi Igusa a generalized Morse function on an n-dimensional manifold M is a smooth function with only Morse and birth-death singularities and a framed function is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of the function f. In …
We give a new proof of Markov's classical theorem relating any two closed braid representations of the same knot or link. The proof is based upon ideas in a forthcoming paper by the authors, "Stabilization in the braid groups". The new proof of the classical Markov theorem is used by Nancy Wrinkle in her forthcoming ma…
The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key…
Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.
problem Classify and understand nondegenerate fibrations of Euclidean spaces.
method Topological and geometric analysis of nondegenerate fibrations, including continuity at infinity.
result Prove that every germ of a nondegenerate fibration extends to a global fibration.
Introduces new stability concept for Fano fibrations.
problem Stability of Fano fibrations with singularities.
method Introduces f-stability and shows its implications. result Fibered semi log canonical singularities are restricted.
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.
Study of line fibrations in R^3, focusing on non-skew cases.
problem Classify line fibrations in R^3, especially non-skew ones.
method Develop parallel plane pushoff technique to study non-skew fibrations.
result Developed a technique to generate nonskew fibrations.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum G2 manifolds. Classifies Seifert fibrations of lens spaces.
problem Classifying Seifert fibrations of lens spaces.
method Algorithmic construction of Seifert fibrations over base orbifolds.
result All Seifert fibrations are equivalent to certain standard models.
Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.
problem Understanding achiral Lefschetz fibrations from non-orientable ones.
method Composition of standard orientation double covering map and non-orientable Lefschetz fibration.
result Specification of a monodromy factorization for the composition.
Shells resist three out of six possible loads if simply connected.
problem Understanding the load resistance of shells.
method Formal mathematical analysis of shell strains and deflections.
result The space of strains is three-dimensional for simply-connected shells.
Contact structures are induced by nondegenerate skew fibrations of R^3.
problem Understanding contact structures induced by skew fibrations of R^3.
method Characterizing nondegenerate fibrations and using a recent result on tightness in contact metric 3-manifolds.
result The plane field induced by any nondegenerate fibration is a tight contact structure.
Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…
Same genus-2 fibration structures for specific types found by different researchers.
problem Identifying equivalent genus-2 fibrations of type (4, 3).
method Comparing Lefschetz fibration structures of genus-2 fibrations of type (4, 3).
result Lefschetz fibration structures are the same for genus-2 fibrations of type (4, 3).
Constructs new coassociative fibrations for G2 manifolds.
problem Tackles the construction of new coassociative fibrations for G2 manifolds.
method Constructs fibrations by coassociative 4-folds, relates to hypersymplectic geometry and Donaldson's work.
result Shows natural generalizations of known coassociative fibrations.
Study fibrations over S2 with same singularities, showing monodromies are equivalent up to direct sums.
problem Classifying torus fibrations over S2 up to fibre sum stabilisation. method Analyzing monodromies and using direct sums with certain torus Lefschetz fibrations.
result Global monodromies of fibrations with same singularities are Hurwitz equivalent after direct sums.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.
problem Understanding the topology of Lefschetz fibrations.
method Analyzes upper bounds for the first Betti number, examines monodromy transitivity, and discusses potential fibrations.
result Upper bounds for the first Betti number and insights into Lefschetz fibration properties.
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …
The paper explores properties of shape mmsimpl-fibrators among Hopfian manifolds.
problem Characterizing shape mmsimpl-fibrators among Hopfian manifolds. method Investigation of shape mmsimpl-fibrators among direct products of Hopfian manifolds. result Direct products of Hopfian manifolds with coperfectly Hopfian fundamental groups are shape mmsimpl-fibrators. A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …
Examines properties of holomorphic fibrations in complex geometry.
problem Holomorphic fibrations in complex geometry.
method Analyzes properties of holomorphic fibrations.
result Provides insights into the structure of holomorphic fibrations.
The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.
problem Embedding 4-manifolds into CP^2 x CP^1.
method Proving Lefschetz fibration embeddings of 4-manifolds into CP^2 x CP^1.
result Every closed, connected, orientable 4-manifold admits a smooth Lefschetz fibration embedding into CP^2 x CP^1.
It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. Th…
Study how singular fibrations affect Poisson cohomology in 4D.
problem Understanding Poisson cohomology in singular fibrations.
method Analyzing the monodromy effect on Poisson cohomology groups.
result Describe the effect of monodromy on Poisson cohomology groups.
The paper extends trisection construction for Lefschetz fibrations with (−n)-sections.
problem Constructing trisections for 4-manifolds with specific Lefschetz fibrations.
method Explicit construction of trisections using monodromies of Lefschetz fibrations.
result Trisections for various 4-manifolds constructed from Lefschetz fibrations.
Study Lagrangian fibrations using Jacobians and Prym varieties.
problem Understanding Lagrangian fibrations in symplectic geometry.
method Survey of holomorphic symplectic varieties with Jacobians and Prym varieties as fibres.
result Characterization of Lagrangian fibrations using Jacobians and Prym varieties.
Proves existence of regular Lagrangians via Weinstein Lefschetz fibrations.
problem Existence of regular Lagrangians.
method Weinstein Lefschetz fibrations with a hypothesis.
result Existence of regular Lagrangians can be characterized by Lefschetz fibrations.
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
A fibration of Rn by oriented copies of Rp is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of R3 by skew oriente…
4D Poincaré complexes have simpler fibrations.
problem Complexity of fibrations in 4D Poincaré complexes.
method Showed Spivak normal fibration reducible.
result Reducible Spivak fibrations in 4D Poincaré complexes.
New proof of indecomposable minimal genus-2 Lefschetz fibration.
problem Existence of indecomposable minimal genus-2 Lefschetz fibration.
method Proved existence by counterexample to the converse of Usher's theorem.
result Existence of an indecomposable minimal genus-2 Lefschetz fibration.
In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify…