Study adiabatic limits of G_2-manifold fibrations.
problem Understanding the adiabatic limit of co-associative fibrations.
method Propose a maximal submanifold in a space of indefinite signature.
result Global versions of constructions set up for adiabatic limit.
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…
Paper proposes a new method to improve clustering ensemble performance.
problem Improving clustering ensemble performance by refining co-association matrix.
method Low-rank tensor approximation to derive coherent-link matrix and refine co-association matrix.
result The proposed method achieves breakthrough in clustering performance compared to state-of-the-art methods.
A method for semi-supervised regression under uncertainty combines graph Laplacian and cluster ensemble.
problem Semi-supervised regression with uncertain data structure and noise.
method Combines graph Laplacian regularization and cluster ensemble methodologies; uses low-rank decomposition of co-association matrix.
result Robust, efficient, and scalable method demonstrated through numerical experiments.
Study SO(3)-invariant G2-cobordisms on 6-manifolds.
problem Understanding stable 3-forms and G2-structures on 6-manifolds. method Investigate a bordism relation for closed SL(3;C)-structures via closed G2-structures under SO(3)-symmetry and co-associative conditions. result The bordism relation is irreflexive and induces a more rigid structure.
We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians intersection in the moduli space of special Lagrangian submanifolds in CY threefo…
Study introduces non-orientable 3-submanifolds in G2-manifolds.
problem Characterizing non-orientable three-submanifolds in G2-manifolds.
method Analogy with associative and co-associative cases, Cartan-Kähler theory.
result Some three-manifolds can be modeled on planes in a special G2-orbit.
Given a parallel calibration φ∈Ωp(M) on a Riemannian manifold M, I prove that the φ--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the φ--critical submanifolds are precisely the integral manifolds of a C∞(M)--linear subspace $\sP \subset Ω^p(M…
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.
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.
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.
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.
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.
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.
Proves finiteness of Lagrangian fibrations with specific invariants.
problem Finiteness of hyperkaehler Lagrangian fibrations with fixed invariants.
method Proves finiteness of fibrations with fixed Fujiki constant, discriminant, and line bundle degree.
result Finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension.
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…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riema…
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 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.
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.
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.
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.
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…
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.
Introduces Hurewicz fibrations for embedding maps of orbifold charts.
problem No specific problem stated; focuses on new concept definition.
method Defines E-fibration embedding and studies its properties.
result Introduces and studies properties of E-fibration embedding.
Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.
The paper studies curvature properties of Monge-Ampère fibrations and their existence.
problem Curvature properties and existence of Monge-Ampère fibrations.
method Analyzes a special relative Kähler fibration with a homogenous Monge-Ampère equation.
result Explicit curvature formulas and bounds for holomorphic curvatures of Monge-Ampère fibrations.
New connection found between Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
problem Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
method Established a compact analog of the P = W conjecture for holomorphic symplectic varieties with Lagrangian fibrations.
result Perverse numbers match Hodge numbers of the total space.
We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.