Study of Seifert fibered spaces using surface complexes.
problem Understanding the topology of Seifert fibered spaces.
method Defined surface complex for 3-manifolds, applied to Seifert fibered spaces.
result Surface complex of Seifert fibered spaces contains a subcomplex isomorphic to the curve complex of the base orbifold.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
Study subsurface projections in fibered 3-manifolds and their relation to veering triangulations.
problem Understanding the relationship between subsurface projections and veering triangulations in fibered 3-manifolds.
method Investigate the connections between subsurface projections in curve and arc complexes and Agol's veering triangulation.
result Large-distance subsurfaces in fibers correspond to large simplicial regions in veering triangulations.
Study complex surfaces fibered over Teichmüller curves with Veech fibers.
problem Understanding the geometric and topological properties of complex surfaces.
method Compute invariants via monodromy action on fiber's mod-m homology. result Exact values of invariants for all known algebraically primitive Teichmüller curves.
We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.
The paper studies fibers of maps in totally nonnegative spaces.
problem Understanding the structure of fibers of Chevalley exponentiation maps.
method Cell stratifications, face posets, and homeomorphic CW complexes.
result Fibers are homeomorphic to interior dual block complexes of subword complexes.
The paper studies translation lengths on sphere complexes and related cones.
problem Understanding the translation lengths of monodromies in fibered manifolds.
method Defined the generalized fibered cone and related cones, and proved their properties.
result Proved the generalized fibered cone is a rational slice of Fried's cone, providing bounds for asymptotic translation lengths.
Prism complexes help classify 3-manifolds, especially Seifert fiber spaces.
problem Classifying compact 3-manifolds using prism complexes.
method Introduced prism complexes and criteria for special prism structures.
result Compact 3-manifolds with special prism structures are Seifert fiber spaces.
The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.
problem Understanding the complexity of Heegaard splittings induced by fibered knots.
method Analyzing the monodromy of fibered knots and their impact on Heegaard splittings.
result Minimal genus Heegaard splittings of a three-manifold are unique and can be induced by fibered knots with complex monodromies.
Translation distances in fibered 3-manifolds with boundary are bounded and grow with complexity.
problem Bounding translation distances in fibered 3-manifolds with boundary.
method Using essential surfaces with non-zero slope and analyzing their complexity.
result Translation distances are bounded and grow with complexity, supporting a conjecture.
Generalized complex structures on certain torus bundles are explored.
problem Exploring generalized complex structures on specific torus bundles.
method Analyzing principal torus bundles over complex manifolds with even dimensional fibers and characteristic class of type (1,1).
result Generalized complex structures on these bundles are equivalent to products of complex and symplectic structures in tubular neighborhoods of fibers.
Researchers found a way to measure the complexity of Seifert fibered spaces with boundaries.
problem Measuring the complexity of Seifert fibered spaces with boundaries.
method Relating triangulation complexity to Seifert data and using barycentric subdivision.
result Determined triangulation complexity in terms of Seifert data and showed singular fibres can be made simplicial.
Neural networks reduce DBP complexity in fiber optics.
problem Complexity in digital backpropagation implementations.
method Neural-network-based approach to implement DBP.
result Learned DBP reduces complexity by 32x100 km fiber-optic link.
Upper bounds on translation lengths in fibered 3-manifolds.
problem Bounding translation lengths in fibered 3-manifolds.
method Analyzing asymptotic behavior of pseudo-Anosov monodromies in fibered cones.
result Upper bounds on asymptotic translation lengths derived from fibered faces.
Calculates characteristic classes of flat vector bundles on complex fibers.
problem Calculating characteristic classes of flat vector bundles on complex fibers.
method Constructs odd characteristic classes for proper flat fibrations and calculates the odd real characteristic classes of flat vector bundles on the base.
result Gives a Riemann-Roch-Grothendieck theorem for calculating odd real characteristic classes of flat vector bundles.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Study pseudo-Anosov monodromies in fibered 3-manifolds using asymptotic translation lengths.
problem Understanding the normal generation of pseudo-Anosov monodromies in fibered 3-manifolds.
method Using asymptotic translation lengths on the curve complex and analyzing properties of sequences of fibers and monodromies.
result For most primitive integral classes, pseudo-Anosov monodromies normally generate the mapping class group on the fiber surface.
Study on mod 2 Betti numbers of complex hyperplane arrangements and Milnor fiber homology.
problem Determining mod 2 Betti numbers of complex hyperplane arrangement complements.
method Combining the mod 2 Aomoto complex and the transfer long exact sequence.
result First homology of Milnor fiber has non-trivial 2-torsion for the icosidodecahedral arrangement.
New method determines arrangement combinatorics from Milnor fiber boundary.
problem Determining arrangement combinatorics from Milnor fiber boundary.
method Explicit method using plumbing graph in normal form.
result Milnor fiber boundary determines arrangement combinatorics.
Study on hyperelliptic Lefschetz fibrations, finding singular fiber counts.
problem Determining the minimal number of singular fibers in hyperelliptic Lefschetz fibrations.
method Analyzing complex surfaces and their Lefschetz fibrations over the 2-sphere.
result Minimal number of singular fibers is 2g+4 for even g≥4 and 2g+6 for odd g≥7.
New upper bound for geodesic complexity derived from cut locus decompositions.
problem Understanding geodesic complexity in Riemannian manifolds.
method Study of decompositions of cut loci and their tangent fibers.
result Established a new upper bound for geodesic complexity.
The study examines pseudo-Anosov maps on curve complexes and their asymptotic translation lengths.
problem Analyzing the behavior of pseudo-Anosov maps on curve complexes.
method Using sequences of fibers and monodromies in the fibered cone, the asymptotic translation length of pseudo-Anosov maps is studied.
result The asymptotic translation length of pseudo-Anosov maps on curve complexes behaves asymptotically like 1/∣χ(Rn)∣2. Takamura established a theory on splitting families of degenerations of complex curves. He introduced a powerful method for constructing a splitting family, called a barking family, in which there appear not only a singular fiber over the origin but also singular fibers over other points, called subordinate fibers. In …
Constructs examples of complex 3D shapes with specific properties.
problem Creating fibered three-manifolds with certain characteristics.
method Builds examples using handlebody bundles and polytopes.
result Examples of fibered three-manifolds with specific properties.
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…
Study on minimal surfaces in hyperbolic 3-manifolds, proving non-foliation and existence of many minimal surfaces.
problem Minimal surfaces in hyperbolic 3-manifolds and their foliation properties.
method Analyzing complex length of curves and using computer programs to find examples.
result Existence of hyperbolic 3-manifolds without closed incompressible minimal surfaces foliating the fiber.
We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…
We show that if the monodromy of a 3-manifold M that fibers over the circle has large translation distance in the curve complex, then the rank of the fundamental group of M is 2g+1, where g is the genus of the fiber.
A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibered over M/T, with fiber T. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety X⊂M o…
The paper studies complex curves with translation structures from differential equations.
problem Analyzing the structure of complex curves from differential equations.
method Defined isoresidual fibration and computed characteristics of complex curves.
result Determined Euler characteristic and classified connected components of isoresidual fibers.
In this paper, we first classify singular fibers of proper C∞ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with bo…
Hyperbolic spaces have many cells in their fibers.
problem Finding lower bounds on the topological complexity of fibered spaces
method Using a freedom theorem for ideals in group rings of hyperbolic groups
result For large injectivity radii, there are many cells of dimension k in the fiber p^{-1}(z)
LSTM neural networks improve fiber nonlinearities in coherent systems.
problem Compensating fiber nonlinearities in digital coherent systems.
method Utilization of Long short-term memory (LSTM) neural networks.
result LSTM neural networks provide superior performance compared to digital back propagation, especially in multi-channel scenarios.
Fiber bundles over hyperbolic manifolds with Anosov representations.
problem Understanding the topology of fiber bundles associated with Anosov representations.
method Analyzing the structure of fiber bundles over specific hyperbolic manifolds.
result Fiber bundles over certain hyperbolic manifolds are always smooth fiber bundles.
Essential dimension of hyperkähler manifolds families is at most 1.
problem Essential dimension of complex manifold families over compact bases.
method Proved essential dimension not greater than 1 for hyperkähler manifolds families over compact simply connected bases.
result Essential dimension of hyperkähler manifolds families is at most 1.
New multi-step approach improves fiber nonlinearity compensation efficiency.
problem Fewer steps are traditionally considered better for fiber nonlinearity compensation.
method Carefully designed multi-step machine learning approaches.
result Multi-step approaches lead to better performance-complexity trade-offs.
New operations transform braids into P-fibered braids.
problem Transforming braids into P-fibered braids.
method Satellite operations and full twists.
result Any braid can be transformed into a P-fibered braid.
We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are hyperbolic, free-by-c…
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…
We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away f…
New characteristic classes for fiber bundles via flat connections.
problem Constructing new characteristic classes for fiber bundles.
method Using flat connections with infinite-dimensional Lie algebras of derivations and fiberwise metrics.
result Induced map on cohomology groups is independent of choices and gives Morita-Mumford-Miller classes for surface bundles.
We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a ge…
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy ty…
Simple knots in lens spaces fiber if their order doesn't divide certain Euclidean remainders.
problem Characterizing fibered simple knots in lens spaces.
method Direct, combinatorial, and geometric methods.
result Conditions for fibered simple knots in lens spaces are determined.
Characterizes groups quasi-isometric to RAAGs with geometric actions.
problem Groups quasi-isometric to RAAGs with finite outer automorphism groups.
method Geometric actions on CAT(0) cube complexes with equivariant fibering.
result All such groups admit a geometric action on a CAT(0) cube complex.
Recognition of Seifert fibered spaces with boundary is computationally tractable.
problem Recognizing Seifert fibered spaces with boundary.
method Proving the existence of fundamental horizontal surfaces and normal vertical annuli with bounded total weight.
result The decision problem is in NP and the construction problem is in FNP.
The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surge…
Extends Higgs fields theory to complex fiber bundles.
problem Characterize nonlinear flat connections on complex fiber bundles.
method Representation of extension class by curvature, nonlinear Higgs bundles, and nonabelian Hodge structure.
result Established a faithful functor from nonlinear flat bundles to nonlinear Higgs bundles.