Paper proves periodic orbits for Hamiltonian systems in twisted disc bundles.
problem Existence of non-contractible periodic orbits in twisted disc bundles.
method Invariance of Floer homology under symplectic deformations and computation of Floer homology for cotangent bundles.
result Existence of periodic orbits for any nontrivial atoroidal free homotopy class.
We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…
Modified Engel structures allow complete h-principle for overtwisted discs.
problem Engel structures and their overtwisted discs.
method Engel twist modification and h-principle proof.
result Complete h-principle for overtwisted Engel structures.
Study detects if a circuit bounds a disc using curve intersections.
problem Determining if a circuit bounds an embedded disc.
method Analyzing the group generated by Dehn twists about curves in a circuit.
result Cycle relation between Dehn twists detects disc-boundability.
Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.
problem Analyzing actions of Dehn twists in geometric group theory.
method Application of Dynnikov coordinates to describe orbits and dynamics of Dehn twists in a thrice-punctured disc.
result The action of Dehn twists has a geometric meaning as a piecewise linear Z2-automorphism. The paper solves conjectures related to strong openness and optimal L2 extension.
problem Strong openness of multiplier ideal sheaves and optimal L2 extension. method Solution of conjectures related to Demailly's strong openness and related conjectures.
result Optimal L2 extension implies Berndtsson's positivity of vector bundles. Study on holomorphic discs in bundles over compact surfaces, proving Fredholm regularity under certain conditions.
problem Analyzing holomorphic discs with boundary on surfaces in vector bundles over compact manifolds.
method Proves Fredholm regularity for sections with a single complex point under specific conditions.
result Holomorphic discs are Fredholm regular under certain conditions, including neutral Kähler and symplectic actions.
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in PU(2,1) and using bending-connectedness. result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
Study of Dehn twists on a disc with 3 points, solving conjugacy problem.
problem Solving conjugacy problem for Dehn twists on a disc with 3 marked points.
method Explicit description of orbits of Dehn twists on the Dynnikov plane, relating to homology dynamics.
result Explicit solution to conjugacy problem for Dehn twists, presenting an untwisting algorithm.
Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
Homological stability proved for handlebody mapping class groups.
problem Homological stability for handlebody mapping class groups.
method Categorical framework developed by Randal-Williams and Wahl, allowing for any number of marked discs and boundary points.
result Homology of handlebody groups stabilizes with respect to genus and number of marked discs for all finite degree coefficient systems.
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…
Classifies 5D stationary black hole domains using plumbing and disc bundles.
problem Classifying the topology of 5D stationary black hole domains.
method Using plumbing constructions and disc bundles over the 2-sphere.
result The domain of outer communication is homeomorphic to S4 or a connected sum of S2imesS2's or CP2. Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δg found using Baumkuchen lemma. Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function ρW using symplectic embeddings and recovers the metric when W is the unit disc-cotangent bundle. result The distance function ρW recovers the Riemannian metric when W is the unit disc-cotangent bundle. Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles M→S should satisfy the inequality ∣eM/χS∣⩽1, where eM stands for the E…
The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a fa…
New examples of Schoenflies balls are produced using a 5D approach.
problem Identifying Schoenflies balls that are not standard.
method Using a 5-dimensional perspective, algebraic and geometric handle cancellation.
result New examples of Schoenflies balls not known to be standard are produced.
The paper extends the uniqueness of complete harmonic metrics to subharmonic weights and proves their existence on the unit disc.
problem Finding complete harmonic metrics on Riemann surfaces for subharmonic weights.
method Extending Li-Mochizuki's theorem to subharmonic weights and proving existence on the unit disc.
result Complete harmonic metrics exist on the unit disc for subharmonic weights.
The paper defines and calculates stabilization distances between surfaces in 4-manifolds.
problem Calculating the minimal number of 1-handle stabilizations for surfaces to become isotopic.
method Using homology of cyclic covers and metabelian twisted homology.
result For every nonnegative integer m, there exist pairs of surfaces with a specific stabilization distance.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Johnson kernel generated by specific Dehn twists on surfaces.
problem Identifying the generating set of Johnson kernel for surfaces.
method Analyzing Dehn twists on specific types of surfaces.
result Johnson kernel generated by Dehn twists on specified surfaces.
Field theories help describe twisted bundles on orbifolds.
problem Understanding twisted cohomology and vector bundles on orbifolds.
method Construction of twist functor for 1|1-dimensional EFTs from gerbe data.
result Construction of supersymmetric field theories from twisted vector bundles.
We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface Σ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …
We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.
We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.
This paper gives a construction for all minimal immersions f of the Poincaré disc into the complex hyperbolic plane CH2 which are equivariant with respect to an irreducible representation ρ of a hyperbolic surface group into PU(2,1). We exploit the fact that each such immersion is a twisted conformal …
The Kobayashi-Hitchin correspondence is proven for twisted vector bundles on Kähler manifolds.
problem Proving the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles.
method Proved the correspondence and approximate correspondence for twisted holomorphic vector bundles on compact Kähler manifolds.
result A twisted holomorphic vector bundle is g−polystable if and only if it is g−Hermite-Einstein, and g−semistable if and only if it is approximate g−Hermite-Einstein. Given a closed symplectic 4-manifold (X,ω), we define a twisted version of the Gromov-Taubes invariants for (X,ω), where the twisting coefficients are induced by the choice of a surface bundle over X. Given a fibered 3-manifold Y, we similarly construct twisted Lefschetz zeta functions associated with surface b…
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Researchers create a new metric on complex projective space bundles.
problem Constructing a hyperkähler metric on complex projective space bundles.
method Explicit construction in local coordinates, using holomorphic isomorphism to coadjoint orbits.
result A hyperkähler metric on twisted cotangent bundles of CPn. We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted K-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham coho…
We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twi…
This paper studies Jones-Wenzl idempotents in the twisted I-bundle over the Möbius band.
problem Understanding Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band.
method Analyzing the trace of Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band.
result Uncovering analog properties of Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band that differ from the annulus case.
We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle (E,H0) over a Riemann surface X. It is already known the gradient flow with initial data (A0,φ0) converges to a critical point (A∞,φ∞) of this functional. Using a modified Chern-Wei…
New 2D complex hyperbolic structures found on sphere orbibundles.
problem Locally rigid complex hyperbolic structures on sphere orbibundles.
method Constructing families of complex hyperbolic structures on disc orbibundles.
result Examples of non-locally rigid complex hyperbolic structures.
New complex structure on hyperbolic disc within hyperkaehler space.
problem Complex structure on hyperbolic disc within hyperkaehler space.
method Mostow decomposition and complex adjoint orbit analysis.
result Complex structure on hyperbolic disc differs from natural embedding.
Study fractional structures on bundle gerbe modules using rational homotopy theory.
problem Understanding twisted Chern classes of torsion bundle gerbe modules.
method Sullivan's rational homotopy theory to realize twisted Chern classes at the level of classifying spaces.
result Introduction of fractional U-structures as a universal framework.
New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
The paper proves bounded cohomology properties of Euclidean space groups.
problem Understanding bounded cohomology of transformation groups of Euclidean spaces and discs.
method Analyzes groups of homeomorphisms and diffeomorphisms of Euclidean spaces and discs, proving boundedly acyclic properties.
result Groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, with implications for characteristic classes of bundles. Researchers calculate Hofer-Zehnder capacity for twisted tangent bundles over surfaces.
problem Determining the Hofer-Zehnder capacity for specific geometric configurations.
method Analyzing constant magnetic fields on closed surfaces and using equivariant compactification.
result Explicit calculations and compactifications for phase and configuration spaces.
Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one con…
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
New method for moduli spaces of twisted quiver representations and Higgs bundles.
problem Computing moduli spaces of twisted quiver representations and Higgs bundles.
method Extending twisted A-type quiver representations to any genus using Hitchin stability and deformation theory.
result Explicit geometric identifications of moduli spaces of twisted representations of argyle quivers on P1. Let X be a smooth projective variety over C. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on X admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach …
Paper untwists gauge fields, linking two bundles.
problem Dealing with two symmetry groups in gauge theories.
method Isomorphism between twisted and standard gauge fields on different bundles.
result Local twisted fields can be used in standard gauge theories.