Study symplectic and Hamiltonian actions on irrational ruled surfaces, proving existence and non-existence of extensions.
problem Tackles the extension of symplectic and Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic 4-manifolds.
method Constructs symplectic involutions and cyclic actions, classifies symplectic morphisms, and proves non-extendability of certain actions.
result Shows existence and non-existence of Hamiltonian circle actions for different cyclic actions on irrational ruled symplectic 4-manifolds.
Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.
problem Understanding symplectic embeddings of balls into complex projective spaces, tori, and K3 surfaces.
method Analyzing embeddings with respect to complex structures compatible with the symplectic form and identifying obstructions.
result Symplectic volume is the primary obstruction for the existence of embeddings of balls into certain manifolds.
The paper solves isotopy problems on 4-manifolds and classifies symplectic structures.
problem Isotopy problems on 4-manifolds and uniqueness of symplectic structures.
method Generalization of Dax invariant to embedded closed surfaces, symplectic topology techniques.
result Infinitely many non-isotopic symplectic forms on irrational ruled surfaces.
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our…
In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular…
Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.
problem Existence of elliptic Reeb orbits on real projective 3-space.
method Use of ECH (Embedded Contact Homology) to find distinguished pseudoholomorphic curves.
result Existence of elliptic Reeb orbit proven for some contact forms on RP3. Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.
problem Understanding Kodaira dimensions on almost complex manifolds.
method Using pseudoholomorphic pluricanonical maps, defining new dimensions, and applying probabilistic combinatorics.
result Almost complex structures with top Kodaira dimension are integrable, and for compact 4-manifolds, they have elliptic fibration structures.
Study tiling spaces over irrational tori using diffeological classification.
problem Understanding the structure of tiling spaces over irrational tori.
method Diffeological classification of irrational tori and analysis of fiber bundle structures.
result Inherited diffeological equivalence of one-dimensional tiling spaces over irrational tori.
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
Following a Geometrical Brownian Motion extension into an Irrational Fractional Brownian Motion model, we re-examine agent behaviour reacting to time dependent news on the log-returns thereby modifying a financial market evolution. We specifically discuss the role of financial news or economic information positive or n…
Study irrational pencils on complex manifolds, finding non-finitely generated homology.
problem Understanding the homology of the kernel induced by irrational pencils on complex manifolds.
method Analyzing critical points and homology of fundamental groups of complex manifolds.
result Homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated.
We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z…
Study irrational rotations and construct 2-filling rays on infinite type surfaces.
problem Understanding dense orbits in skew product transformations.
method Skew product transformation with continued fraction analysis.
result Existence of infinite cliques of 2-filling rays on infinite type surfaces.
Zagier's conjecture on knot invariants is proven for irrationals, with applications to quantum modular forms.
problem Proving the continuity of a function related to the figure-eight knot's colored Jones polynomial at irrationals.
method Analyzing the asymptotic behavior of the colored Jones polynomial and using properties of continued fractions.
result The continuity conjecture for the function h(x) holds almost everywhere on the real line, and a smooth approximation is established. Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.
problem Existence and properties of Reeb orbits on three-sphere with torus knots.
method Knot filtered embedded contact homology and obstructions to exact symplectic cobordisms.
result Upper bound on mean action of periodic orbits and linking bounds for Reeb orbits.
We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.
We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable norm in some natural examples.
We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse …
Study infinite symplectic forms on ruled surfaces.
problem Uniqueness of symplectic structures on four-manifolds.
method Provided an infinite family of non-isotopic symplectic forms.
result Symplectic forms on ruled surfaces are pairwise non-isotopic.
The paper explores spaces of Kähler and symplectic forms on 4-manifolds.
problem Investigating the properties of Kähler and symplectic forms on 4-manifolds.
method Analyzing the uniqueness, connectedness, and openness of spaces of Kähler forms and introducing holomorphically tamed symplectic forms.
result Formulated a parallel question for holomorphically tamed symplectic forms and related it to Kähler-type symplectic forms.
Symplectic forms match on circle pattern space.
problem Matching symplectic forms on circle pattern space.
method Pullback of symplectic forms to circle pattern space.
result Symplectic forms on circle pattern space coincide.
A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be C0-approximated by PL symplectic manifolds. Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
New coefficient detects irrational rotation behavior on infinite-type surfaces.
problem Detecting irrational rotation behavior on surfaces of infinite type.
method Introducing a new quasimorphism, the Dehn twist coefficient, and proving its properties.
result The Dehn twist coefficient can have image all of R for some infinite-type surfaces.
In this paper, we prove that for every Finsler n-dimensional sphere (Sn,F) with reversibility $\lm$ and flag curvature K satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincaré …
Characterizes density-valued symplectic forms on multisymplectic manifolds.
problem Understanding density-valued symplectic forms on multisymplectic manifolds.
method Intrinsic characterization and Darboux-type theorems.
result Proves Darboux-type theorems for density-valued symplectic forms.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.
problem Characterizing Lagrangian submanifolds in adjoint semisimple orbits.
method Analyzing real flags and orbits of real forms with respect to symplectic forms.
result Classification of infinitesimally tight Lagrangian submanifolds in the compact case and Lagrangian submanifolds in the complex case.
The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions n=4r−2. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…
The paper classifies symplectic forms on R^4 and determines invariants under symplectomorphisms.
problem Classifying symplectic forms on R^4 under symplectomorphisms.
method Using pfaffian and sum function invariants, the paper provides a complete description of orbit spaces and determines global invariants.
result The paper provides a complete classification of symplectic forms on R^4 under symplectomorphisms, providing necessary conditions for intertwining.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectom…
Method resolves 4D symplectic orbifolds using complex geometry.
problem Resolving symplectic orbifolds in 4 dimensions.
method Combining complex geometry techniques with symplectic form gluing.
result Examples of 4D symplectic orbifolds successfully resolved.
In this paper, we prove that on every Finsler n-sphere (Sn,F) for n≥6 with reversibility λ and flag curvature K satisfying (λ+1λ)2<K≤1, either there exist infinitely many prime closed geodesics or there exist [2n]−2 closed geodesics possessing irrational average indices. If in add…
The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.
problem Constructing symplectic forms on 4-manifolds with rational symplectic forms.
method Using branched coverings and holomorphic line bundles, the paper constructs symplectic forms that are Kähler in a neighborhood of the 2-skeleton of the manifold.
result The paper proves the existence of a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of the manifold.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
Symplectic forms from two phase spaces are proven equivalent.
problem Equivalence of symplectic forms from different phase spaces.
method Proof of equivalence for theories over space-time with boundary.
result Symplectic forms derived from canonical and covariant phase spaces are equivalent.
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…
In this paper, we prove that for every Finsler n-sphere (Sn,F) for n≥3 with reversibility λ and flag curvature K satisfying (λ+1λ)2<K≤1, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigen…
A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural r…
We study the geometry of the Margulis region associated with an irrational screw translation g acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of g on its boundary which plays the role of an invariant horoball for a translation in dimensions ≤3. Th…
We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a 2n-dimensional manifold. In the C-analytic category this set consists of the Martinet hypersurface Σ2, the restriction of the singular symplectic form ω to TΣ2 and the kern…
We give a method to lift (2,0)-tensors fields on a manifold M to build symplectic forms on TM. Conversely, we show that any symplectic form $\Om$ on TM is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three (2,0)-tensor fields associated to $\Om$.
The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.
problem Classifying K-contact forms with specific properties on 3-manifolds.
method Analyzing the Reeb vector field and its orbits, proving diffeomorphism results, and relating to spectral invariants.
result Compact 3-manifolds carrying such K-contact forms are diffeomorphic to lens spaces with specific orbit properties.
The paper defines and proves a new property for symplectic manifolds.
problem The study introduces a new property for symplectic manifolds.
method Defines and proves the L2-hard Lefschetz property for complete symplectic manifolds. result Proves that a complete symplectic manifold satisfies the L2-hard Lefschetz property if and only if every class of L2-harmonic forms contains a L2 symplectic harmonic form. The paper explores complex geometries of 3-forms on symplectic 6-manifolds.
problem Understanding the geometry of 3-forms on symplectic 6-manifolds.
method Investigation of geometries associated with 3-forms of various orbital types.
result Rich geometric structures attached to unstable 3-forms from Calabi-Yau degeneration.