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.
Study of symplectic trivialization and reduction of bundles with symmetry and connection.
problem Symplectic trivialization and reduction of bundles with symmetry and connection.
method Analysis of the Tulczyjew's triplet with an Ehresmann connection.
result Trivializations and reductions of iterated tangent and cotangent bundles.
In this paper, we prove the existence of certain symplectic conifold transitions on all CP1-bundles over symplectic 4--manifolds, which generalizes Smith, Thomas and Yau's examples of symplectic conifold transitions on trivial CP1-bundles over Kähler surfaces. Our main result is to determine the diffeomorphis…
In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurf…
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…
For a closed symplectic manifold (M,ω), a compatible almost complex structure J, a 1-periodic time dependent symplectic vector field Z and a homotopy class of closed curves γ we define a Floer complex based on 1-periodic trajectories of Z in the homotopy class γ. We suppose that the closed 1-form iZtω…
In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.
A study of symplectic actions of a finite group G on smooth 4-manifolds is initiated. The central new idea is the use of G-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set str…
New construction provides non-trivial representations for geometric quantisation.
problem Geometric quantisation of non-integral symplectic structures.
method Construction from Noncommutative Differential Geometry adapted to diffeology.
result The construction provides non-trivial representations.
We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…
Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
problem Understanding symplectic Torelli groups of rational surfaces.
method Using positivity condition, type of cohomology class, and Lagrangian spherical classes.
result Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…
We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds t…
Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles fo…
In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of class…
It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic…
A surjective submersion π:M→B carrying a field of simplectic structures on the fibres is symplectic if this Poisson structure is minimal. A symplectic submersion may be interpreted as a family of mechanical systems depending on a parameter in B. We give some conditions to find a closed form which represent the…
We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quand…
New insights into symplectic loops and their flux groups.
problem Understanding symplectic loops and their properties.
method Analyzing symplectic diffeomorphisms and their orbits.
result Flux of symplectic loops vanishes for contractible orbits.
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. New groups can't be fundamental groups of symplectic Calabi-Yau manifolds.
problem Proving certain groups cannot be fundamental groups of symplectic Calabi-Yau manifolds.
method Using the method of Friedl and Vidussi, they showed these groups cannot be fundamental groups of SCY manifolds.
result The Lodha--Moore groups and their n-adic generalizations cannot be fundamental groups of any SCY manifold. In this paper we use the Lubotzky alternative for finitely generated linear groups to determine which 4-manifolds admitting a free circle action can be endowed with a symplectic structure with trivial canonical class. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-m…
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…
We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the cond…
We introduce the secondary Stiefel-Whitney class w~2 of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class w~2 vanishes. Using this, we give a detailed description of the combinatorial str…
In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
We consider symplectic Floer homology in the lowest nontrivial dimension, that is to say, for area-preserving diffeomorphisms of surfaces. Particular attention is paid to the quantum cap product; we show that it distinguishes the trivial element of the mapping class group from any nontrivial one.
We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the cla…
Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…
Study symplectic scalar curvature on supermanifolds.
problem Define and analyze symplectic scalar curvature on supermanifolds.
method Introduced two families of odd super-Fedosov structures using graded symmetric and non-symmetric connections.
result Found non-trivial odd symplectic scalar curvature for the second family.
The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.
problem Analyzing the isotopy of C-symplectic structures and their applications.
method Proves an analogue of Moser's isotopy theorem for families of C-symplectic structures.
result Locally trivial degenerate twistorial deformation over the base of holomorphic Lagrangian fibrations.
Study of adjoint orbits in simplest non-trivial Lie algebra case.
problem Geometric properties of adjoint orbits in sl(2,R). method Analysis of adjoint orbits, showing three possibilities: hyperboloids or cones.
result Just three possibilities for adjoint orbits: hyperboloids or cones.
Kähler complexity one Hamiltonian T-manifolds have trivial paintings.
problem Understanding the structure of Kähler complexity one Hamiltonian T-manifolds.
method Proving the existence of a trivial painting for compact, connected Kähler complexity one Hamiltonian T-manifolds.
result Every compact, connected Kähler complexity one Hamiltonian T-manifold has a trivial painting.
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. An important question with a rich history is the extent to which the symplectic category is larger than the Kaehler category. Many interesting examples of non-Kaehler symplectic manifolds have been constructed. However, sufficiently large symmetries can force a symplectic manifold to be Kaehler. In this paper, we solve…
In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow Σs∞ at a singular point (X0,T0) of a symplectic mean curvature flow Σt or of a Lagrangian mean curvature flow Σt is …
Paper proves non-extendability of quasimorphism and non-vanishing of Reznikov's class.
problem Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class.
method Analyzing specific symplectic manifolds like blow-ups of tori and products of surfaces.
result Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class for certain symplectic manifolds.
We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obt…
Study reveals new geometric structures for magnetic field Hamiltonian systems.
problem Understanding Hamiltonian systems in magnetic fields.
method Investigation of symplectic-Haantjes geometry.
result Non-trivial symplectic-Haantjes manifolds found.
We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologicall…
The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.
problem Generalizing convex and star-shaped concepts to symplectic vector spaces.
method Study of variational problems for symplectically convex and star-shaped curves.
result Extremal points of the variational problem are rigid multiply traversed conics for a range of parameters.
Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a non-trivial homomorphism to the group SO(3). In particular, Y cannot be a homotopy-sphe…
We consider the Lie algebra consisting of all derivations on the free associative algebra, generated by the first homology group of a closed oriented surface, which kill the symplectic class. We find the first non-trivial abelianization of this Lie algebra and discuss its relation to unstable cohomology classes of the …
We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and th…
In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morp…
Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. π_0(S) is generated by the cla…