Related papers: Distributions associated to almost complex structu…

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes…

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex…

Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $\omega$ on a manifold M into a multiple of itself, $f^* \omega = \eta \omega$. We assume $\omega$ is bounded, as some of the results may fail otherwise.…

For each compact almost Kahler manifold $(X,\om,J)$ and an element A of $H_2(X;Z)$, we describe a closed subspace $\ov{\frak M}_{1,k}^0(X,A;J)$ of the moduli space $\ov{\frak M}_{1,k}(X,A;J)$ of stable J-holomorphic genus-one maps such that…

In this letter, we used a combination of DEM and the multi-sphere method to investigate the random packing dynamics of $\Sigma_{2v}(2\pi/3)$-triplets. These triplets consist of three overlapping primary spheres, forming a bent structure…

On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of…

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the…

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a…

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent…

We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time…

Let $G=H\ltimes K$ denote a semidirect product Lie group with Lie algebra $\mathfrak g=\mathfrak h \oplus \mathfrak k$, where $\mathfrak k$ is an ideal and $\mathfrak h$ is a subalgebra of the same dimension as $\mathfrak k$. There exist…

Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted…

We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$. We define the notion of almost complex blow-up of $(M,J)$ at $x$. We prove the existence of almost complex blow-ups at $x$ under suitable assumptions on the almost…

We study symplectic structures on K\"ahler surfaces with p_g = 0. We give an example of a projective surface which admits a symplectic structure which is not compatible with any K\"ahler metric.

The space of the structure (0,3)-tensors of the covariant derivatives of the structure endomorphism and the metric on almost contact B-metric manifolds is considered. A known decomposition of this space in orthogonal and invariant subspaces…

On a manifold with an almost contact metric structure $(\varphi,\vec\xi,\eta,g,X,D)$ the notions of the interior and the $N$-prolonged connections are introduced. Using the $N$-prolonged connection, a new almost contact metric structure is…

There exist non-degenerate 3-form $d\omega_I$, $\omega_I(X,Y)=g(IX,Y)$, for each leftinvariant almost Hermitian structure $(g,I)$, where $g$ is Killing-Cartan metric on the $M=S^3\times S^3=SU(2)\times SU(2)$. Known \cite{H1}, that…