Related papers: On Non-Separating Contact Hypersurfaces in Symplec…

Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M.

In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the…

We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically…

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not…

While the Hamiltonian group actions on closed symplectic manifolds have been widely explored throughout the last couple of decades, the study on Hamiltonian group actions on symplectic manifolds with a contact type boundary has started only…

We study constructions of contact forms on closed manifolds. A notion of strong symplectic fold structure is defined and we prove that there is a contact form on $M \x X$ provided that $M$ admits such a structure and $X$ is contact. This…

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in…

Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding…

We provide examples of contact manifolds of any odd dimension $\geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations.

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.

We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure…

A presymplectic structure on odd dimensional manifold is given by a closed 2-form which is nondegenerate, i.e., of maximal rank. We investigate geometry of presymplectic manifolds. Some basic theorems analogous to those in symplectic and…

According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $\pi$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric…

In this paper we introduce a technique, called rim surgery, which can change a smooth embedding of an orientable surface of positive genus and nonnegative self-intersection in a smooth 4-manifold while leaving the topological embedding…

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be…

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…

In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such…

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of…

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to…