Related papers: More counterexamples to Lagrangian Poincar\'e recu…

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {\v{S}}evera. We study the relationship of odd symplectic…

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a…

We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then it is quasiconvex if and only if it is polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove that…

In this paper we construct and classify Lagrangian T^3-fibrations on non compact symplectic manifolds with singular fibres of prescribed topological type. This contributes to the understanding of the structure of the singular fibres that…

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori…

The Hamiltonian shape invariant of a domain $X \subset \mathbb R^4$, as a subset of $\mathbb R^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or…

We construct counterexamples to inverse problems for the wave operator on domains in $\mathbb{R}^{n+1}$, $n \ge 2$, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which…

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian…

In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually…

We present the topological foundations for the solvability of Multiplicative Cousin problems formulated on an axially symmetric domain $\Omega \subset \mathbb H.$ In particular, we provide a geometric construction of quaternionic Cartan…

Let (M,\omega) be a four dimensional compact connected symplectic manifold. We prove that (M,\omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of…

In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincar\'e and Lagrange-Poincar\'e reduction. In fact, we show that solutions of the Hamilton-Poincar\'e…

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal…

Examples of nonformal simply connected symplectic manifolds are constructed.

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable…

The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out…

To find all two-dimensional equivariant symplectic submanifolds in symplectic toric manifolds, we combine the convex geometry of Delzant polytopes with local equivariant symplectic models and obtain a criterion for determining when a…

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane…

In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In…