How large is the shadow of a symplectic ball?

Consider the image of a 2n-dimensional unit ball by an open symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection into a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.Comment: Final version, identical to the published one. Added comments about the relationship with a conjecture of Viterbo and related references. Some of the results of this paper are contained in our previous preprint arXiv:1105.2931, which is no longer updated and will never become a published articl

Rock flows

Rock flows are defined as forms of spontaneous mass movements, commonly found in mountainous countries, which have been studied very little. The article considers formations known as rock rivers, rock flows, boulder flows, boulder stria, gravel flows, rock seas, and rubble seas. It describes their genesis as seen from their morphological characteristics and presents a classification of these forms. This classification is based on the difference in the genesis of the rubbly matter and characterizes these forms of mass movement according to their source, drainage, and deposit areas

Symplectic Lefschetz fibrations on S^1 x M^3

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper18.abs.htm