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

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 Some of the results of this paper are contained in our previous preprint arXiv:1105.2931, which is no longer updated and will never become a published articl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

A model for simulating speckle-pattern evolution based on close to reality procedures used in spectral-domain OCT

A robust model for simulating speckle pattern evolution in optical coherence tomography (OCT) depending on the OCT system parameters and tissue deformation is reported. The model is based on application of close to reality procedures used in spectral-domain OCT scanners. It naturally generates images reproducing properties of real images in spectral-domain OCT, including the pixelized structure and finite depth of unambiguous imaging, influence of the optical spectrum shape, dependence on the optical wave frequency and coherence length, influence of the tissue straining, etc. Good agreement with generally accepted speckle features and properties of real OCT images is demonstrated.Comment: 13 pages, 6 figure

Universal manifold pairings and positivity

Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a sesquilinear pairing p= with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if is non-zero whenever x is non-zero. If this is the case, we call the pairing p positive. The question arises for each dimension d=0,1,2,.... We find p(d) positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly s-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d=3+1. There is a further physical implication of this paper. Whereas 3-dimensional Chern-Simons theory appears to be well-encoded within 2-dimensional quantum physics, eg in the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper53.abs.htm