Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.Comment: Final version to appear in CMP. Two new remarks have been added as well as some numerical calculations for metric entrop

Spontaneous activity regulates Robo1 transcription to mediate a switch in thalamocortical axon growth

Developing axons must control their growth rate to follow the appropriate pathways and establish specific connections. However, the regulatory mechanisms involved remain elusive. By combining live imaging with transplantation studies in mice, we found that spontaneous calcium activity in the thalamocortical system and the growth rate of thalamocortical axons were developmentally and intrinsically regulated. Indeed, the spontaneous activity of thalamic neurons governed axon growth and extension through the cortex in vivo. This activity-dependent modulation of growth was mediated by transcriptional regulation of Robo1 through an NF-κB binding site. Disruption of either the Robo1 or Slit1 genes accelerated the progression of thalamocortical axons in vivo, and interfering with Robo1 signaling restored normal axon growth in electrically silent neurons. Thus, modifications to spontaneous calcium activity encode a switch in the axon outgrowth program that allows the establishment of specific neuronal connections through the transcriptional regulation of Slit1 and Robo1 signaling

RIEMANNIAN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE

Of special interest in the history of Riemannian geometry have been manifolds with positive sectional curvature. In these notes we want to give a survey of this subject and some recent developments. We start with some historical developments. 1. History and Obstructions It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales ” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time. Here we only make a few comments. Curvature of surfaces in 3-space had been studied previously by a number of authors and was defined as the product of the principal curvatures. But Gauss was the first to make the surprising discovery that this curvature only depends on the intrinsic metric and not on the embedding. Here one finds for example the formula for the metric in the form ds2 = dr2 + f(r, θ) 2dθ2. Gauss showed that every metric on a surface has this form in ”normal ” coordinates and that it has curvature K = −frr/f. In fact one can take it as the definition of the Gauss curvature and proves Gauss’s famous ”Theorema Egregium” that the curvature is an intrinsic invariant and does not depend on the embedding in R3. He also proved a local version of what we nowadays call the Gauss-Bonnet theorem (it is not clear what Bonnet’s contribution was to this result), which states that in a geodesic triangle ∆ with angles α, β, γ the Gauss curvature measures the angle ”defect”: Kdvol = α + β + γ − π Nowadays the Gauss Bonnet theorem also goes under its global formulation for a compact surface