Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematician

Microtubule-associated protein 6 mediates neuronal connectivity through Semaphorin 3E-dependent signalling for axonal growth.

Structural microtubule associated proteins (MAPs) stabilize microtubules, a property that was thought to be essential for development, maintenance and function of neuronal circuits. However, deletion of the structural MAPs in mice does not lead to major neurodevelopment defects. Here we demonstrate a role for MAP6 in brain wiring that is independent of microtubule binding. We find that MAP6 deletion disrupts brain connectivity and is associated with a lack of post-commissural fornix fibres. MAP6 contributes to fornix development by regulating axonal elongation induced by Semaphorin 3E. We show that MAP6 acts downstream of receptor activation through a mechanism that requires a proline-rich domain distinct from its microtubule-stabilizing domains. We also show that MAP6 directly binds to SH3 domain proteins known to be involved in neurite extension and semaphorin function. We conclude that MAP6 is critical to interface guidance molecules with intracellular signalling effectors during the development of cerebral axon tracts

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold -- only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rate

Three topics in quantum communication: error filtration, quantum string flipping, photon pair generation in periodically poled twin hole fibers

We briefly present three topics in quantum communication. First of all a method, called error filtration, to reduce errors during quantum communication. We apply this method to quantum cryptography in a noisy environment, and describe an experimental realisation thereof. Second we describe an experimental realisation of quantum string flipping, in which two parties that do not trust each other want to generate a string of random bits. Finally we present the experimental realisation of a source of photon pairs at telecommunication wavelengths based on parametric fluorescence in periodically poled twin hole fibers

Three topics in quantum communication: error filtration, quantum string flipping, photon pair generation in periodically poled twin hole fibers

We briefly present three topics in quantum communication. First of all a method, called error filtration, to reduce errors during quantum communication. We apply this method to quantum cryptography in a noisy environment, and describe an experimental realisation thereof. Second we describe an experimental realisation of quantum string flipping, in which two parties that do not trust each other want to generate a string of random bits. Finally we present the experimental realisation of a source of photon pairs at telecommunication wavelengths based on parametric fluorescence in periodically poled twin hole fibers