Equivariant discretizations of diffusions, random walks, and harmonic functions

For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons–Sullivan discretizations of the processes and the associated function theory

Bottom of spectra and coverings of orbifolds

We discuss the behaviour of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds

An estimate for the measure theoretic entropy of geodesic flows

A new proof and a generalization of the Osserman-Sarnak estimate for the measure theoretic entropy of geodesic flows is presente

Rank rigidity for CAT(0) cube complexes

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor

Bottom of spectra and amenability of coverings

For a Riemannian covering $\pi\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of $M_0$

Entropy of semiclassical measures for nonpositively curved surfaces

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced in appendix A of a previous work (arXiv:0809.0230, version 2

Dual-tip-enhanced ultrafast CARS nanoscopy

Coherent anti-Stokes Raman scattering (CARS) and, in particular, femtosecond adaptive spectroscopic techniques (FAST CARS) have been successfully used for molecular spectroscopy and microscopic imaging. Recent progress in ultrafast nanooptics provides flexibility in generation and control of optical near fields, and holds promise to extend CARS techniques to the nanoscale. In this theoretical study, we demonstrate ultrafast subwavelentgh control of coherent Raman spectra of molecules in the vicinity of a plasmonic nanostructure excited by ultrashort laser pulses. The simulated nanostructure design provides localized excitation sources for CARS by focusing incident laser pulses into subwavelength hot spots via two self-similar nanolens antennas connected by a waveguide. Hot-spot-selective dual-tip-enhanced CARS (2TECARS) nanospectra of DNA nucleobases are obtained by simulating optimized pump, Stokes and probe near fields using tips, laser polarization- and pulse-shaping. This technique may be used to explore ultrafast energy and electron transfer dynamics in real space with nanometre resolution and to develop novel approaches to DNA sequencing.Comment: 11 pages, 6 figure

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of the Dirac operator on such a manifold has $r$ small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface

Plans for Aeroelastic Prediction Workshop

This paper summarizes the plans for the first Aeroelastic Prediction Workshop. The workshop is designed to assess the state of the art of computational methods for predicting unsteady flow fields and aeroelastic response. The goals are to provide an impartial forum to evaluate the effectiveness of existing computer codes and modeling techniques, and to identify computational and experimental areas needing additional research and development. Three subject configurations have been chosen from existing wind tunnel data sets where there is pertinent experimental data available for comparison. For each case chosen, the wind tunnel testing was conducted using forced oscillation of the model at specified frequencie