Tropical curves, graph complexes, and top weight cohomology of M_g

We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight cohomology of M_g and also with the genus g part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least 7. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of another theorem of Willwacher, that homology of the graph complex vanishes in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J. Amer. Math. So

The polarisation of afterglow emission reveals GRB jet structure

We numerically compute light and polarisation curves of gamma-ray burst afterglows for various configurations of the jet luminosity structure and for different dynamical evolutions. We especially consider the standard homogeneous ``top hat'' jet and the ``universal structured jet'' with power-law wings. We also investigate a possible more physical variation of the ``top hat'' model: the ``Gaussian jet''. The polarisation curves for the last two jet types are shown here for the first time together with the computation of X-ray and radio polarised fluxes. We show that the lightcurves of the total flux from these configurations are very similar to each other, and therefore only very high quality data could allow us to pin down the underlying jet structure. We demonstrate instead that polarisation curves are a powerful means to solve the jet structure, since the predicted behaviour of polarisation and its position angle at times around the jet break are very different if not opposite. We conclude that the afterglow polarisation measurements provide clear footprints of any outflow energy distribution (unlike the lightcurves of the total flux) and the joint analysis of the total and polarised flux should reveal GRBs jet structure.Comment: 16 pages, 18 figures, MNRAS, temp, 321. Light curves and polarisation curves for a Gaussian jet added. Cartoon of the three jet structures adde

Guided Surface Plasmon Mode of Semicircular Cross Section Silver Nanoridges

Tightly confined plasmon waveguide modes supported by semicircular cross section top silver nanoridges are investigated in this paper. Mode field profiles, dispersion curves, propagation distances, confinement factors, and figure-of-merits of semicircular top silver nanoridge plasmon waveguide mode are calculated for different radii of curvature at different wavelengths. It is found that semicircular top silver nanoridges support tightly confined quasi-TEM plasmon waveguide modes. Semicircular top silver nanoridge mode has longer propagation distance and higher figure-of-merit than that of the cylindrical silver nanowire of the same radius of curvature.Comment: 21 pages, 11 figure