Fourier Mukai Transforms for Gorenstein Schemes

We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also contemplate a criterion for equivalence. We offer a proof that is new even if we restrict to the smooth case. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre also is it. These results imply the invertibility of the usual relative Fourier-Mukai transform for an elliptic fibration as a direct corollary.Comment: Final version. To appear in Advances in Mathematic

Status of neutrino oscillations 2018: first hint for normal mass ordering and improved CP sensitivity

We present a new global fit of neutrino oscillation parameters within the simplest three-neutrino picture, including new data which appeared since our previous analysis~\cite{Forero:2014bxa}. In this update we include new long-baseline neutrino data involving the antineutrino channel in T2K, as well as new data in the neutrino channel, data from NO$\nu$A, as well as new reactor data, such as the Daya Bay 1230 days electron antineutrino disappearance spectrum data and the 1500 live days prompt spectrum from RENO, as well as new Double Chooz data. We also include atmospheric neutrino data from the IceCube DeepCore and ANTARES neutrino telescopes and from Super-Kamiokande. Finally, we also update our solar oscillation analysis by including the 2055-day day/night spectrum from the fourth phase of the Super-Kamiokande experiment. With the new data we find a preference for the atmospheric angle in the upper octant for both neutrino mass orderings, with maximal mixing allowed at $\Delta\chi^2 = 1.6 \, (3.2)$ for normal (inverted) ordering. We also obtain a strong preference for values of the CP phase $\delta$ in the range $[\pi,2\pi]$, excluding values close to $\pi/2$ at more than 4$\sigma$. More remarkably, our global analysis shows for the first time hints in favour of the normal mass ordering over the inverted one at more than 3$\sigma$. We discuss in detail the origin of the mass ordering, CP violation and octant sensitivities, analyzing the interplay among the different neutrino data samples.Comment: Updated neutrino oscillation analysis using the most recent results from T2K, NO$\nu$A, RENO and Super-Kamiokande. 17 pages, 8 figures, 1 tabl

Dark matter local density determination: recent observations and future prospects

This report summarises progress made in estimating the local density of dark matter ($\rho_{\mathrm{DM,\odot}}$), a quantity that is especially important for dark matter direct detection experiments. We outline and compare the most common methods to estimate $\rho_{\mathrm{DM,\odot}}$ and the results from recent studies, including those that have benefited from the observations of the ESA/Gaia satellite. The result of most local analyses coincide within a range of $\rho_{\mathrm{DM,\odot}} \simeq \text{0.4--0.6}\,\mathrm{GeV/cm^3} = \text{0.011--0.016}\,\mathrm{M_\odot / pc^3}$, while a slightly lower range of $\rho_{\mathrm{DM,\odot}} \simeq \text{0.3--0.5}\,\mathrm{GeV/cm^3} = \text{0.008--0.013}\,\mathrm{M_\odot / pc^3}$ is preferred by most global studies. In light of recent discoveries, we discuss the importance of going beyond the approximations of what we define as the Ideal Galaxy (a steady-state Galaxy with axisymmetric shape and a mirror symmetry across the mid-plane) in order to improve the precision of $\rho_{\mathrm{DM,\odot}}$ measurements. In particular, we review the growing evidence for local disequilibrium and broken symmetries in the present configuration of the Milky Way, as well as uncertainties associated with the Galactic distribution of baryons. Finally, we comment on new ideas that have been proposed to further constrain the value of $\rho_{\mathrm{DM,\odot}}$, most of which would benefit from Gaia's final data release.Comment: 37 pages, 3 tables, 1 figure. Invited report article for Reports on Progress in Physic

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved