We study the effects of induced interactions on the pairing gap, transition temperature, and chemical potential of a quasi-two-dimensional Fermi gas of atoms with spin-orbit coupling. We find that these mean-field parameters are significantly modified when induced interactions are taken into account. We also investigate the implications of induced interaction corrections for the BEC-BCS crossover driven by spin-orbit coupling, which happens even for small (compared to the Fermi energy) values of the binding energy.

The simple finite Lie superalgebras D(2, 1; α), G(3), D(4, 1), D(2, 2), A(3, 1) and F (4) admit D-module representations, given by a set of differential operators of a single variable t ∈ R, at a critical value of the scaling dimension λ. These superalgebras are one-dimensional N -extended superconformal algebras with N = 4 (D(2, 1; α)), N = 7 (G(3)) and N = 8 (the remaining ones). The critical D-module reps induce invariant actions in the Lagrangian framework for superconformal mechanics in D target dimensions. The N = 8 critical scalings λ = 1 D−4 are linked to the D-dimensional harmonic functions with D = 1, 2, 3, 5, 6, 7, 8. This talk is based on J. Math. Phys. 53 (2012) 043513 (arXiv:1112.0995), J. Math. Phys. 53 (2012) 103518 (arXiv:1208.3612) and some extra material.

We review in these notes the status of the construction of the Pierre Auger Observatory and present the first Physics results, based on the data collected during the first year and a half of operation. These results are preliminary, once the work to understand the systematics of the detectors are still underway. We discuss the cosmic ray spectrum above 3 EeV, based on the measurement done using the Surface Detector and the Fluorescence Detector, both, components of the observatory. We discuss, as well, the search for anisotropy near the Galactic Center and the limit on the photon fraction at the highest energies.

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