On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

Interacting Preformed Cooper Pairs in Resonant Fermi Gases

We consider the normal phase of a strongly interacting Fermi gas, which can have either an equal or an unequal number of atoms in its two accessible spin states. Due to the unitarity-limited attractive interaction between particles with different spin, noncondensed Cooper pairs are formed. The starting point in treating preformed pairs is the Nozi\`{e}res-Schmitt-Rink (NSR) theory, which approximates the pairs as being noninteracting. Here, we consider the effects of the interactions between the Cooper pairs in a Wilsonian renormalization-group scheme. Starting from the exact bosonic action for the pairs, we calculate the Cooper-pair self-energy by combining the NSR formalism with the Wilsonian approach. We compare our findings with the recent experiments by Harikoshi {\it et al.} [Science {\bf 327}, 442 (2010)] and Nascimb\`{e}ne {\it et al.} [Nature {\bf 463}, 1057 (2010)], and find very good agreement. We also make predictions for the population-imbalanced case, that can be tested in experiments.Comment: 10 pages, 6 figures, accepted version for PRA, discussion of the imbalanced Fermi gas added, new figure and references adde

The masterpieces of John Forbes Nash Jr.

In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader

Involvement of regional lymph nodes after penetration of Schistosoma mansoni cercariae in naive and infected mice

The parotid lymph nodes of naive and previously infected Balb/c mice were studied after, respectively, infection and re-infection with cercariae of Schistosoma mansoni via the ears. Schistosomula were able to pass through the lymph node by following the lymph flow or by penetrating the veins of the medullary cords. The number of nodal mast cells was higher from day 2 to 6 of primary infection; and from day 5 to 11 of re-infection. The amount of degranulating mast cells was significantly higher at day 4 of infection and at day 1 of re-infection. Eosinophils characterized the nodal inflammatory processes observed after day 5 in both primarily-infected and re-infected mice. However, only in the latter the eosinophils were able to adhere to the larval surface. In primarily-infected mice, no intranodal larva presented signs of degeneration. In contrast, in re-infected animals, some degenerating larvae were found inside eosinophilic infiltrates. The eosinophils reached the nodal tissue by migrating through the high endothelial venules and their collecting veins

Sequences affecting the V(D)J recombinational activity of the IgH intronic enhancer in a transgenic substrate.

The immunoglobulin heavy chain intronic transcriptional enhancer (E mu) is part of a complex cis-regulatory DNA region which has notably been shown to modulate V(D)J rearrangements of associated variable gene segments. We have used recombination substrates comprised of the E mu enhancer together with various lengths of additional downstream mu sequences to assess the individual contribution of those sequences to the V(D)J recombinational regulatory activity. Surprisingly, in the absence of large amounts of mu sequences, substrate rearrangements were not detected in Southern blot analyses of the lymphoid tissues from independent transgenic mice, but were readily detectable following transfection into cultured pre-B cells. A short mu segment which includes matrix association regions (MARs) was not sufficient to restore high levels of rearrangements within the reporter transgenes. However, additional experiments demonstrated that the mu sequences are dispensable for V(D)J recombination in transgenic thymuses, implying a suppressive effect exerted by the vector sequences left in the transgenic insert, when they are attached near the E mu regulatory region. This suppression of V(D)J recombination, which correlates with an hypermethylation of the transgenes, is discussed in view of previously reported transgenic and gene targeting experiments