The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

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Intuitively, the classical variants of the Lefschetz Section Theorem relate a complex algebraic variety X to the intersection of X with a hyperplane H transversal to X (or, alternatively, to an ample divisor D of X). They are tremendously useful to compute invariants of the variety.

However, Lefschetz Section Theorems also hold for spaces that are constructed combinatorially rather than algebraically. Among other things, I will introduce Lefschetz theorems for — certain real subspace arrangements and their complements, — toric arrangements and their complements and, — matroids and smooth tropical varieties (joint work with Anders Björner). These theorems translate results of Lefschetz, Hamm-Le, Andreotti–Frankel, Bott–Thom, Akizuki–Kodaira–Nakano and Kodaira–Spencer to a combinatorial setting.

Entanglement is an intriguing property in quantum mechanics. Maldacena and Susskind have recently conjectured that the entanglement of EPR pair is interpreted to an ER bridge or a wormhole. This conjecture is called « EPR = ER ». Indeed, it is known that, from the holographic point of view, there is a wormhole on the world-sheet minimal surface corresponding to a (EPR) pair of accelerating quark and anti-quark. Therefore we discuss the causal structure on the world-sheet minimal surface of other scattering particles.

I will present recent calculations of the ultraviolet limit of multi-loop scattering amplitudes in N=4 supergravity. These calculations are performed using a squaring relationship between integrands for Yang-Mills amplitudes and for gravity amplitudes. I will discuss in detail the procedure we use to extract ultraviolet divergences from Feynman integrals, and I will present the three- and four-loop ultraviolet divergences in four-graviton scattering. I will also show how the four-loop divergence might be interpreted in terms of the U(1) duality anomaly of the theory.

We start by the example of the classical Lefschetz fixed point theorem for an isometry of a compact manifold. The Lefschetz number is a localized index as a result of the pairing on the level of K-theory. The localized indices can be used to produce topological invariants of the manifold such as the Lefschetz numbers.

Relativistic jets are often launched from the vicinity of accreting black holes. They are observed to be produced in stellar-mass black-hole binary systems and are believed to be the fundamental part of the gamma-ray burst phenomenon. Powerful relativistic jets are also ejected by accreting supermassive black holes in some active galactic nuclei. There is no doubt that the jet-launching mechanism is related to accretion onto black holes, but there has been no general agreement as to the ultimate source of energy of these spectacular high energy phenomena. In principle, relativistic jets can be powered either by the black hole gravitational pull or by its rotation (spin), with large-scale magnetic fields invoked as energy extractors in both cases. In the context of strongly magnetized jets Blandford & Znajek (1977) proposed a model of electromagnetic extraction of black hole’s rotational energy based on the analogy with the classical unipolar induction phenomenon. The physical meaning of this process has been subject to a long controversy. I will show that the Blanford-Znajek process is a Penrose process of black-hole energy extraction. I will first consider the case of arbitrary fields or matter described by an unspecified, general energy-momentum tensor and show that the necessary and sufficient condition for extraction of a black hole’s rotational energy is analogous to that in the mechanical Penrose process: absorption of negative energy and negative angular momentum. I will show that a necessary condition for the Penrose process to occur is for the Noether current to be spacelike or past directed (timelike or null) on some part of the horizon. In the particle (« mechanical ») case, the general criterion for the occurrence of a Penrose process reproduces the standard result. For stationary, force-free electro-magnetic field one recovers the condition obtained by Blandford and Znajek in their original article. In the case of relativistic jet-producing « magnetically arrested disks » I will show that the negative energy and angular-momentum absorption condition is obeyed when the Blandford-Znajek mechanism is at work, and hence the high energy extraction efficiency up to ~300 % found in recent numerical simulations of such accretion flows results from tapping the black hole’s rotational energy through the Penrose process. I will show how black-hole rotational energy extraction works in this case by describing the Penrose process in terms of the Noether current.

Current knowledge of Mercury orbit is mainly brought by the direct radar ranging obtained from the 60s to 1998 and five Mercury flybys made by Mariner 10 in the 70s, and MESSENGER made in 2008 and 2009. On March 18, 2011, MESSENGER became the first spacecraft orbiting Mercury. The radioscience observations acquired during the orbital phase of MESSENGER drastically improved our knowledge of the Mercury orbit. An accurate MESSENGER orbit is obtained by fitting one-and-half years of tracking data using GINS orbit determination software. The systematic error in the Earth-Mercury geometric positions, also called range bias, obtained from GINS are then used to fit the INPOP dynamical modeling of the planet motions. An improved ephemeris of the planets is then obtained, INPOP13a, and used to perform general relativity test of PPN-formalism. Monte Carlo simulations will be introduced for estimating the most significant levels of GR violations in terms of PPN parameters and their correlated parameter (oblateness of the sun) and of time-varying Gravitational constant G.