In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bound on the curvature tensor on a given space-like hypersuface. I will present the proof of this conjecture, which sheds light on the specific nonlinear structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

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.