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.

Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of BunG, and on trace formulas. Work with Will Sawin.

The notion of Cohomological Hall algebra (COHA) was introduced in our joint paper with Maxim Kontsevich 10 years ago. It can be thought of as a mathematical incarnation of the notion of BPS algebra envisioned by physicists Harvey and Moore in the 90's.

Mathematically, COHA is an associative algebra structure on the cohomology of the moduli stack of objects of a 3-dimensional Calabi-Yau category with coefficients in a certain constructible sheaf. Interesting categories can be of geometric or algebraic origin (sheaves on Calabi-Yau 3-folds, quivers with potential, etc.).

In the talk I plan to discuss actions of COHA on the cohomology of certain instanton moduli spaces (spiked instantons of Nekrasov). This gives a relationship of COHA with affine Yangians and more recent "vertex algebras at the corner" introduced  by Gaiotto and Rapcak.

Recent advances in two-loop superstring theory will be discussed, including the structure of supermoduli space, the spontaneous supersymmetry breaking on Calabi-Yau orbifolds, and the matching of the D6 R4 effective low energy corrections to supergravity with predictions from supersymmetry and duality.

I show how massless supergravity theories can evade the theorems that try to establish that there are « No Solitons without Horizons. » The explicit construction of smooth, horizonless BPS solitons will be reviewed and some of their physical properties will be discussed. The possible role of such solitons in describing black-hole microstates will be examined.

The talk will review the birth of GR on the background of historical documents, and in particular on the basis of Einstein’s extant manuscripts. It will show how Einstein’s work emerged from a transformation of the knowledge of classical physics in a process that involved an interaction between the development of mathematical formalism and its physical interpretation. It will also emphasize the role of philosophical reflections for both the heuristics and the interpretation of the theory.