Dans ce cours nous étudierons plusieurs modèles de graphes aléatoires allant du plus classique (le modèle d'Erdös-Renyi introduit en 1960) aux plus récents (les cartes planaires aléatoires étudiées depuis le début des années 2000). Le fil conducteur du cours sera la notion de convergence locale et les propriétés des graphes limites dits dilués.

Contenu du cours :

– Modèle d'Erdös-Renyi, transition de phase et propriétés de base
– Convergence locale et "méthode objective" d'Aldous et Steele
– Arbre couvrant minimum et théorème de Frieze 
– Graphes aléatoires unimodulaires
– Limites locales d'arbres aléatoires
– Limites locales de cartes aléatoires (construction, épluchage, théorème de Benjamini-Schramm)

The goal of this talk is to give a pedagogical introduction to Bell’s theorem and its implication for our view of the physical world, in particular how it establishes the existence of non-local effects or of actions at a distance. We also discuss several misunderstandings of Bell’s result and we will explain how the de Broglie-Bohm theory allows us to understand, to some extent, what non-locality is.

Dans cet exposé, je présenterai certains résultats mathématiques  récents concernant la limite de champ moyen pour des systèmes quantiques en  interaction, le lien avec la condensation de Bose-Einstein et la théorie de  Bogoliubov.

In this talk I will advocate the need to explore the properties of Fermi statistics. Although most investigations have been done with electrons in quantum conductors, the effects presented here are relevant for any fermionic systems.
In phase coherent conductors electrons behave as effective non interacting Fermions (Landau quasiparticles). Reducing the lateral size of ultra-clean conductors to achieve perfect control of the electron transmission has enable experiments revealing fundamental and sometime unexpected properties of the Fermi sea: conductance quantization, noiseless current, electron anti-bunching, continuous single electron injection from a voltage biased a contact, spontaneous creation of spin entangled electron-hole pairs in tunnel barrier, to cite a few.
Further Fermi statistics entanglement properties can be investigated in a linear electron optics scheme, mimicking quantum optics with electrons replacing photons. This requires realizing single electron sources, a challenging issue as contrary to photons, which propagate in vacuum, single electrons must be launched on top of a Fermi sea prone to generate extra excitations. However a magic combination of Fermi statistics and wave properties allow time-resolved single electron to be created in the form of minimal excitation states called Leviton. Single electron sources, either time or energy resolved, have been recently used in single electron partitioning experiments and to measure two-(or more) electron Hong Ou Mandel correlations. If time permits some possible extensions to anyonic quasi-particles like the Laughlin quasi-particles of the Quantum Hall Effect will be discussed.

In my talk I will discuss a family of matrix models, which describes the generating functions of intersection numbers on moduli spaces both for open and closed Riemann surfaces. Linear (VirasoroW-constraints) and bilinear (KPMKP integrable hierarchies) equations follow from the matrix model representation.

It has been realized recently that the c=1 conformal block of 4 point function in Liouville CFT is related to the Tau function of the Painlevé 6 integrable system. Here we propose a general construction: starting from a very general integrable system (a Hitchin system: the moduli space of flat G-connections over a Riemann surface, with G an arbitrary semi-simple Lie group), we define some « amplitudes », and we show that these amplitudes satisfy all the axioms of a CFT: they satisfy OPEs, Ward identities and crossing symmetry. The construction is very geometrical, by defining a notion of « quantum spectral curve » attached to a flat connection, defining homology and cohomology on it, and showing that amplitudes satisfy Seiberg-Witten like relations, and behave well under modular transformations.
So this link between CFTs and integrable systems unearths a new and beautiful quantum geometry.

This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.

Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds’ liftings.

I will discuss theoretical and computational models which describe evolution of complex patterns in a living organism towards specific « target morphology ». As a proof-of-concept, two working models of patterns’ regeneration will be presented.

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014, arXiv:1601.07904, arXiv:1610.05308.

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

The emergence of very long time scales on the gravity side of the AdS/CFT correspondence has led to the introduction of the notion of complexity in the research of quantum gravity. Various notions of complexity have been studied in the area of Quantum Information as well as in Quantum Field Theory. I will discuss, in this context, complexities and the time scales of their evolution following in the various definitions. The physics involved will be stressed where it is understood.

Complementary series are families of unitary representations of certain semisimple Lie groups and of groups of automorphisms of homogeneous trees. I wil recall their definition for semisimple groups and explain how to prove their existence for groups of automorphisms of trees by a method which allows to extend this construction in order to build new representations of free groups.