This is an informal, yet hopefully entertaining talk by a non-expert. After a short (and undoubtedly
incomplete) discussion of some basic puzzles confronting present-day cosmology, I recall the Friedmann Equations for the evolution of a homogeneous, isotropic universe and then proceed to describe the quantum-mechanical state of the universe shortly after inflation. This leads me to speculate about a possible mechanism, based on the chiral magnetic effect, for the growth of intergalactic magnetic fields in the early universe. This mechanism is a fourdimensional cousin of the quantum Hall effect. I then discuss ideas how to unify models of Dark Matter and Dark Energy into a single theory, which, in addition, might shed light on the origin of the matter-antimatter asymmetry of the universe.

Periods, numerical as algebraic integrals, and abstract, associated to de Rham comparison isomorphism, are fundamental in Physics and Mathematics. … Why are they so much related!?

Various considerations regarding their geometric and dynamic interpretation will be provided, together with thoughts requiring further readings and study.

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

I will talk about loop infrared effects in de Sitter QFT. Namely about their types, physical meaning and origin and also about their resumation and physical consequences. The talk is based on arXiv:1701.07226.

Thurston conjectured that quasi-Fuchsian manifolds are uniquely determined by the induced hyperbolic metrics on the boundary of their convex core and Mess extended this conjecture to the context of globally hyperbolic anti de-Sitter spacetimes. In this talk I will discuss a universal version of Thurston and Mess' conjectures: any quasisymmetric homeomorphism from the circle to itself is obtained on the convex hull of a quasicircle in the boundary at infinity of the 3-dimensional hyperbolic (resp. anti-de Sitter) space. We will also discuss a similar result for convex domains bounded by surfaces of constant curvature K in (-1,0) in the hyperbolic setting and of curvature K in (-∞,-1) in the anti de-Sitter setting with a quasicircle as their asymptotic boundary.

(This is joint work in progress with F. Bonsante, J. Danciger and J.-M. Schlenker.)

In this talk I present the phase diagram of a U(N)^2 x O(D) invariant fermionic planar matrix quantum mechanics (equivalently tensor or complex SYK models) in the new large D limit dominated by melonic graphs. The Schwinger-Dyson equations can have two solutions describing either a "large" black hole phase a la SYK or a "small" black hole with trivial IR behavior. In the strongly coupled region of the mass-temperature plane, there is a line of first order phase transitions between the small and large black hole phases. This line terminates at a new critical point which can be studied numerically in detail. The critical exponents are non-mean-field and different on the two sides of the transition. If time allows, I will compare this to purely bosonic unstable and stable melonic models.

During pre-implantation development, the mammalian embryo forms the blastocyst, which will implant into the uterus. The architecture of the blastocyst is essential to the specification of the first mammalian lineages and to the implantation of the embryo. Consisting of an epithelium enveloping a fluid-filled cavity and the inner cell mass, the blastocyst is sculpted by a succession of morphogenetic events. These deformations result from the changes in the forces and mechanical properties of the tissue composing the embryo. Using microaspiration, live-imaging, genetics and theoretical modelling, we study the biophysical and cellular changes leading to the formation of the blastocyst. In particular, we uncover the crucial role of acto-myosin contractility, which generates periodic waves of contractions, compacts the embryo, controls the position of cells within the embryo and influences fate specification.

A variety of spin systems, including Heisenberg and XXZ models, can be represented in terms of random loops in d+1 dimensions. The (conjectured) phase transitions are expressed in terms of the occurrence of macroscopic loops. Moreover, the macroscopic loops are expected to obey a Poisson–Dirichlet distribution. For Zd such results are lacking. Here we present progress on mean-field models: on the complete graph and on regular tress. Based on ongoing work with Jürg Fröhlich and Daniel Ueltschi.

I will talk about the "cohesive module" introduced by Jonathan Block. Concepturally it is a complex of vector bundles with a superconnection. I will describe the geometric nature of this object. In particular, on complex manifolds, the cohesive module gives a DG enhancement of the bounded derived category of coherent sheaves. Nevertheless, it applies to many other places.

In this talk we will start by explaining a spectacular analogy between topological spaces and higher category theory : In [1] Michael Batanin built the globular weak higher groupoid fundamental for any topological space X, wished by Alexander Grothendieck. To do such construction he used the coendomorphism operad associated to the coglobular object of disks in Top, the category of topological spaces. He was able to built the weak higher groupoid fundamental functor thanks to the contractibility of this operad. In the other hand the author [2] built a coglobular object in the category of higher operads, such that algebras for the first object are weak infini-categories, algebras for the second object are weak infini-functors, etc. With such coglobular object we also get a coendomorphism operad, built itself with operads instead of topological spaces. We conjecture that this operad is contractible like the topological one. If this is true then the globular weak higher category of globular weak higher categories exist. Also we will explain how to build the free cubical contractible higher operad which algebras are cubical weak higher categories, and if we have time we will explain how from this we obtain the cubical weak higher groupoid fundamental functor for topological spaces.

[1] Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak-n-categories, Advances in Mathematics, 1998.

 

[2] Camell Kachour, Steps toward the weak higher category of weak higher categories in the globular setting, Category and Algebraic Structures with Applications, 2015.

In the recent development of modular geometry on toric noncommutative manifolds (Connes-Moscovici 2014), metrics are parametrized by self-adjoint elements in the ambient C*-algebra, whose exponential are called Weyl factors. Local invariants, such as the Riemannian curvature, are encoded in the coefficients of certain heat kernel expansion. The new ingredient, purely due to noncommutativity, is the the inner automorphism generated by the Weyl factor, whose corresponding derivation can be viewed as a noncommutative differential.  From analytic point of view, curvature is designed to measure the commutators of covariant derivatives. In this talk, we will discuss some intriguing spectral functions which define the interplay between the inner automorphisms and the classical differentials.  I recently found that hypergeometric functions and its multivariable generalization are the building blocks. Geometric applications such as Gauss-Bonnet theorem lead to some functional relations/equations between them which are still begging for more conceptual understanding.