Séminaires mathématique 2015 - 2016

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  Jeudi 26 Mai 2016, à 14h, salle 114

Antoine Géré (Dept. of Mathematics, University of Genova) : Perturbatively finite gauge models on the noncommutative three-dimensional space $\mathbbR^3_\lambda$

We show that natural noncommutative gauge theory models on $\mathbbR^3_\lambda$ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of $\mathbbR^3_\lambda$ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differential calculus. Restricting ourselves to positive actions with covariant coordinates as field variables, a suitable gauge-fixing leads to a family of matrix models with quartic interactions and kinetic operators with compact resolvent. Their perturbative behavior is then studied. We find that the amplitudes of the ribbon diagrams for the models of a subfamily of these matrix models for which the interactions have a symmetric form are finite to all orders in perturbation. This result extends finally to any of the models of the whole family of matrix models obtained from the above gauge-fixing. The origin of this result is then discussed.

  Jeudi 12 Mai 2016, à 14h, salle 114

Timothy Budd (NBI, Copenhagen, Denmark) : Geometry of random planar maps with high degrees

Many classes of random planar maps, i.e. planar graphs embedded in the sphere modulo homeomorphisms, possess scaling limits (in a Gromov-Hausdorff sense) described by a universal random continuum metric space known as the Brownian map. One way to escape this universality class is to consider certain Boltzmann planar maps with carefully tuned weights on the faces. Le Gall and Miermont have shown that the scaling limits of these fall into a larger class of continuum random metric spaces, often referred to as the stable maps. In this talk I will consider the geometry of the dual (in the planar map sense) of these Boltzmann planar maps, which shows quite different scaling behavior. In particular, I will discuss the growth of the perimeter and volume of certain geodesic balls of increasing radius based on the examination of a peeling process. For a particular range of parameters, in the so-called the dilute phase, precise scaling limits may be obtained for these processes, which suggests possibility of a non-trivial scaling limit in the Gromov-Hausdorff sense. Finally I will discuss a more detailed scaling limit in terms of growth-fragmentation processes. Based on work with Jean Bertoin, Nicolas Curien, and Igor Kortchemski.

  Jeudi 14 Avril 2016, à 14h, salle 114

Antonio Duarte Pereira (UFF, Niteroi, Brazil - AEI, Golm, Germany) : Non-perturbative features of Yang-Mills theories and the Gribov problem

In this talk, I will review the Refined Gribov-Zwanziger framework designed to deal with the so-called Gribov problem in Yang-Mills theories and its standard BRST soft breaking. I will show that, within this scenario, the BRST transformations are modified in the non-perturbative regime in order to be a symmetry of the model. This fact has been supported by recent lattice simulations and opens a new avenue for the investigation of non-perturbative effects in Yang-Mills theories.

  Jeudi 18 Février 2016, à 14h, salle 114

Karim Noui (LMPT, Tours - APC, Univ. Paris 7) : Towards a classification of healthy scalar-tensor theories

Theories with higher order time derivatives in the Lagrangian generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. Nonetheless, in some situations, this fate can be avoided and the Ostrogradski ghost does not exist. This is the case for Horndeski theories which are scalar-tensor theories, candidates for an alternative explanation of dark energy. However, there are not the only "safe" higher-derivative scalar-tensor theories. In this talk, I will present a way to classify ghost-free scalar tensor theories, focussing mainly on theories that depend quadratically on the second derivatives of a scalar field. This talk is based on works in collaboration with D.Langlois.

  Jeudi 21 Janvier 2016, à 14h, salle 114

Alessandro Giuliani (Dept. of Mathematics and Physics, University of Roma Tre) : Height fluctuations and universality relations in interacting dimer models

We consider weakly interacting dimer models on the two-dimensional square lattice. By constructive renormalization group techniques, we compute the multipoint dimer correlations and all the moments of the height function. In particular, we rigorously establish the asymptotic equivalence between the height function and the massless Gaussian free field (GFF). The stiffness K=K(\lambda) of the GFF is shown to be a non trivial analytic function of the interaction strength \lambda between neighboring dimers. We also prove one of the Haldane relations adapted to the present context, namely that K *equals* the dimer-dimer critical exponent X_+. We briefly discuss a possible connection between this universality relation and the Miller-Sheffield duality between the GFF and SLE_\kappa. Joint work with V. Mastropietro and F. Toninelli.

  Jeudi 10 Decembre 2015 à 14h, salle 114

Giovanni Landi (Dept. of Mathematics, University of Trieste) : Sigma-model solitons on noncommutative spaces

We use results from time-frequency analysis and Gabor analysis to construct new classes of sigma-model solitons over the Moyal plane and over noncommutative tori, taken as source spaces, with a target space made of two points. A natural action functional leads to self-duality equations for projections in the source algebra. Solutions, having non-trivial topological content, are constructed via suitable Morita duality bimodules.

  Jeudi 3 Décembre 2015, à 14h, salle 110 (Salle Cosmo)

Valentin Bonzom (LIPN, Université Paris 13) : 3D holography : from discretum to continuum

We consider a very simple example of discretized quantum gravity, 3D gravity with zero cosmological constant on the solid torus, with arbitrary boundary fluctuations. In contrast with similar calculations performed directly in the continuum, we work with a boundary at finite distance from the torus axis. We show that after taking the continuum limit on the boundary — but still keeping finite distance from the torus axis — the one—loop correction is the same as the one recently found in the continuum by Barnich and collaborators for an asymptotically flat boundary. We further calculate Hamilton’s principal function to quadratic order in the boundary fluctuations and look for a boundary field theory which would reproduce it. We identify a boundary theory with a Liouville—like coupling to the boundary metric. Integrating out this dual field reproduces the (boundary diffeomorphism invariant part of) Hamilton’s principal function. The considerations here show that bulk—boundary dualities might also emerge at finite boundaries and that discrete approaches can be helpful in identifying such dualities.

  Jeudi 12 Novembre 2015, à 14h, salle 114

Bertrand Delamotte (LPTMC, Université Paris VI) : What can be learnt from the nonperturbative renormalization group ? A personal selection of topics

Wilson’s renormalization group has long been both regarded as a nice conceptual framework and as a much too complicated method to compute anything. As for this second aspect, decisive progresses were made in the beginning of the 90’s and many nonperturbative results have been obtained since then. After a general presentation of the method, a selection of some of them will be presented. Among things that are out of reach of perturbation theory, I shall select : (i) the computation of non-universal quantities such as a critical temperature or a phase diagram and (ii) of fixed points that are not connected to the gaussian. I shall also show that critical exponents can be very accurately computed and a discussion of the convergence of the approximation schemes used will follow.

  Mercredi 14 Octobre 2015 à 15h30, salle 114

Johannes Thurigen (AEI, Golm, Germany) : Generalizing tensor models and group field theories to arbitrary boundary graphs

Group field theory is a generalization of matrix models to arbitrary spacetime dimension. Providing also geometric data in terms of holonomies, it gives at the same time also a completion of Spin foam models and a second quantized reformulation of Loop quantum gravity. Nevertheless, the focus in group field theory has been on simplicial space so far, corresponding to regular boundary graphs. In contrast, the spin network states in the kinematical Hilbert space of Loop quantum gravity have support on arbitrary graphs. In this presentation I will discuss two proposals for generalizing Group field theory to arbitrary boundary graphs : a straightforward extension in terms of multiple group fields as well as a more tractable generalization using a dual-weighting mechanism on a single field. While these constructions apply equally well to tensor models, they shed in particular a new light on the implementation of simplicity constraints in Spin foams and Group field theory.

  Jeudi 8 Octobre 2015 à 14h30, salle 114

Fabien Besnard (Pole de Recherche, EPF) : On the interplay between noncommutativity and causality

Noncommutative geometry is based on the observation that geometric notions can be defined in an algebraic way, and that the algebraic definition remains meaningful and gains in generality when the assumption of commutativity is removed. In this talk we will explain how order-theoretic notions can be introduced into this scheme, giving rise to noncommutative ordered spaces. We will show on the simple though important example of almost-commutative manifolds that two causally connected events cannot be arbitrarily close together when the algebra is not commutative. Since, as we will argue, noncommutative ordered spaces can be expected to show up in Lorentzian noncommutative geometry as well as in quantum gravity, this result might have some relevance in both areas. We will start from scratch, give many examples, and compare our framework with another approach, based on Lorentzian spectral triples, which is due to Franco.