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Abstract: The number F(h) of imaginary quadratic fields with class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h x is less or equal to y 100 was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h→∞ and determined its average order.
For odd h we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. [...]

Recording during the thematic meeting: « Dynamics and Graphs over Finite Fields: Algebraic, Number Theoretic and Algorithmic Aspects» the March 31, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract : We discuss subshifts of finite type (tilings) that combine virtually opposite properties, being at once very simple and very complex. On the one hand, the combinatorial structure of these subshifts is rather simple: we require that all their configurations are quasiperiodic, or even that all configurations contain exactly the same finite patterns (in the last case a subshift is transitive, i.e., irreducible as a dynamical system). On the other hand, these subshifts are complex in the sense of computability theory: all their configurations are non periodic or even non-computable, or all their finite patterns have high Kolmogorov complexity, the Turing degree spectrum is rather sophisticated, etc.
We start with the simplest example of such centaurisme ² with an SFT that is minimal and contains only aperiodic (and quasiperiodic) configurations. Then we discuss how far these heterogeneous properties can be strengthened without getting mutually exclusive.
This is a joint work with Bruno Durand (Univ. de Montpellier).

Recording during the thematic meeting : "Computability, Randomness and Applications" the June 22, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Résumé : Dans la deuxième partie de ce cours nous considérerons un problème lié, celui des algorithmes compétitifs. Dans le cadre de l'algorithmique « en-ligne », les caractéristiques d'une instance d'un problème ne sont découvertes qu'au fur et à mesure du traitement de l'instance (comme on ne découvre l'histoire d'un livre qu'au fur et à mesure où on en lit des pages). Ne pas connaître à l'avance toutes les caractéristiques d'une instance interdit souvent - mais pas toujours - de construire un algorithme optimal. Nous montrerons, entre autres, comment utiliser la technique de l'adversaire pour établir une borne inférieure au facteur de compétitivité de tout algorithme en-ligne (cette fois-ci en dehors de toute notion de complexité).

Enregistré pendant le session thématique « Algorithme et programmation » le 3 mai 2016 au Centre International de Rencontres Mathématiques (Marseille, France)

Réalisation: Guillaume Hennenfent

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Recording during the thematic Jean-Morlet Chair - Doctoral school: "Introduction to relative aspects in representation theory, Langlands functoriality and automorphic forms" the May 19, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract : k-abelian singletons in connection with Gray codes for Necklaces. This work is based on [1]. We are interested in the equivalence classes induced by k-abelian equivalence, especially in the number of the classes containing only one element, k-abelian singletons. By characterizing k-abelian equivalence with k-switchings, a sort of rewriting operation, we are able to obtain a structural representation of k-abelian singletons. Analyzing this structural result leads, through rather technical considerations, to questions of certain properties of sets of vertex-disjoint cycles in the de Bruijn graph dBΣ(k−1) of order k−1. Some problems turn out to be equivalent to old open problems such as Gray codes for necklaces (or conjugacy classes). We shall formulate the problem in the following. [...]

Recording during the thematic meeting: « Combinatorics on Words
» the March 16, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract : Recall that normality is a elementary form of randomness: an infinite word is normal to a given alphabet if all blocks of symbols of the same length occur in the word with the same asymptotic frequency. We consider a notion of independence on pairs of infinite words formalising that two words are independent if no one helps to compress the other using one-to-one finite transducers with two inputs. As expected, the set of independent pairs has Lebesgue measure 1. We prove that not only the join of two normal words is normal, but, more generally, the shuffling with a finite transducer of two normal independent words is also a normal word. The converse of this theorem fails: we construct a normal word as the join of two normal words that are not independent. We construct a word x such that the symbol at position n is equal to the symbol at position 2n. Thus, x is the join of x itself and the subsequence of odd positions of x. We also show that selection by finite automata acting on pairs of independent words preserves normality. This is a counterpart version of Agafonov's theorem for finite automata with two input tapes.
This is joint work with Olivier Carton (Universitéé Paris Diderot) and Pablo Ariel Heiber (Universidad de Buenos Aires).

Recording during the thematic meeting : "Computability, Randomness and Applications" the June 21, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.
I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.

Recording during the thematic meeting: "Spectral theory of novel materials" the April 19, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: This talk is based on a work jointly with Timothy Budd (Copenhagen), Nicolas Curien (Orsay) and Igor Kortchemski (Ecole Polytechnique).
Consider a self-similar Markov process X on [0,∞) which converges at infinity a.s. We interpret X(t) as the size of a typical cell at time t, and each negative jump as a birth event. More precisely, if ΔX(s)=−y [is less than] 0, then s is the birth at time of a daughter cell with size y which then evolves independently and according to the same dynamics. In turn, daughter cells give birth to granddaughter cells each time they make a negative jump, and so on.
The genealogical structure of the cell population can be described in terms of a branching random walk, and this gives rise to remarkable martingales. We analyze traces of these mar- tingales in physical time, and point at some applications for self-similar growth-fragmentation processes and for planar random maps.

Recording during the thematic meeting : "Random trees and maps: probabilistic and combinatorial aspects" the June 7, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: We consider the family of rooted planar maps MΩ where the vertex degrees belong to a (possibly infinite) set of positive integers Ω. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner

Recording during the thematic meeting : "Random trees and maps: probabilistic and combinatorial aspects" the June 8, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: Condensed matter electronic systems endowed with odd time-reversal symmetry (TRS) (a.k.a. class AII topological insulators) show topologically protected phases which are described by an invariant known as Fu-Kane-Mele index. The construction of this in- variant, in its original form, is specific for electrons in a periodic background and is not immediately generalizable to other interesting physical models where different forms of TRS also play a role. By exploiting the fact that system with an odd TRS (in absence of disorder) can be classified by Quaternionic vector bundles, we introduce a Quaternionic topological invariant, called FKMM-invariant, which generalizes and explains the topological nature of the Fu-Kane-Mele index. We show that the FKMM-invariant is a universal characteristic class which can be defined for Quaternionic vector bundles in full generality, independently of the particular nature of the base space. Moreover, it suffices to discriminate among different topological phases of system with an odd TRS in low dimension. As a particular application we describe the complete classification over a big class of low dimensional involutive spheres and tori. We also compare our classification with recent results concerning the description of topological phases for two-dimensional adiabatically perturbed systems.
Joint work with: K. Gomi.

Recording during the thematic meeting: "Spectral theory of novel materials" the April 19, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety X play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a point, or even a reduced point if there are no infinitesimal deformations. In this case one says that X is rigid, respectively inifinitesimally rigid.
A basic example is projective space, which is the only example in dimension 1. In the case of surfaces, infinitesimally rigid surfaces are either the Del Pezzo surfaces of degree ≥ 5, or are some minimal surfaces of general type.
As of now, the known surfaces of the second type are all projective classifying spaces (their universal cover is contractible), and have universal cover which is either the ball or the bidisk (these are the noncompact duals of P2 and P1×P1 ), or are the examples of Mostow and Siu, or the Kodaira fibrations of Catanese-Rollenske.
Motivated by recent examples constructed with Dettweiller of interesting VHS over curves, which we shall call BCD surfaces, together with ingrid Bauer, we showed the rigidity of a class of surfaces which includes the Hirzebruch-Kummer coverings of the plane branched over a complete quadrangle.
I shall also explain some results concerning fibred surfaces, e.g. a criterion for being a K(π,1)-space; I will finish mentioning other examples and several interesting open questions.

Recording during the thematic meeting: "Topology of complex algebraic varieties" the June 1, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: Superfluids are remarkable because they lack mechanisms of viscous dissipations, and because vorticity is concentrated in thin vortex lines - a property which arises from the existence and uniqueness of a macroscopic wave function. In this talk I shall review recent experiments and numerical simulations which reveal analogies and differences between the flow of ordinary fluids and the flow of superfluids. In particular, I shall describe conditions under which, in a homogeneous isotropic turbulent superfluid away from boundaries, the distribution of kinetic energy over the length scales is similar to the classical Kolmogorov distribution, and new insight into the properties of superfluid flow near boundaries.

Recording during the thematicmeeting : "New challenges in mathematical modelling and numerical simulation of superfluids" the June 27, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: We propose a platform for finding the global minimum of XY Hamiltonian with polariton graphs. We derive an approximate analytic solution to the spinless complex Ginzburg-Landau equation that describes the density and kinetics of a polariton condensate under incoherent pumping. The analytic expression of the wavefunction is used as the building block for constructing the XY Hamiltonian of two-dimensional polariton graphs. We illustrate examples of the quantum simulator for various classical magnetic phases on some simple lattice geometries: linear, triangular, square.

Recording during the thematicmeeting : "New challenges in mathematical modelling and numerical simulation of superfluids" the June 30, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Résumé : Les processus de fragmentation sont des modèles aléatoires pour décrire l'évolution d'objets (particules, masses) sujets à des fragmentations successives au cours du temps. L'étude de tels modèles remonte à Kolmogorov, en 1941, et ils ont depuis fait l'objet de nombreuses recherches. Ceci s'explique à la fois par de multiples motivations (le champs d'applications est vaste : biologie et génétique des populations, formation de planètes, polymérisation, aérosols, industrie minière, informatique, etc.) et par la mise en place de modèles mathématiques riches et liés à d'autres domaines bien développés en Probabilités, comme les marches aléatoires branchantes, les processus de Lévy et les arbres aléatoires. L'objet de ce mini-cours est de présenter les processus de fragmentation auto-similaires, tels qu'introduits par Bertoin au début des années 2000s. Ce sont des processus markoviens, dont la dynamique est caractérisée par une propriété de branchement (différents objets évoluent indépendamment) et une propriété d'auto-similarité (un objet se fragmente à un taux proportionnel à une certaine puissance fixée de sa masse). Nous discuterons la construction de ces processus (qui incluent des modèles avec fragmentations spontanées, plus délicats à construire) et ferons un tour d'horizon de leurs principales propriétés.

Enregistré pendant les "Journées ALEA" le 10 mars 2016 au Centre International de Rencontres Mathématiques (Marseille, France)

Réalisation: Guillaume Hennenfent

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Résumé : De nombreux problèmes d'optimisation sont NP-complets. Nous ne connaissons pas de problème NP-complet qui admette un algorithme optimal de résolution s'exécutant en temps polynomial en la taille de l'instance (sinon P=NP serait établi), et l'intuition commune est que P =/= NP. Pour ces problèmes, la recherche de solutions optimales peut donc être prohibitive. Les algorithmes d'approximation offrent un compromis intéressant: par définition, ils s'exécutent en temps polynomial et fournissent des solutions dont la qualité est garantie. Nous introduirons la notion d'algorithme d'approximation et de schéma d'approximation en temps polynomial, et nous illustrerons ces notions sur de nombreux exemples. Nous montrerons également comment établir qu'un problème n'admet pas d'algorithme d'approximation (à moins que P=NP), ou comment établir une borne inférieure au facteur d'approximation de tout algorithme d'approximation (sauf si P=NP).

Enregistré pendant le session thématique « Algorithme et programmation » le 3 mai 2016 au Centre International de Rencontres Mathématiques (Marseille, France)

Réalisation: Guillaume Hennenfent

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