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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: Sometimes, you feel you're spoilt for choice: there are so many good predictors that you could use! Why select and focus on just one? I will review the framework of robust online aggregation (also known as prediction of individual sequences or online aggregation of expert advice). This setting explains how to combine base forecasts provided by ensemble methods. No stochastic modeling is needed and the performance achieved is comparable to the one of the best (constant convex combination of) base forecast(s). I will illustrate the technology on various data sets, including electricity consumption and exchange rates. More importantly, I will point out open issues, both on the theoretical and on the practical sides.

Recording during the Thematic month on statistics - Week 1: "Statistical learning" the February 4,2016 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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The study of efficient communication communication using tools from information theory has led to many interesting results for basic problems in the last few decades. In this mini-course, we shall learn the basic concepts of communication complexity and see some its applications to distributed computing, data structures and complexity theory, stressing the central role played by information theory in this arena.


Recording during the thematic meeting: «Nexus of Information and Computation Theories » theJanuary 26, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

Abstract: In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will discuss an infinite dimensional Hamilton-Jacobi equation for the quasi-potential of stationary non equilibrium states, fluctuation-dissipation relationships, the underlying Hamiltonian structure, a relation with work and Clausius inequality, a large deviations functional for the current flowing through a system.

Recording during the thematic meeting: "Nonequilibrium: physics, stochastics and dynamical systems" the January 19, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: 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: In classical probability theory, the law of large numbers and the central limit theorem provide sharp guarantees on how the average of a large number of independent and identically distributed random variables concentrates around its mean. It has however come to be discovered that ​such​ ​behavior is shared by ​many ​functions of independent random variables that don't depend too much on any one variable. Quantifying this ​concentration ​has become an important research area and has found applications in numerous fields.

In this tutorial, we will explore some of the fundamental inequalities illustrating measure concentration. Specific topics include variance bounds, the entropy method, and the transportation method. We will also discuss some applications of these results.

Background: A basic familiarity with probability theory will be helpful​;​ a graduate-level first course in probability theory will be more than sufficient. We will spend some time reviewing this ​​material and no pre-requisites are assumed.


Recording during the thematic meeting: «Nexus of Information and Computation Theories » theJanuary 28, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

Abstract: The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

Recording during the thematic meeting: "Nonequilibrium: physics, stochastics and dynamical systems" the January 20, 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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Recording during the thematic meeting: «Nexus of Information and Computation Theories » theJanuary 26, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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: A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.

Recording during the thematic meeting: "Spectrum of random graphs" the January 7, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Abstract: Random forests are among the most popular supervised machine learning methods. One of their most practically useful features is the possibility to derive from the ensemble of trees an importance score for each input variable that assesses its relevance for predicting the output. These importance scores have been successfully applied on many problems, notably in bioinformatics, but they are still not well understood from a theoretical point of view. In this talk, I will present our recent works towards a better understanding, and consequently a better exploitation, of these measures. In the first part of my talk, I will present a theoretical analysis of the mean decrease impurity importance in asymptotic ensemble and sample size conditions. Our main results include an explicit formulation of this measure in the case of ensemble of totally randomized trees and a discussion of the conditions under which this measure is consistent with respect to a common definition of variable relevance. The second part of the talk will be devoted to the analysis of finite tree ensembles in a constrained framework that assumes that each tree can be built only from a subset of variables of fixed size. This setting is motivated by very high dimensional problems, or embedded systems, where one can not assume that all variables can fit into memory. We first consider a simple method that grows each tree on a subset of variables randomly and uniformly selected among all variables. We analyse the consistency and convergence rate of this method for the identification of all relevant variables under various problem and algorithm settings. From this analysis, we then motivate and design a modified variable sampling mechanism that is shown to significantly improve convergence in several conditions.

Recording during the Thematic month on statistics - Week 1: "Statistical learning" the February 2,2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Recording during the thematic meeting: «Nexus of Information and Computation Theories » theJanuary 27, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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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