Les $q$-séries (parfois appelées séries basiques hypergéométriques) sont des séries construites en utilisant les $q$-factorielles $(a;q)_n := (1-a)(1-aq)...(1-aq^{n-1}).$ On les retrouve dans de nombreux domaines des mathématiques tels que la combinatoire, la théorie des nombres, la théorie des groupes et la physique mathématique. Sous l'influence de Ramanujan, les $q$-séries ont souvent été étudiées en relation avec les partitions d'entiers. Nous commencerons par une introduction générale aux $q$-séries et étudierons quelques identités classiques, puis nous verrons...

Dans les années 1970, William Tutte développa une approche algébrique, basée sur des «invariants», pour résoudre une équation fonctionnelle qui apparait dans le dénombrement de triangulations colorées. La transformée de Laplace de la distribution stationnaire du mouvement brownien réfléchi dans des cônes satisfait une équation similaire. Pour être applicable, cette méthode requiert l’existence de deux fonctions appelées respectivement invariant et fonction de découplage. Tous les modèles ont des invariants mais on démontre que l’existence de...

Angel and Schramm ont étudié en 2003 la limite locale des triangulations uniformes. La loi limite, appelée UIPT (pour Uniform Infinite planar Triangulation) a depuis été pas mal étudiée et est plutôt bien comprise. Dans cet exposé, je vais expliquer comment on peut obtenir un résultat analogue à celui d’Angel et Schramm mais lorsque les triangulations ne sont plus uniformes mais distribuées selon un modèle d’Ising. Une partie importante de la preuve consiste à étudier...

Le but de ce cours sera de présenter quelques techniques liées aux processus de Schur, dans le cadre le plus simple de la mesure de Plancherel sur les partitions d'entiers. La mesure de Plancherel est une mesure sur l'ensemble des partitions d'un entier n, où une partition donnée apparaît avec une probabilité proportionnelle au carré de son nombre de tableaux de Young standard. Cette mesure apparaît très naturellement en lien avec le fameux problème de...

Le but de ce cours sera de présenter quelques techniques liées aux processus de Schur, dans le cadre le plus simple de la mesure de Plancherel sur les partitions d'entiers. La mesure de Plancherel est une mesure sur l'ensemble des partitions d'un entier n, où une partition donnée apparaît avec une probabilité proportionnelle au carré de son nombre de tableaux de Young standard. Cette mesure apparaît très naturellement en lien avec le fameux problème de...

We present a novel methodology for causal inference based on an invariance principle. It exploits the advantage of heterogeneity in larger datasets, arising from different experimental conditions (i.e. an aspect of "Big Data"). Despite fundamental identifiability issues, the method comes with statistical confidence statements leading to more reliable results than alternative procedures based on graphical modeling. We also discuss applications in biology, in particular for large-scale gene knock-down experiments in yeast where computational and statistical...

Twistor spaces of K3 surfaces are non-Kähler compact complex manifolds which play a fundamental role in the moduli theory of K3 surfaces. They come equipped with a holomorphic submersion to the complex projective line which under the period map corresponds to a twistor line in the K3-period domain. In this talk I will explain how one can view a twistor line as a certain base point in the linear cycle space of the period domain....

Let $(X,0)$ be an ICIS of dimension $2$ and let $f :(X,0)\to\mathbb{C} ^2$ be a map germ with an isolated instability. Given $F : (\mathcal{X} , 0) \to (\mathbb{C} \times \mathbb{C}^2, 0)$ a stable unfolding of $f$, we look to the invariants related to the family $f_s$ and we find relations between them. We obtain necessary and sufficient conditions for $F$ to be Whitney equisingular. (Joint work with B. Orfice-Okamoto and J. N. Tomazella)

Start the video and click on the track button in the timeline to move to talk 1, 2 and to the discussion. - Talk 1: Paul Apicella - Striatal dopamine and acetylcholine mechanisms involved in reward-related learning The midbrain dopamine system has been identified as a major component of motivation and reward processing. One of its main targets is the striatum which plays an important role in motor control and learning functions. Other subcortical neurons...

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case. Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the...

We study the entanglement of radical extensions over the rational numbers, and describe their Galois groups as subgroups of the full automorphism group of the multiplicative groups involved. A character sum argument then yields the densities (under GRH) for a wide class of primitive root problems in terms of simple ‘local’ computations.

In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates. We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity. BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local...

We will report on an ongoing project to understand geometric Langlands in genus one, in particular a version that depends only on the topology of the curve (as appears in physical descriptions of the subject). The emphasis will be on the realization of the automorphic and spectral categories as the center/cocenter of the affine Hecke category. We will mention work with D. Ben-Zvi and A. Preygel that accomplishes this on the spectral side, then focus...

In this talk we introduce a class of statistics for spatial data that is observed on an irregular set of locations. Our aim is to obtain a unified framework for inference and the statistics we consider include both parametric and nonparametric estimators of the spatial covariance function, Whittle likelihood estimation, goodness of fit tests and a test for second order spatial stationarity. To ensure that the statistics are computationally feasible they are defined within the...

Data collection and subsequent interpretation plays an important role in many ecological problems. Quantities such as the total population size and/or average population density are often evaluated based on data collected as a result of a sampling procedure. Accurate evaluation of the above quantities is crucial in ecological applications where they are used for making decision about means of control. Examples include management of pest insects in agricultural fields, prevention of plant diseases and control...

Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we develop the $L^q$-$L^r$ decay estimates of the evolution operator $T(t,s)$ as...

A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should...

Les missions des correspondants Mathrice pour l'agenda, l'annuaire...

Cet exposé présente un certain nombre d'exemples d'exercices de programmation sur les arbres qui peuvent être effectués dans les premières années d'université. Les programmes étant eux-mêmes des arbres, écrire des programmes qui opèrent sur des arbres permet d'écrire des programmes qui opèrent sur d'autres programmes.

Using a Lagrangian framework, we study overdamping phenomena in gyroscopic systems composed of two components, one of which is highly lossy and the other is lossless. The losses are accounted for by a Rayleigh dissipative function. We prove that selective overdamping is a generic phenomenon in Lagrangian systems with gyroscopic forces and give an analysis of the overdamping phenomena in such systems. Central to the analysis is the introduction of the notion of a dual...

In this talk, we will first review some of the analogies between full groups of measure-preserving equivalence relations and the symmetric group over the integers, which have been used by A. Eisenmann and Y. Glasner to provide interesting examples of invariant random subgroups (IRSs) of the free group. We will then see how the notion of cost, introduced by G. Levitt, naturally enters this picture. After that, we will explain how a stronger analogy between...

We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method. We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero....

There are five superstring theories, all formulated in 9+1 spacetime dimensions; lower-dimensional theories are studied by taking some of the spatial dimensions to be compact (and small). One of the remarkable features of this setup is that the same lower-dimensional theory can often be realized by pairing different superstring theories with different geometries. The focus of these lectures will be on the mathematical implications of some of these physical “dualities.” Our main focus from the...

In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed. Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 dimension, of the type occurring in traffic jams. Simple examples of such...