Reception Young Scientists
Plenary & Named Lectures
Alexei Borodin (Massachusetts Institute of Technology)
The goal of the talk is to survey the emerging field of integrable probability, whose goal is to identify and analyse exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.
Michael Cranston (UCI) Itô Prize Lecture
'Overlaps and Pathwise Localization in the Anderson Polymer Model'
Bénédicte Haas (University of Paris-Dauphine)
'Random trees constructed by aggregation'
We study a general procedure that builds random continuous trees by attaching recursively a new branch on a uniform point of the pre-existing tree. This encompasses the famous "line-breaking" construction of the Brownian tree of Aldous. Our aim is to see how the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness and Hausdorff dimension. Based on a joint work with Nicolas Curien (Université Paris-Sud).
Alan Hammond (UC Berkeley)
'Self-avoiding polygons and walks: counting, joining and closing'
'Measure valued processes and Skorokhod maps applied to them in studying the asymptotics of non Markovian queueing systems'
Measure valued processes where the measure is defined on the Borel sets of the real line have been used in the past decade to analyze and study the limits of queues with non exponential service times.
One such example is the many servers queue where the state space consists of two parts. One the number of customers in the system and the other a measure valued process that puts a unit mass on the times in service of the customers that are being served. This couple is Markovian and serves as a tool to study Law of Large Numbers and Central Limit Theorems of the system as the number of servers tends to infinity. In the former the limit is described by a PDE applied to measures, and in the latter an SPDE driven by a Gaussian martingale.
Another example comes from systems with impatient customers and there one introduces a measure that puts a unit mass on the waiting times of customers in the queue or on their residual patience when it is known. We introduce a Skorohod map acting on the space of right continuous with left limits functions with values in the space of measures on [0,∞). The motivation is to treat queuing systems in which one schedules tasks by prioritizing according to a continuous parameter. One such well-known models is the earliest-deadline-first (EDF). In this lecture I'll show how to apply the tool to obtain new formulations of fluid model equations and Weak Law of Large Numbers-scale convergence results for the EDF single server queues with time varying arrival and service rates, and the EDF many servers queue (as the number of servers tends to infinity).
Michel Ledoux (University of Toulouse) Schramm Lecture
'Stein's method, logarithmic Sobolev and transport inequalities'
Stein’s method is a most powerful tool towards probabilistic approximation results, while logarithmic Sobolev inequalities yield sharp concentration inequalities within the entropy method. The talk will emphasize recent connections between Stein's approximation method and logarithmic Sobolev inequalities via a new class of functional inequalities involving entropy, Fisher
information and Stein’s kernel. The new inequalities improve upon the classical logarithmic Sobolev and Talagrand transportation cost inequalities, and produce both bounds for normal entropic convergence and extended concentration inequalities. The investigation is developed within Markov semigroup interpolation and iterated gradients, leading in addition to abstract Malliavin-type arguments extending the Wiener chaos analysis.
Joint work with I. Nourdin and G. Peccati.
Régine Marchand (University of Lorraine)
'Random growth models with possible extinction'
In first-passage percolation, a typical random growth model, the growth is asymptotically governed by a shape theorem, and the proof relies on the use of a subadditive ergodic theorem. In models where extinction is possible, exploiting subadditive arguments is less simple.
In this talk, we will explain a way to obtain convergence results for random growth models with possible extinction. As a guideline, we will focus on the simple question of counting open paths in oriented percolation.
This is based on a joint work with Olivier Garet and Jean-Baptiste Gouéré.
Grégory Miermont (ENS, Lyon) IMS Medallion Lecture
'Random maps and Brownian surfaces'
I will explain how to obtain a random surface of a given topology as a scaling limit of random maps, focusing in particular on the sphere and disc topologies, and discuss the geometric and universality properties of the objects thus defined. This is based on joint work with Jérémie Bettinelli.
Jason Miller (Massachusetts Institute of Technology)
'Liouville quantum gravity and the Brownian map'
Sandrine Péché (University of Paris-Diderot)
'Universal versus Non universal features in random matrix theory via deformed ensembles'
We will review some recent results about universal/non universal asymptotic spectral properties of random matrices in the limit where the dimension grows to infinity. The talk focuses on deformed Gaussian ensembles of random matrices, which is the simplest ensemble of non Gaussian random matrix.
Christophe Sabot (University of Lyon)
'A random Schrödinger operator associated with the Vertex Reinforced Jump Process and the Edge Reinforced Random Walk'
The Vertex Reinforced jump Process (VRJP) is a continuous time self-interacting process with a linear reinforcement on sites (as a function of the local time). It is closely related to the famous Edge reinforced random walk (ERRW) introduced by Diaconis in the 80's. On finite graphs it can be represented as a mixture of Markov jump processes and the mixing distibution is a marginal of a supersymmetric field investigated by Disertori, Spencer, Zirnbauer.
Several results can be infered from this connection, as a phase transition between recurrence and transience in dimension d>2.
In this talk, we will review these recent results and present new developments. In particular we will introduce a random Schrödinger operator with a 2-decorrelated potential from which the mixing field can be obtained. The random potential of this operator comes from a new exponential family that can be viewed as a multivariate generalization of the inverse Gaussian.
Interesting phenomenons appear by extending this representation to infinite graphs. In particular, the transience of the VRJP is signed by the existence of a positive generalized eigenfunction of the random Schrödinger operator. From this we can infer a functional CLT at weak disorder for the VRJP and ERRW in dimension d>2.
Based on joint works with Pierre Tarrès, Margherita Disertori and Xiaolin Zeng.
Scott Sheffield (Massachusetts Institute of Technology) IMS Medallion Lecture
'Chinese dragons and mating trees: tales from the random geometry sphere'
What is the right way to think of a "random surface" or a "random planar graph"? How can one explain the dendritic patterns that appear in snowflakes, coral reefs, lightning bolts, and other physical systems, as well as in toy mathematical models inspired by these systems? How are these questions related to random walks and random fractal curves? To conformal matings of Julia sets? To string theory? To statistical mechanics? To continuum random trees?
I will survey several recent results in random planar geometry that address these issues. In doing so, I will introduce several random objects that are in some sense "canonical." They appear in many contexts and are characterized by special symmetries, much the way the deterministic sphere is a canonical object, characterized by symmetry. These random objects include curves, metrics, distributions, measures, surfaces, growth patterns, and planar trees. A recurring theme of the talk will be that these canonical structures are much more closely and intricately linked than we previously realized.
Based on joint work with Bertrand Duplantier and Jason Miller.
Andrew Stuart (University of Warwick)
'Data Assimilation -- New Challenges in Random and Stochastic Dynamical Systems'
The proliferation of data, together with carefully crafted mathematical models, means that in many areas of science and engineering the data and model should be considered in conjunction. Taking this perspective leads to new and interesting challenges in mathematical analysis. Since the data is often noisy, and in some cases the model is uncertain, interaction with probability arises naturally. This talk will be devoted to studying the conjunction of data and model in the context of time-evolving problems, a subject frequently referred to as data assimilation.
Predicting the state of a chaotic dynamical system whose initial condition is uncertain is difficult even in short time intervals. However in the presence of partial and noisy observations of the system, the question arises as to whether the initial uncertainty can be kept small in the infinite time horizon. We show that studying this problem leads to interesting questions relating to nonlinear Markov chains in discrete time. We also describe continuous time limits, leading to new nonlinear stochastic PDEs, such as families of damped-driven interacting Navier-Stokes equations, coupled through their empirical covariance. Theorems and numerical illustrations concerning this subject will be presented.
Terence Tao (UCLA) Doob Lecture
'Universality for random matrices and random polynomials'
We survey some of the progress in recent years in understanding the universality phenomenon for various models of random matrices (in particular, Wigner matrices) and random polynomials (in particular, Kac or Weyl type polynomials).
Augusto Teixeira (IMPA)
'Percolation and local isoperimetric inequalities'
Boris Tsirelson (Tel Aviv University) Lévy Lecture
'Moderate deviations for random complex zeroes'
The normal approximation can work on the logarithmic scale moderately far from the center of a distribution, which is the so-called Moderate Deviations Principle, well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Nevertheless it can be proved for stationary random comples zeroes, as will be explained.