Francesca Cottini
Gaussian limits for polynomial chaos and 2d directed polymers
Monday, May 22nd, 1:10-1:50, MNO 1.050.

Abstract

We will present a general and novel criterion, based only on second moment assumptions, to show the convergence towards a Gaussian limit for polynomial chaos, i.e. multi-linear polynomials of independent random variables. This result is motivated by the study of 2d directed polymers and of the related 2d Stochastic Heat Equation, whose asymptotic behavior has been widely investigated in recent years. In particular, we will show how this criterion allows us to recover the existing results in a simpler way and, furthermore, to obtain new information on our models of interest.

Pierre Perruchaud [website]
A story involving random functions, infinite-dimensional Brownian motion, Morse singularities, and the Chern-Gauss-Bonnet theorem
Monday, May 8th, 1:10-1:50, MNO 1.040.

Abstract

As an illustrative first example, consider the zero set of a smooth function on the sphere. If the function is close to the constant 1, then the zero locus is empty ; likewise if it is close to the constant -1. However, by the intermediate value theorem, if we deform the first function into the second, then at some point the zero set is going to be non-empty. Under reasonable assumptions, it will be first be a point, then a loop, then it will grow in size, then possibly the topology will become richer (more connected components), then it will start to shrink, until it is just a loop, shrinking to a point, and disappearing. The bulk of my talk will focus on this question: how does the topology and geometry of the zero set of a Brownian motion of smooth sections of a bundle behave? It is motivated, among other results, by the probabilistic proof of the Chern-Gauss-Bonnet theorem of L. Nicolaescu, linking the metric quantities associated to the covariance of a Gaussian random field to the topological properties of a generic zero set, a motivation I will try to convey.

Georgi Baklicharov [website]
Assumption-Lean Quantile Regression
Monday, April 24th, 1:10-1:50, MNO 1.040.

Abstract

Quantile regression is a powerful tool for detecting varying associations across different parts of the dependent variable's distribution. However, when using quantile regression to parameterize the conditional association between an exposure and an outcome, given covariates, two potential issues are often ignored. Firstly, the exposure coefficient estimator may not converge to a meaningful quantity when the model is misspecified, and secondly, variable selection methods may induce excess uncertainty, rendering inferences overly optimistic. In this work, we address these issues by introducing a nonparametric main effect estimand that still captures the (conditional) association of interest, even when the quantile model is misspecified. This estimand is estimated using the efficient influence function under the nonparametric model, allowing for the incorporation of data-adaptive procedures such as variable selection and machine learning. Our approach provides a flexible and reliable method for detecting associations that is robust to model misspecification and excess uncertainty induced by variable selection methods.

Mingkun Liu [website]
Random multi-geodesics on hyperbolic surfaces of large genus
Monday, April 17th, 1:10-1:50, MNO 1.040.

Abstract

On a hyperbolic surface, a closed geodesic is said to be simple if it does not intersect itself, and a multi-geodesic is a disjoint union of simple closed geodesics. In this talk, I will explain how to pick a random multi-geodesic, and present an attempt to answer the following question: what is the shape of a random multi-geodesic on a hyperbolic surface of large genus? We will see that it looks like a random permutation, and in particular, the average lengths of its first three largest connected components are approximately, 75.8%, 17.1%, and 4.9%, respectively, of the total length. This is joint work with Vincent Delecroix.

Michele Stecconi [website]
Riemannian geometry encoded in a Gaussian field
Monday, March 6th, 1:10-1:50, MNO 1.050.

Abstract

There is a one to one correspondence between Gaussian fields on a manifold and immersions of the manifold in a separable Hilbert space, up to orthogonal transformations. The Gaussian field provides a whole new language to talk about the geometry of the manifold and offers new interpretations of some things. In particular, the metric, the volume, the second fundamental form, the Levi-Civita connection and the Riemann tensors can all be described easily in terms of the Gaussian field. Moreover, the Gauss-Bonnet theorem (“the Euler characteristic equals the integral of the Pfaffian of the curvature”) can proved and generalized as a consequence of Kac-Rice formula.

Francesca Pistolato
A brief introduction to the MRW Model
Monday, February 20th, 1:10-1:50, MNO 1.040.

Abstract

We will introduce the Monochromatic Random Wave Model as proposed by Zelditch in 2009. We will review the results known so far about its nodal length on different spaces and address a small-scale reduction principle that exploits an analogous CLT proved by Dierickx, Nourdin, Peccati, Rossi in 2022.

Giovanni Peccati [website]
SOS OSSS
Monday, February 6th, 1:10-1:50, MNO 1.010.

Abstract

The OSSS inequality (O'Donnel, Saks, Servedio & Schramm, 2005) is a bound on the first absolute centered moment of random variables defined on the discrete cube (endowed with a product measure), whose proof is based on the use of stopping sets and randomized algorithms. When restricted to Boolean functions, it yields a spectacular improvement of the discrete Poincaré inequality, which is often (drastically) better than the celebrated Talgrand's \(L^1\text{–}L^2\) estimates. The OSSS inequality is at the core of the Duminil-Copin/Raoufi/Tassion “revolutionary” theory of sharp phase transitions in percolation (2017-2020), and is nowadays considered a fundamental tool by several communities. Although such a result has been adapted to a variety of situations (non-product measures and configuration spaces, for instance) several basic questions around it continue to stay open, for instance: (i) can one prove an OSSS inequality without assuming the existence of a randomized algorithm?; (ii) can one prove a genuine \(L^p\) OSSS inequality for \(p>1\)?
The aim of my talk is to state and prove the original OSSS inequality, and to illustrate some open problems surrounding it.

Rafał Martynek [website]
On the expected suprema of positive processes
Monday, December 12th, 1:10-1:50, chalk room (MNO).

Abstract

In this talk I would like to present the result conjectured by M. Talagrand concerning the expected suprema of so-called selector processes which are the sums of biased Bernoulli variables multiplied by non-negative coefficients. Recently, J. Park and H. T. Pham proved it and extended it to positive empirical processes. Together with W. Bednorz and R. Meller we managed to provide alternative arguments for both of the results and apply it also to positive infinitely divisible processes. I would like to discuss the highlights of the main construction and fill the WiP part with some further questions.

Louis Gass
Zeros and critical points of Gaussian fields: a moment-based approach?
Monday, November 28th, 1:10-1:50, MNO 1.050.

Abstract

The study of the zero set associated with a (vector valued) Gaussian field is a very classical topic in probability. In this talk, I will present a moment-based approach in order to study local and global properties of this zero set. The analysis is based on the Kac-Rice formula, which gives an integral expression for the p-th moment of the nodal volume of a random field. I will present a conjecture on the finiteness of the p-th moment and its connection with multivariate interpolation theory. The finiteness of the first moments leads to interesting properties concerning local attraction/repulsion of critical points of random fields.

Akram Heidari
Parameter estimation of discretely observed interacting particle systems
Monday, November 7th, 1:10-1:50, MNO 1.050.

Abstract

In this talk, we consider the problem of joint parameter estimation for drift and volatility coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval \([0,T]\), we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step \(\Delta_n\) goes to 0 and the number of particles \(N\) goes to infinity, and asymptotically normal when additionally, the condition \(\Delta_nN\to0\) holds. The talk is based on joint work with C. Amorino, V. Pilipauskaite and M. Podolskij.

Laurent Loosveldt
Wavelet-type approximation of Hermite processes
Monday, October 17th, 1:10-1:50, MNO 1.020.

Abstract

In recent years, wavelet analysis has proved to be an efficient tool to study stochastic processes. It allows to express some of them as a random series involving smooth functions. This feature is particularly appreciated when it comes to numerically simulate such processes or study paths regularity. For instance, one can cite the wavelet-type series of fractional Brownian motion or Rosenblatt process. These two processes belong to the class of Hermite processes, fractional Brownian motion being of order 1 and Rosenblatt process of order 2. As of today, there is no wavelet-type representation for Hermite processes of higher order. The aim of this talk is to fill this gap. It is a joint work with Antoine Ayache and Julien Hamonier from the University of Lille.