Mini-courses
In the first lecture, we begin by introducing and reviewing tools essential for compact inference. Specifically, we delve into topological data analysis methods that utilize the Hausdorff distance, along with its robust Wasserstein-based versions. Additionally, we explore the theory of metric measure spaces as conceptualized by Gromov. In our second lecture, we examine geometric-based clustering methods, ranging from k-means-based techniques to topological data analysis methods. We will also consider spectral clustering. During the third lecture, our focus shifts to statistical tests on geometric spaces, such as directional spaces. We particularly address the issue of uniformity testing and its relationship to hyperuniformity.
Hyperuniformity in point processes refers to the property where large subsamples exhibit low variance in the number of points, as well as in other linear statistics. This phenomenon has been actively studied and sought after for several decades in mathematics, condensed matter physics and statistical physics, encompassing models from determinantal processes, random matrices, random Gaussian functions or Gibbs measures. In this minicourse, we introduce formally this phenomenon in the framework of random stationary measures. We also derive the main mathematical examples coming from random matrices and statistical physics, and the macroscopic properties of hyperuniformity: good transport properties, and rigidity. We will also discuss the relations with other fields, and the numerical aspects, such as the problem of detecting and estimating hyperuniformity, or that of producing large hyperuniform samples in a reasonable time.
Talks
Jonas Jalowy: Box-Covariances of Hyperuniform Point Processes
When hyperuniform point processes are characterized by specific variance asymptotics of growing boxes, a natural question arises: What is the (relative) covariance between such two boxes? In this talk, we discuss a curious (non-)universal covariance structure of hyperuniform point processes depending on their class/exponent and identify a limiting Gaussian 'coarse grained process' counting the number of points in large boxes as a function of the position of the box. This talk is based on a joint work with Hanna Stange.
Gabriel Mastrili: Estimating the hyperuniformity exponent of spatial point processes.
Stationary hyperuniform point processes exhibit reduced fluctuations compared to a Poisson process. This property is quantified by the hyperuniformity exponent, which governs the decay of the variance of linear statistics and the spectral measure (or structure factor in physics) near the origin. This presentation introduces an estimator for the hyperuniformity exponent and examines its asymptotic behavior.
Jérémie Bigot: Regularized estimation of Monge-Kantorovich quantiles for spherical data
We introduce a regularized estimator built from entropic optimal transport, by extending the definition of the entropic map to the spherical setting. We propose a stochastic algorithm to directly solve a continuous OT problem between the uniform distribution and a target distribution on the sphere, by expanding Kantorovich potentials in the basis of spherical harmonics
TBA
Ottmar Cronie: Papangelou conditional intensities for marked Gibbs processes and their applications to point process operations
Operations, such as thinnings, superpositions and jitterings, are central in the study of point processes. Among other things, such operations can be used in different statistical settings. In this talk we start by illustrating how these operations can be obtained by first applying marking to a point process and then applying a mapping of the obtained marked point process. Once this has been established, we proceed to focus on operations on Gibbs processes. To this end, we provide a new characterisation of Papangelou conditional intensities of marked Gibbs processes. We then proceed to providing different distributional characteristics of Gibbs process on which operations have been applied.
Anthéa Monod: Fréchet Means in Varying Geometries
The Fréchet mean is a central concept in metric geometry, representing a generalized notion of an average for a given set of points; it is a key measure of central tendency as a barycenter for a given set of points in a general metric space. Unlike in Euclidean spaces, where the mean is uniquely defined as the barycenter, the Fréchet mean in general metric spaces is determined by solving an optimization problem and may not be unique. In this talk, I will overview some recent geometric and algorithmic results on the Fréchet mean studied in algebraic settings—specifically, in persistent homology, a key methodology in topological data analysis derived from algebraic topology; tropical geometry, a piecewise linear, combinatorial and polyhedral variant of algebraic geometry that has gained prominence in applications; and in polytope normed spaces, which encompass and generalize the tropical setting.
David Vernotte: Fractal behavior for nodal lines of planar continuous Gaussian fields at criticality.
Percolation for continuous Gaussian fields can be understood as a natural generalization of classical Bernoulli
percolation. Instead of working with a random configuration on a discrete lattice, one introduces $f$ a continuous, centered, and stationary Gaussian field on the Euclidean space $\mathbb{R}^2$. We are interested in the random excursion set associated to the field $f$ at the critical level $\ell_c=0$ (that is, we consider the random set of points where the field $f$ takes positive values). It is known that at arbitrarily large scales, rectangles are crossed by the excursion set with a probability that stays bounded away from $0$ and $1$. We are interested in the length of these crossings (when they exist). We adapt an argument developed by Aizenman and Burchard to show that, with high probability, any crossing of a rectangle at scale $\lambda$ must have a length of order at least $\lambda^s$ where $s>1$ is an exponent depending only on the law of the field.
Nina Javerzat: A fast algorithm for 2d Rigidity Percolation
Rigidity Percolation (RP) attracted much attention in the Soft Matter community, as an elegant framework to understand the emergence of solidity in media that not present any long-range structural order. The solidification of amorphous systems like gels, fiber networks or living tissues can indeed be understood by focusing on locally rigid structures --clusters, that grow and coalesce until one eventually percolates the whole system, giving it macroscopic mechanical stability. I will summarize the state of the art about the RP transition and its universality class, in the physics and mathematics literatures. I will then present new theoretical results that clarify the rigid clusters' coalescing mechanisms, and how they differ from the standard Connectivity Percolation (CP) problem. I will explain how to use these results to design an exact algorithm to simulate the RP transition, that scales almost linearly in the system's size. I will end with perspectives on how this can be used to understand better the RP problem.
Laure Marêché: A journey to zero-temperature U-Ising dynamics with frozen sites
The zero-temperature U-Ising dynamics with frozen sites is a model of statistical mechanics in which any element of ℤ² has a state + or -, and can change state only if there are enough elements of ℤ² with the target state around it. In this talk we will relate how this model came into study. We will begin with the physical motivation of the original Ising model, then present several models which spawned from it and led to the introduction of the zero-temperature U-Ising dynamics with frozen sites, explaining along the way the questions that were studied and the answers that were obtained.
Valentina Ros: Gaussian random fields in high dimensions: geometry through random matrix theory
Gaussian random fields defined on high-dimensional spaces emerge naturally as energy landscapes for complex physical systems, fitness landscapes for biological systems and, ore recently, cost or loss landscapes in computer science applications. They tend to be highly non-convex, with plenty of stationary points (local minima, maxima or saddles). The local curvature of Gaussian landscapes in the vicinity of their stationary points is described by random matrices belonging to invariant ensembles, deformed by both additive and multiplicative finite-rank perturbations. I will discuss two properties of these perturbed invariant ensembles, namely (i) the joint large deviation functions of their minimal eigenvalue and eigenvector, (ii) the typical overlap between eigenvectors of pairs of such matrices, correlated with each others. I will illustrate how these properties are relevant when addressing specific questions on the random landscape, such as (i) what is the statistical distribution of saddle points surrounding a given local minimum of the landscape, and (ii) what is the landscape profile along paths interpolating between two of its local minima. The motivation for these problems relies on the study of how these landscapes are explored dynamically by local stochastic dynamics, as I will briefly discuss.
Alessia Caponera: Fréchet Means in Varying Geometries
The Fréchet mean is a central concept in metric geometry, representing a generalized notion of an average for a given set of points; it is a key measure of central tendency as a barycenter for a given set of points in a general metric space. Unlike in Euclidean spaces, where the mean is uniquely defined as the barycenter, the Fréchet mean in general metric spaces is determined by solving an optimization problem and may not be unique. In this talk, I will overview some recent geometric and algorithmic results on the Fréchet mean studied in algebraic settings—specifically, in persistent homology, a key methodology in topological data analysis derived from algebraic topology; tropical geometry, a piecewise linear, combinatorial and polyhedral variant of algebraic geometry that has gained prominence in applications; and in polytope normed spaces, which encompass and generalize the tropical setting.
Michael McAuley: Topological functionals of smooth Gaussian fields
Smooth Gaussian fields arise in a variety of contexts in both pure and applied mathematics. While their geometric properties are well understood, their topological features pose deeper mathematical challenges. In this talk, I will outline the classical theory that describes the geometric behaviour of Gaussian fields, before turning to more recent developments aimed at understanding their topology using the Wiener chaos expansion.
