10:00am - 12:00pmIntegral and algebraic geometric methods in the study of Gaussian random fields
Chair(s): David Ginsbourger (Idiap Research Institute and University of Bern, Switzerland), Jean-Marc Azaïs (Institut de Mathématiques de Toulouse)
Integral and algebraic geometry are at the heart of a number of contributions pertaining to the study of Gaussian random fields and related topics, not only from probabilistic and statistical viewpoints but also from the realm of interpolation and function approximation. This minisymposium will gather a team of junior researchers and established experts presenting original research results reflecting diverse challenges of geometrical and applied geometrical nature primarily involving Gaussian fields.
These encompass the study of geometrical and topological properties of sets implicitly defined by random fields such as zeros of random polynomials, excursion sets, as well as integral curves stemming for instance from filament estimation. Also, Gaussian field approximations dedicated to the estimation of excursion probabilities and more general geometric questions will be tackled, as well as algebraic methods in sparse grids for polynomial and Gaussian process interpolation.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Asymptotic normality for the Volume of the nodal set for Kostlan-Shub-Smale polynomial systems
Jean-Marc Azaïs
Institut de Mathématiques de Toulouse
We study the asymptotic variance and the CLT, as the degree goes to infinity, of the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system.
Euler characteristic and bicovariogram of random excursions
Raphaël Lachieze-Rey
Université Paris Descartes
The Euler characteristic of a planar compact smooth set is a privileged tool of geometric analysis as it is a local quantity carrying information on various macroscopic features of the set topology. We indicate here how to compute it directly from the covariograms of the set, i.e. the volumes of the intersection of the set with translated copies of itself. This approach works under hypotheses of $mathcal{C}^{1,1}$ regularity, and can be applied to excursions of $mathcal{C}^{1,1}$ bivariate functions. In the realm of random sets or fields, this identity gives the mean Euler characteristic in terms of the third order marginals.
Bayesian approach to filament estimation with a latent Gaussian random field model
Wolfgang Polonik1, Johannes Krebs2
1UC David, 2UC Davis
Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process. We survey existing results and present expansions of these results to larger classes of underlying sampling schemes and to the multivariate case.
On the universality of roots of random polynomials
Guillaume Poly
Université de Rennes I
A classic question in random polynomial theory is to determine whether the distribution of the roots depends on the distribution of the random coefficients. We will explore this question both for algebraic and trigonometric models and point out some important differences between the two regimes.