Integral Geometry Day in Paris
Organizers: Andreas Bernig (Goethe University Frankfurt/Main) and Vadim Lebovici (Sorbonne University)Administrative manager at IMJ-PRG: Sandrine Bédé
Where:
Salle 502 Couloir 15-25, 5ème étage (monter par la tour 25) Institut de Mathématiques Jussieu - Paris Rive Gauche, 4 Place Jussieu, 75005 Paris
PLAN
When: 2nd, October 2026
Abstract The Integral Geometry Day in Paris wishes to gather the integral geometers from France and neighbors to strengthen the connections between the various poles of this field: geometric probabilities, geometry of singularities and algebraic geometry.
Speakers Hermine Biermé (Université de Tours) Antoine Commaret (Inria Centre at Université Côte d'Azur) Agnès Desolneux (ENS Paris-Saclay) (canceled)Schedule
--------------------09h15--09h30 Welcome
--------------------09h30--10h30 Talk: Francesca Pistolato - Lipschitz-Killing curvatures for excursion sets of spin spherical random fields
Abstract: In this talk, we study the geometry of excursion sets of spin spherical random fields by means of their Lipschitz-Killing curvatures. Without requiring isotropy, it is possible to compute explicitly the expectation of these functionals in terms of the spin parameter and the level of the excursion. Moreover, we will see how to derive an explicit chaotic decomposition for the second curvature, the surface area, providing a small step forward in the study of second-order asymptotics in the high-frequency regime. The talk is based on joint works with M. Stecconi.--------------------10h30-11h00 Break
--------------------11h00-12h00 Talk: Antoine Commaret - Continuity of the normal cycle with respect to C0 convergence
Abstract: The normal cycle NX of a (possibly singular) subset X of Rd is essentially the integral current associated with the unit normal bundle of X. It allows one to recover second order information such as the curvature measures of X or their tensorial variantes. It is a crucial object in integral geometry, especially in the theory of smooth valuations developed by Alesker, where it is used to define smooth, isometry-invariant valuations. Fu's uniqueness result on normal cycles may be seen as definition of the broadest possible class of subsets of Rd for which a normal cycle can be defined; yet the extent of this class remains to be understood.In this talk I will introduce two distinct topologies on compact subsets of Rd, coming from the homotopy distance and the acyclic convergence. We show that normal cycles are continuous with respect to these topologies on sets with smooth boundaries, which may come as a surprise since these convergence do not a priori provide any control on the convergence of tangent spaces. As a corollary, we show that every compact set definable in an o-minimal structure admits a normal cycle.
--------------------12h00-14h00 Lunch--------------------
--------------------14h00-15h00 Talk: Hermine Biermé - The anisotropy of Gaussian random fields through their Lipschitz-Killing curvature densities
Abstract: I will present a joint work with Agnès Desolneux on the geometry of excursion sets of smooth stationary Gaussian fields. In this work, we focus on the effect of anisotropy on their Lipschitz-Killing curvature densities (close from area, perimeter and Euler characteristics of excursion sets for 2D fields) and propose new geometrical spectral moments. Using isoperimetric inequalities, they allow to define a new geometrical index of anisotropy.--------------------15h00-15h30 Break
--------------------15h30-16h30 Talk Agnès Desolneux - The geometry of pixelized Gaussian random fields
Abstract: I will present a joint work with Hermine Biermé. In applications, 2D random fields are observed on grids of pixels, and the question we will address is what is the relationship between the (discrete) geometry of the observed excursion sets and the true (continuous) geometry as described by Hermine in her talk ? Is there a convergence when the pixel size goes to 0 ? I will also explain how a single excursion set with unknown level can allow to recover a lot of information about the underlying Gaussian field. --------------------Image credits: Marianne Le Vexier