Webinar of the team UMI - PRISMA (http://www.umi-prisma.polito.it/)

PRISMA webinars follow a colloquium-style format designed to foster exchange and discussion within the Italian probability and statistics community. Each session features two speakers, who give two closely connected 30-minute talks providing the community with a perspective on their research area. Over the past few years, recordings of the seminars have been made available on the UMI YouTube channel:


The next event is scheduled for Tuesday, Oct 6, 2026. The speakers will be Michele Stecconi (Université du Luxembourg) and Anna Paola Todino (Università del Piemonte Orientale), who will speak on:

Statistics on Yau's conjecture: variance asymptotics and Wiener-Itô chaos decomposition for Gaussian nodal volumes

according to the following schedule

16:00 Stecconi (Statistics on Yau’s conjecture: variance asymptotics)
16:30 Break and discussions
16:45 Todino (The Wiener-Itô chaos decomposition for Gaussian nodal volumes)
17:15 Conclusions and discussions

The abstract can be found below. The seminars will be streamed on Webex at the following link:

https://unimib.webex.com/unimib-it/j.php?MTID=mad4e0bc0268687d5172a5217fffbd59a

Numero Riunione:
2744 110 0273

Password Riunione:
Abu39JQkaQ4

We look forward to seeing many of you there!

Luciano Campi, Maurizia Rossi

Abstract (Stecconi). Yau's conjecture predicts that the nodal set (the set of zeroes) of a Laplace eigenfunction on an n-dimensional compact Riemannian manifold has volume proportional to the frequency. Berry's conjecture suggests a statistical viewpoint: at high frequency, eigenfunctions should look locally like monochromatic random waves. We give Yau's conjecture a probabilistic formulation and study how much the nodal volume of a random wave fluctuates around the predicted value, depending on the spectral window. For wide windows, of size proportional to the frequency, the variance has exactly the order 1/N, where N is the number of eigenfunctions in the window, on every manifold. For the monochromatic window, on manifolds without conjugate points (negatively curved ones, for instance), we obtain an upper bound smaller than the square of the best one previously known. This improvement reveals that Berry's cancellation holds on such manifolds: the nodal volume of monochromatic waves fluctuates strictly less than it does for wider ones, and strictly less than the volume of any non-zero level set. The proofs rely fundamentally on a new Wiener chaos decomposition of nodal volumes, discussed in the subsequent talk. 

Abstract (Todino). Analyzing the fluctuations of Gaussian zero sets often requires decomposing the nodal volume into orthogonal components known as Wiener-Itô chaos. Historically, this approach has suffered from severe computational complexity, especially in high dimensions, restricting its applicability to highly symmetric spaces. In this talk, we introduce a new, remarkably simpler formula for the chaos expansion of smooth Gaussian fields on arbitrary Riemannian manifolds. We will show how this new perspective drastically reduces the complexity of variance computations, moving from products of many Hermite polynomials to just four, and removes isotropy requirements. 

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Maurizia Rossi

Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano-Bicocca