Dear colleagues,

we are happy to announce the following two talks

*Giacomo Cherubini* (INdAM & Univ. Roma La Sapienza) *From arithmetic random waves to universal quadratic forms*

*Abstract: *Arithmetic random waves are eigenfunctions of the Laplacian on the torus R^2/Z^2, written as sums of sines and cosines with random coefficients. In agreement with Yau's conjecture, we know that the length of the nodal set (i.e. the zero set) is of size proportional to the square root of the eigenvalue and in 2013, Krishnapur-Kurlberg-Wigman proved an asymptotic formula for its fluctuations. In this talk we will give an overview of the situation for the torus R^n/Z^n, highlighting the link between eigenspaces and quadratic forms and arriving at the statement of the "290-Theorem", which characterizes when a positive definite integral quadratic form represents all positive integers.

*Riccardo W. Maffucci* (Univ. Coventry) *Nodal intersections and correlation structure for random waves*

*Abstract: *This talk is based on joint works with V. Cammarota, D. Marinucci, and M. Rossi. Several recent papers studied the ensemble of Laplace Toral eigenfunctions, and their randomisation ‘arithmetic waves’, introduced in 2007 by Oravecz-Rudnick-Wigman. These waves are related to the arithmetic of writing a number as the sum of d squares, where d is the dimension. One question is the nodal (zero) volume of the wave, in the high energy limit. Restrictions of the wave to submanifolds have also attracted recent attention. We considered the length of intersection between the nodal set, and a fixed reference surface in 3 dimensions. Expectation, and variance in a general scenario were prior work. In the generic setting we prove a CLT. We will discuss (finer order) variance and (non-Gaussian) limiting distribution in the case of ‘static’ surfaces (e.g. sphere). We will also discuss recent developments on the correlation structure between different functionals of the wave. Under certain assumptions, there is asymptotic full correlation between intersection length and nodal area.

Date and time: Thursday* May 9, 14:00-15:30 (Rome time zone)* Place: *Aula 3014, Dip. Matematica e Applicazioni, Univ. Milano-Bicocca, Via R. Cozzi 55, Milano*

*Webex meeting link:* https://unimib.webex.com/unimib-it/j.php?MTID=m1c4d20c1f6c3199f3682809e0f4af...

*Meeting number:* 2742 624 6987

*Meeting password:* mmYN7WusD46

*These talks are part of the* *(PMS)^2: Pavia-Milano Seminar series on Probability and Mathematical Statistics*

*organized jointly by the universities Milano-Bicocca, Pavia and Milano-Politecnico.*

Participation is free and welcome!

Best regards The organizers (Carlo Orrieri, Maurizia Rossi, Margherita Zanella)