Per il ciclo di Seminari (PMS)^2, organizzato dalle Università di Milano-Bicocca, Pavia e Milano-Politecnico, Giacomo Cherubini (INdAM & Università Roma La Sapienza) e Riccardo W. Maffucci (Università Coventry) terranno i seguenti seminari:
Relatore: Giacomo Cherubini
Titolo: 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 ofsines and cosines with random coefficients. In agreement with Yau's conjecture, we know that the length of the nodalset (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 thetorus 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.
Relatore: Riccardo W. Maffucci
Titolo: 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 papersstudied the ensemble of Laplace Toral eigenfunctions, and their randomisation ‘arithmetic waves’, introduced in 2007by 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 ofthe wave to submanifolds have also attracted recent attention. We considered the length of intersection between thenodal set, and a fixed reference surface in 3 dimensions. Expectation, and variance in a general scenario were priorwork. In the generic setting we prove a CLT. We will discuss (finer order) variance and (non-Gaussian) limitingdistribution in the case of ‘static’ surfaces (e.g. sphere). We will also discuss recent developments on the correlationstructure between different functionals of the wave. Under certain assumptions, there is asymptotic full correlationbetween intersection length and nodal area.
Data e ora: Giovedì 9 maggio, 14:00 - 15:30 (fuso orario di Roma)
Luogo: Aula 3014, Dipartimento di Matematica e Applicazioni (Edificio U5), Università di Milano-Bicocca.
Informazioni per partecipare online
Link Riunione Webex: https://unimib.webex.com/unimib-it/j.php?MTID=m1c4d20c1f6c3199f3682809e0f4af2a2
Numero Riunione: 27426246987
Password: mmYN7WusD46
La partecipazione è libera e gradita!