Cari Colleghi
Venerdì 25 gennaio si terrà presso il Dipartimento di Matematica dell'Università di Roma Tor Vergata (Aula D'Antoni) il mini-workshop "The Geometry of Random Fields"; il programma è il seguente:
9.30-10.15 Dmitry Belyaev (Oxford University)
Geometry of smooth Gaussian fields and percolation
Abstract: Gaussian fields and their level sets are of significant mathematical interest, but they also appear in many other areas of science: oceanography, cosmology and engineering, just to name a few. In this talk I will give an gently introduction to the geometric theory of smooth Gaussian fields, will discuss conjectures about their connection with percolation models and will talk about recent progress in this area.
10.15-11.00 Igor Wigman (King's College London)
Points on nodal lines with given direction
Abstract: We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.
11.00-11.30 Coffee Break
11.30-12.15 Anne Estrade (Université Paris Descartes)
On Berry's dislocation lines in 3D framework
Abstract: We study the length of dislocation lines that are given as nodal lines of a complex Gaussian random wave indexed by $\R^3$. We consider random waves that are eigenfunction of the 3D Euclidean Laplacian, isotropically distributed or not. We are particularly interested in the expectation and the variance of the length and their behavior as the associated eigenvalue goes to infinity. We compare the results with the similar situation in 2D that has been extensively studied since Berry's paper in 2002. This is a work in progress with Federico Dalmao and Jose Leon (Universidad de la Republica, Uruguay).
12.15- 13.00 Maurizia Rossi (Università di Pisa)
Title: Nodal lengths of random spherical harmonics.
Abstract: Random spherical harmonics are Gaussian eigenfunctions of the spherical Laplacian; in this talk we study the geometry of their excursion sets, in particular the high-energy behavior of their boundary length. For the nodal case we show that the length is asymptotically equivalent, in the L^2-sense, to the sample trispectrum, obtaining as a by-product a quantitative Central Limit Theorem for it. The length at non-zero level, being asymptotically equivalent to the (random) norm of the eigenfunction, is proved to be Gaussian and independent of the nodal length in the high-energy limit. Removing the effect of the eigenfunctions norm (which play no role in the nodal case indeed) we prove that the partial correlation between nodal and boundary lengths is asymptotically one. This talk is based on joint works with D. Marinucci and I. Wigman.
Grazie per l'attenzione, Domenico Marinucci