Zero sets of random functions

Informations projet

RandomZeroSets

N° de convention de subvention: 692616

DOI

10.3030/692616

Projet clôturé

Date de signature de la CE 3 Mai 2016

Date de début 1 Octobre 2016

Date de fin 30 Septembre 2023

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 658 750,00

Contribution de l’UE

€ 1 658 750,00

Coordonné par

TEL AVIV UNIVERSITY
Israel

Periodic Reporting for period 5 - RandomZeroSets (Zero sets of random functions)

Période du rapport: 2022-10-01 au 2023-09-30

Résumé du contexte et des objectifs généraux du projet

The project is focused on zero sets of random functions. This is a rapidly growing area that pertains to pure mathematics (which is a fundamental science) and lies at the crossroads of analysis, probability theory, and mathematical physics. Various instances of zero sets of random functions have been used to model phenomena in quantum chaos, real algebraic geometry, theory of entire functions, and theory of random point processes. The challenging problems addressed in the proposal connect classical fields of mathematics (analysis, probability theory, mathematical physics, and number theory) with such rapidly developing disciplines as spectral geometry, statistical topology, percolation theory.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

We have investigated a variety of topics, from zero sets of random spherical harmonic, of discrete harmonic functions and of linear combinations of Laplacian eigenfunctions, to rigidity of random processes, zeroes of Taylor series with random and psuedo-random coefficients,
and measurable entire functions.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

Our work led us to answers to several long-standing open problems (e.g. Nadirashavili and Nevai conjectures) and to several quite unexpected discoveries concerning behavior of discrete harmonic functions and of linear combination of Laplace eigenfunctions.

Nodal portrait of a random spherical harmonic of degree 80 (Dmitry Beliaev)

Nodal portrait of a random spherical harmonic of degree 40 (Dimtry Beliaev)