Zero sets of random functions

Projektinformationen

RandomZeroSets

ID Finanzhilfevereinbarung: 692616

DOI

10.3030/692616

Projekt abgeschlossen

EK-Unterschriftsdatum 3 Mai 2016

Startdatum 1 Oktober 2016

Enddatum 30 September 2023

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 658 750,00

EU-Beitrag

€ 1 658 750,00

Koordiniert durch

TEL AVIV UNIVERSITY
Israel

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

Berichtszeitraum: 2022-10-01 bis 2023-09-30

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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)