Obiettivo The subject of this proposal lies at the crossroads of analysis, additive combinatorics, number theory and fractal geometry exploring equidistribution phenomena for random walks on groups and group actions and regularity properties of self-similar, self-affine and Furstenberg boundary measures and other kinds of stationary measures. Many of the problems I will study in this project are deeply linked with problems in number theory, such as bounds for the separation between algebraic numbers, Lehmer's conjecture and irreducibility of polynomials.The central aim of the project is to gain insight into and eventually resolve problems in several main directions including the following. I will address the main challenges that remain in our understanding of the spectral gap of averaging operators on finite groups and Lie groups and I will study the applications of such estimates. I will build on the dramatic recent progress on a problem of Erdos from 1939 regarding Bernoulli convolutions. I will also investigate other families of fractal measures. I will examine the arithmetic properties (such as irreducibility and their Galois groups) of generic polynomials with bounded coefficients and in other related families of polynomials.While these lines of research may seem unrelated, both the problems and the methods I propose to study them are deeply connected. Campo scientifico natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics Parole chiave spectral gap random walks on groups local limit theorem Bernoulli convolution self-similar measure Furstenberg boundary measure generic polynomial irreducibility of polynomials Programma(i) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Argomento(i) ERC-2018-STG - ERC Starting Grant Invito a presentare proposte ERC-2018-STG Vedi altri progetti per questo bando Meccanismo di finanziamento ERC-STG - Starting Grant Istituzione ospitante THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Contribution nette de l'UE € 1 334 109,00 Indirizzo TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge Regno Unito Mostra sulla mappa Regione East of England East Anglia Cambridgeshire CC Tipo di attività Higher or Secondary Education Establishments Collegamenti Contatta l’organizzazione Opens in new window Sito web Opens in new window Partecipazione a programmi di R&I dell'UE Opens in new window Rete di collaborazione HORIZON Opens in new window Costo totale € 1 334 109,00 Beneficiari (1) Classifica in ordine alfabetico Classifica per Contributo netto dell'UE Espandi tutto Riduci tutto THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Regno Unito Contribution nette de l'UE € 1 334 109,00 Indirizzo TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge Mostra sulla mappa Regione East of England East Anglia Cambridgeshire CC Tipo di attività Higher or Secondary Education Establishments Collegamenti Contatta l’organizzazione Opens in new window Sito web Opens in new window Partecipazione a programmi di R&I dell'UE Opens in new window Rete di collaborazione HORIZON Opens in new window Costo totale € 1 334 109,00