Projektbeschreibung
Neue Ansätze zur Beschreibung der Nullstellen von Zufallspolynomen
Zufallspolynome sind in der Physik von Interesse, da sie zur höchst genauen Beschreibung des Verhaltens von Coulomb-Gasen und bestimmten Bose-Gasen angewendet werden können. Ziel des EU-finanzierten Projekts RandPol ist die Untersuchung der Nullstellen von Zufallspolynomen – eines der ältesten und grundlegendsten Probleme der Mathematik. Klassische Probleme auf der Makroebene sind: Wo können Nullstellen lokalisiert werden? Gibt es umfassende Beschreibungen des Verhaltens der isolierten Nullstellen? Das Forschungsteam möchte außerdem das Verhalten von Zufallspolynomen auf der Mikroebene untersuchen. Insbesondere sollen die lokale Nullstellenverteilung und die Interaktion zwischen benachbarten Nullstellen mit renormierter Energie in Zusammenhang gebracht werden. Dieser ist der Schlüssel für die Beschreibung des lokalen Verhaltens der Teilchen in einem Coulomb-Gas.
Ziel
The aim of this project is to study the zeros of random polynomials. Random polynomials appeared in the 1930's with the pioneering works of Bloch, Polya and Kac. The study of random polynomials gained a strong interest of the physicists since the 1990's due to their connexion with interaction particle systems such as electron gases (also called Coulomb gases). Random polynomials were first considered as a toy model for more complicated systems, but in the 2000's they appeared to exactly describe the behavior of certain Bose Gases.
We want to understand the behavior of random polynomials at the macroscopic and microscopic scale. At the macroscopic scale, several questions naturally arise: Where can we locate the zeros? Are there isolated zeros far away from the rest of them? Can we describe the behavior of those isolated zeros?
Very recent breakthrough (2019) were made in both localisation and isolated zeros for specific models of random pomynomials with some symmetry constraints. Our goal is to obtain general results on these questions. We plan to develop new ideas to tackle these questions, without relying on the symmetry structure.
At the microscopic scale, we want to understand the local repartition of the zeros and the interaction between neighboors. This question lead to adapt concepts from mathematical physics such as the microscopic renormalized energy to the study of zeros of random polynomials. Recent studies also linked the local behavior of zeros of random polynomials to the zeros of Random Analytic Functions, which have a lot of connexions to several domains in mathematics (Analysis, Image processing, random matrix theory, mathematical physiqucs).
The concept of renormaliazed energy is very recent (2017), and was introduced by Leblé and Serfaty to understand the local behavior of particles from a Coulomb gas. We think that this approach can lead to a new understanding of the behavior of zeros of random polynomials.
Wissenschaftliches Gebiet
Schlüsselbegriffe
Programm/Programme
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Aufforderung zur Vorschlagseinreichung
Andere Projekte für diesen Aufruf anzeigenFinanzierungsplan
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Koordinator
1211 Geneve
Schweiz