(Homological Projective Duality)-invariance of the Tate, Beilinson and Riemann conjectures

Informations projet

HPD-inv of TBR

N° de convention de subvention: 892994

DOI

10.3030/892994

Projet clôturé

Date de signature de la CE 17 Avril 2020

Date de début 1 Septembre 2020

Date de fin 31 Août 2022

Financé au titre de

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coût total

€ 224 933,76

Contribution de l’UE

€ 224 933,76

Coordonné par

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 1 - HPD-inv of TBR ((Homological Projective Duality)-invariance of the Tate, Beilinson and Riemann conjectures)

Période du rapport: 2020-09-01 au 2022-08-31

Résumé du contexte et des objectifs généraux du projet

Three of the most important conjectures in mathematics - the Tate conjecture, the Beilinson conjecture and the generalized Riemann hypothesis - concern the location and order of the zeros/poles of the L-functions. These conjectures play a central role in mathematics. For example, in the particular case of an elliptic curve, the Beilinson conjecture reduces to the Birch and Swinnerton-Dyer conjecture, and in the particular case of a point, the generalized Riemann hypothesis reduces to the famous Riemann hypothesis. These are two of the seven Millennium Prize Problems. The overall objective of this project is to prove that the aforementioned conjectures of Tate, Beilinson, and Riemann, are invariant under homological projective duality.

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

The researcher proved that the generalized Riemann hypothesis is invariant under homological projective duality in the sense of Kuznetsov. This result was published in the article - Noncommutative Riemann hypothesis, Proceedings of the American Mathematical Society 150 (2022), 2385-2404 - and was presented at several conferences, workshops and seminars. In the case where the base field is a totally imaginary number field, the researcher proved moreover that the Tate and Beilinson conjectures are also invariant under homological projective duality in the sense of Kuznetsov. In order to achieve the aforementioned results the researcher develop, among other tools, a general theory of noncommutative L-functions which is of independent interest.

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)

The results obtained in this project deepen the current understanding of the Tate, Beilinson and Riemann conjectures. Moreover, they opened the door to many other research directions and applications. Furthermore, it is expected that the results obtained in this project will influence the mathematical community to learn (more) about noncommutative L-functions and to adopt them as part of their toolkit.

Logo of the Youtube channel "Noncommutativity for everyone"