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CORDIS

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

Project description

A variety of novel proofs address some of the most important conjectures in mathematics

Many people might remember factoring polynomial equations in secondary school math classes. Not everyone uses them in their job, but polynomial equations are relevant to a variety of fields from finance and electronics to chemistry, physics and engineering. Algebraic varieties represent solutions of a system of polynomial equations in real or complex number space, and they are the subject of three of the most important conjectures in mathematics: the Tate, Beilinson and Riemann conjectures. The EU-funded HPD-inv of TBR project is enhancing the descriptions associated with these three conjectures and applying the resulting proof to additional important topics regarding algebraic varieties.

Objective

This project will be carried out at the Warwick Mathematics Institute under the supervision of Prof. John Greenlees; I have worked for the past eight years at MIT and I will move in 2020 to the University of Warwick as an Associate Professor. 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 associated to algebraic varieties. 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 Riemann hypothesis. These are two of the seven Millenium Prize Problems. The first objective of this project is to prove that the conjectures of Tate, Beilinson, and Riemann, are invariant under homological projective duality in the sense of Kuznetsov. The second objective is to combine this invariance result with the different homological projective dualities in the literature in order to obtain not only a proof of the conjectures of Tate and Beilinson in numerous new cases but also an equivalence between the generalized Riemann hypothesis of very different algebraic varieties. These objectives will greatly improve the state-of-the-art of the Tate and Beilinson conjectures and will considerably deepen our understanding of the generalized Riemann hypothesis. In order to achieve them, I will combine Kontsevich's noncommutative viewpoint on algebraic geometry with mathematical tools from several different areas (e.g. algebraic topology, derived categories, algebraic K-theory, etc). This will enhance my creative and innovative potential, will foster my professional maturity and independence, will diversify my technical skills, and also will enable me to receive advanced training. Hence, this project is directly aligned with the MSCA-IF-EF-RI objectives.

Coordinator

UNIVERSITY OF WARWICK
Net EU contribution
€ 224 933,76
Address
KIRBY CORNER ROAD UNIVERSITY HOUSE
CV4 8UW COVENTRY
United Kingdom

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Region
West Midlands (England) West Midlands Coventry
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 224 933,76