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Project

Absolute geometry and zeta functions (1518717N)

The Deninger-Manin program aims at an Algebraic Geometry over the “ield with one element F1”which allows one to mimic Weil’ solution of the Riemann Hypothesis for function fields of curves in positive characteristic, to the classical Riemann Hypothesis. This mysterious “ield”already was mentioned in a 1957 paper of Tits, in which symmetric groups are seen as linear groups over F1.

In 1992, Deninger described of a category of motives that would admit a translation of Weil’ proof to the hypothetical curve of integers. He showed that a certain Lefschetz-type formula would hold in which a terms "h2" appears. Manin proposed that this mysterious term be interpreted as the affine line over F1. The Riemann Hypothesis became a main motivation to search for geometry over F1.

Since 2005, this new and emerging field started to grow rapidly, with several scheme theories over F1 being independently initiated by Connes—onsani, Deitmar, Lorscheid and others. Connes and Consani wrote a large number of papers on this subject, and many deep problems arise in their program.

In this proposal, I want to attack foundational questions that are central in F1-theory, including a conjecture about zeta functions which is addressed in the Deninger-Manin program, the further development of F1-scheme theory to enable this approach, and questions of Connes and Consani relating the hyperstructure of the adèle class space of a global field to Singer actions of projective spaces.

Date:1 Jan 2017 →  31 Dec 2019
Keywords:geometry
Disciplines:Other natural sciences, Other biological sciences