Noncommutative geometry and motives (a quoi servent les endomotifs?)

TL;DR

This paper surveys the development of pure motives in algebraic geometry, explores recent advances in noncommutative geometry, and introduces endomotives with applications in number theory.

Contribution

It presents a new theory of endomotives and discusses its relevance and applications in number theory within the context of noncommutative geometry.

Findings

Introduction of the endomotives theory

Connections between motives and noncommutative geometry

Applications in number theory

Abstract

This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number-theory.