Motives--Abstract Art of Numbers, Shapes, and Categories.

In arithmetic geometry, we study problems of numbers by transforming them into problems of shapes called algebraic varieties (geometric objects). Cohomology theories extract the information of algebraic varieties as linear data. Various cohomology theories have been developed to study different aspects of algebraic varieties. However, it is widely believed that there is a universal theory called the motive theory, which unifies these cohomology theories. This article gives an overview of the motive theory and presents attempts by the author and his collaborators to generalize it. [ABSTRACT FROM AUTHOR]