Why you should care about motives

Period numbers are complex numbers defined as values of integrals over rational algebraic differential forms over domains of intergration satisfying certain algebraicity conditions, again over the rational or algebraic numbers. Theere are several definitions in the literature, but luckily they all give the same set, even a countable algebra containing all algebraic numbers. We take it as given that periods are very interesting. In this expository talk we want to explain how the language of motives is intrinsically related to periods. They are best understood as entries of the comparison matrices between singular cohomology and algebraic de Rham cohomology not only of algebraic varities, but more generally of motives. This point of view gives a lot of structure to the period algebra. The category of motives itself is characterised as representations of a pro-algebraic group, the motivic Galois group. This generalises the ordinary Galois group. At least conjecturally it is related to the period algebra in the same way as the Galois group is related to the field of algebraic numbers. Along the way we will give an introduction to Nori's definition of the abelian category of motives