## Numbers as functions [1]

**Conférence grand public de Yuri Manin [2]**

The analogies between numbers and functions of one variable were known long ago: integers share common properties with polynomials over rational numbers or over finite fields; p-adic numbers are similar to formal series.

During the last two decades, such analogies have been developed and became background of two fascinating new chapters of algebra/number theory: Alexandru Buium's "Arithmetic Differential Equations" and a collective endeavor "Algebraic Geometry Over a Field of Characteristic One".

In the talk, I will present motivations, examples and some basic constructions of both theories, stressing their interrelations.