Blueprints - towards absolute arithmetics

The definition of a blueprint was initially motivated by Jacques Tits' idea of descending Chevalley
groups to F1: there was a need to go beyond Deitmar's F1-geometry attached to monoids by
including a relation on the formal sums in the elements of a monoid. Meanwhile it became clear that blueprints not only yield a satisfactory answer to Tits' problem, with applications to (idempotent
and tropical) semirings, but provide the natural background tounderstand certain aspects
of geometry---buildings and their appartments; canonical bases and total positivity; cluster algebras;
moduli of quiver representations and quiver Grassmannians---as well as arithmetics: the Riemann zeta function finds a reinterpretation in terms of a locally blueprinted space X, which can be seen as the
compactification of Spec Z. Most striking, the fibre product X x X behaves indeed like a surface.

In this talk, I will give an introduction to blueprints, focussing on their arithmetic applications.