Poor man's adeles and multiple zeta values

The "poor man's adeles" of the title is the informal name of the ring $ \Prod_p\bigl(\Bbb Z/p\Bbb Z_p\bigr)/\Oplus_p\bigl(\Bbb Z/p\Bbb Z_p\bigr)$ whose elements are "numbers" having a well-defined value modulo almost every prime number. It turns out that examples of elements of this ring show up in many places in mathematics. In the lecture I will describe several examples of this, most notably a finite-field version of the well-known multiple zeta values invented by Euler and much studied in recent years (this part is joint work with Masanobu Kaneko), but also examples coming from areas as different as quantum invariants of homology 3-spheres and transition matrices between different bases of the space of solutions of a linear differential equation with regular singularities.
(joint colloquium of Laboratory of Algebraic Geometry and Laboratory of Mirror Symmetry).