Canonical functions and rational approximations.

Associating a “canonical form” associated to a semi-algebraic region in the complex points of an algebraic variety, whose poles lie along its boundary, has recently risen in popularity. In this talk I will introduce the notion of "canonical function" associated to two such regions. In brief, it is a rational function which has *poles* along the boundary of the first region, and *zeros* along the boundary of the second region. Associated to this data is a canonical Picard-Fuchs recurrence relation, periods and mysterious symmetry groups. Examples constructed in this way provide rare mathematical gems: for instance Apéry’s famous approximations to zeta(2), zeta(3), surprising connections with modular forms, and much more besides.