Mechanising the theory of periods: Algorithms for univariate period integrals

Speaker: Emre Sertöz
Abstract:It is impossible to design an algorithm capable of computing with arbitrary real or complex numbers: there are simply too many of them. By contrast, many numbers arising in the sciences belong to a countable set, conjecturally accessible to computation.  These are the periods — values of integrals of geometric origin, with examples ranging from the circumference of the unit circle to Feynman integrals with rational momenta.

Conjectures of Grothendieck, Kontsevich, and Zagier imply that not only the periods themselves, but also the algebraic relations between them are of geometric origin. This suggests that a substantial part of transcendental number theory falls within the purview of algebraic geometry.

Even if these conjectures hold, the algorithmic challenge remains: can we effectively determine all relations among a given set of periods? In the univariate case — so-called 1-periods — Huber and Wüstholz (2022) proved that all relations are indeed of algebraic-geometric origin. In joint work with Joël Ouaknine (MPI-SWS) and James Worrell (Oxford), we turned this result into an algorithm that finds all linear relations among such integrals, and in particular decides if a given univariate period is transcendental.