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While computing with arbitrary real or complex numbers is algorithmically impossible, many numbers arising in mathematics and physics belong to a countable and conjecturally computable subset: the periods. These values are of algebro-geometric origin and encompass examples from π to values appearing in Feynman diagrams with rational momenta.
Conjectures of Grothendieck, Kontsevich, and Zagier suggest that identities involving periods must themselves have geometric origins, placing significant portions of transcendental number theory within the scope of algebraic geometry. Recently, Huber and Wüstholz (2022) proved that linear relations between univariate periods (1-periods) do indeed fall within the purview of algebraic geometry.
In joint work with Joël Ouaknine (MPI-SWS) and James Worrell (Oxford), we give an algorithm that makes this geometric characterization effective: it determines all linear relations over the algebraic numbers among any given finite tuple of 1-periods and decides whether individual periods are transcendental. The algorithmic approach interweaves computational techniques across multiple domains, combining symbolic and rigorous numerical methods in ways that demand robust mathematical software infrastructure.