Abstract:

A.Grothendieck conjectured that a vector bundle with connection on a curve over a number field has
all flat sections given by algebraic functions iff the p-curvature vanishes for almost all primes p.
In 1984 D.Chudnovsky and G.Chudnovsky proved Grothendieck's Algebraicity Conjecture in the abelian
case: if the derivative of logarithm of a series with integer coefficients is algebraic, then the
series itself is algebraic. I'll talk about a surprising application of this result in the theory
of large random unitary matrices, based on the notion of algebraic noncommutative series introduced
by N.Chomsky and M.Schützenberger.