Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

My first talk explains the classification of algebraic solutions to some very special algebraic differential equations. In this second talk I'll explain a conjectural classification of algebraic solutions to arbitrary algebraic differential equations, generalizing the Grothendieck-Katz $p$-curvature conjecture, and some evidence for this conjecture obtained in joint work with Josh Lam. Loosely speaking, the conjecture says that a solution to a (possibly non-linear!) algebraic differential equation is algebraic if and only if only finitely many primes appear in the denominators of the coefficients of its Taylor expansion at a non-singular point; the "only if" direction was proved by Eisenstein in 1852. We prove the conjecture for some broad classes of algebraic differential equations--including the Painlevé VI equation and Schlesinger system--and initial conditions of algebro-geometric interest.