Speaker: Sergei Yakovenko, Weizmann Institute, Rehovot, Israel Date: 10:30 09/04/2021 Title:How transcendental are Periods? Abstract: Periods (integrals of rational 1-forms over algebraic 1-cycles) in general are transcendental functions of the relevant parameters, since even a single integration destroys algebraicity. However, it turns out that their behavior (in particular, the number of isolated zeros when restricted on one-dimensional lines in the parameter space) is similar to that of algebraic functions (in particular, the number of zeros is explicitly bounded from above). This implies an explicit upper bound for an infinitesimal flavor of the Hilbert 16th problem. The main engine of the proof is a subtle difference between systems of first order linear ordinary differential equations on the projective line and linear equations of higher order with rational coefficients.