CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

October 6, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Seminar presented by Akitoshi Takayasu (University of Tsukuba, Japan)

In this talk, we introduce a numerical method for rigorously finding the monodromy matrix of hypergeometric differential equations. From a base point defined by fundamental solutions, we analytically continue the solution on a contour around a singular point of the differential equation using a rigorous integrator. Depending on the contour we obtain the monodromy representation of fundamental solutions, which represents the fundamental group of the equation. As an application of this method, we consider a Picard-Fuchs type hypergeometric differential equation arising from a polarized K3 surface. The monodromy matrix shows a deformation of homologically independent 2-cycles for the surface along the contour, which is regarded as a change of characterization for the K3 surface. This is joint work with Naoya Inoue (University of Tsukuba) and Toshimasa Ishige (Chiba University).