Computational Approach to Riemann Surfaces

Christian Klein (University of Bourgogne)

Tuesday 29th October, 2013 15:00-16:00 Maths 515


We present a fully numerical approach to compact Riemann surfaces starting from plane algebraic curves. The code in Matlab computes for a given algebraic equation in two variables the branch points and singularities, the holomorphic differentials and a base of the homology. The monodromy group for the surface is determined via analytic continuation of the roots of the algebraic equation on a set of contours forming the generators of the fundamental group. The periods of the holomorphic differentials are computed with Gauss-Legendre integration along these contours. The Abel map is obtained in a similar way. The performance of the code is illustrated for many examples. As an application we study quasi-periodic solutions to certain integrable partial differential equations.

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