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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Computational approach to compact Riemann surfaces
- Christian Klein (Université de Bourgogne)
DTSTART;TZID=Europe/London:20191213T113000
DTEND;TZID=Europe/London:20191213T123000
UID:TALK135691AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135691
DESCRIPTION:A purely numerical approach to compact Riemann sur
faces starting from plane algebraic curves is pres
ented. The critical points of the algebraic curve
are computed via a two-dimensional Newton iteratio
n. The starting values for this iteration are obta
ined from the resultants with respect to both coor
dinates of the algebraic curve and a suitable pair
ing of their zeros. A set of generators of the fun
damental group for the complement of these critica
l points in the complex plane is constructed from
circles around these points and connecting lines o
btained from a minimal spanning tree. The monodrom
ies are computed by solving the de ning equation o
f the algebraic curve on collocation points along
these contours and by analytically continuing the
roots. The collocation points are chosen to corres
pond to Chebychev collocation points for an ensuin
g Clenshaw-Curtis integration of the holomorphic d
ifferentials which gives the periods of the Rieman
n surface with spectral accuracy. At the singulari
ties of the algebraic curve\, Puiseux expansions c
omputed by contour integration on the circles arou
nd the singularities are used to identify the holo
morphic differentials. The Abel map is also comput
ed with the Clenshaw-Curtis algorithm and contour
integrals. A special approach is presented for hyp
erelliptic curves in Weierstrass normal form.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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