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SUMMARY:Computational approach to compact Riemann surfaces - Christian Kle
 in (Université de Bourgogne)
DTSTART:20191213T113000Z
DTEND:20191213T123000Z
UID:TALK135691@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:A purely numerical approach to compact Riemann surfaces starti
 ng from plane algebraic curves is presented. The critical points of the al
 gebraic curve are computed via a two-dimensional Newton iteration. The sta
 rting values for this iteration are obtained from the resultants with resp
 ect to both coordinates of the algebraic curve and a suitable pairing of t
 heir zeros. A set of generators of the fundamental group for the complemen
 t of these critical points in the complex plane is constructed from circle
 s around these points and connecting lines obtained from a minimal spannin
 g tree. The monodromies are computed by solving the de ning equation of th
 e algebraic curve on collocation points along these contours and by analyt
 ically continuing the roots. The collocation points are chosen to correspo
 nd to Chebychev collocation points for an ensuing Clenshaw-Curtis integrat
 ion of the holomorphic differentials which gives the periods of the Rieman
 n surface with spectral accuracy. At the singularities of the algebraic cu
 rve\, Puiseux expansions computed by contour integration on the circles ar
 ound the singularities are used to identify the holomorphic differentials.
  The Abel map is also computed with the Clenshaw-Curtis algorithm and cont
 our integrals. A special approach is presented for hyperelliptic curves in
  Weierstrass normal form.
LOCATION:Seminar Room 1\, Newton Institute
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