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Current Research on Riemann Surfaces and Algebraic Curves


Seminar Notes. FSU Seminar on Computations on Curves. Fall 2002.

Seminar Notes.  FSU Seminar on Computations on Curves. Spring 2002.  Contains preliminary expositions on moduli of path families on algebraic curves.

Mika Seppälä: Myrberg’s numerical uniformizaton of hyperelliptic curves. In press (Ann. Acad. Sci. Fenn.). 16 pages.
Solution to the Numerical Uniformization Problem for Hyperelliptic Curves; this is the Hilbert’s 22nd Problem for hyperelliptic curves

Peter Buser and Mika Seppälä:  Triangulations and homology of Riemann surfaces.  In press (Proc. AMS).
An algorithmic method to pass from triangulations to homology producing a short homology basis for a triangulated Riemann surface.

Peter Buser and Mika Seppälä: Short homology bases for Riemann surfaces.  In press (Topology).
Shows the existence of a homology basis (of a  hyperbolic Riemann surface)  consisting of curves with lengths bounded by the genus and by the homological systole.  


Published papers related to current research

Patrizia Gianni, Mika Seppälä, Robert Silhol, and Barry Trager: Riemann surfaces, algebraic curves  and their period matrices.    -J. Symbolic Comput. 26 (1998), no. 6, 789-803. [ORIGINAL ARTICLE]

Peter Buser and Mika Seppälä:  Computing on Riemann Surfaces.  -In Topology and Teichmüller Spaces, World Scientific Publishing Co, 5 - 30 (1996), 365-372.
An overview to computational problems in the theory of Riemann surfaces and algebraic curves including a proof for the convergence of the Myrberg algorithm.  

Mika Seppälä: Computational Conformal Geometry. -Progr. Math., 143, Birkhäuser Verlag, 365 - 372 (1996). An overview the work of the European HCM project.

Semmler, K.-D., and M. Seppälä:  Numerical uniformization of hyperelliptic curves. -Proceedings of ISSAC 95.
Partial solution to Hilbert’s 22nd problem for hyperelliptic curves.


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