Abstract
Today the techniques related to Riemann surfaces are widely used in different branches of mathematics, theoretical physics, industry and even in medicine. Unfortunately, many achievements of the theory remain on paper since theoretical formulae contain special functions that few scholars try to compute. In the present note we describe several useful techniques for the solution of equations in the moduli spaces of algebraic curves. Conceptually all the methods we talk about belong to the 19th century but their actual implementation have become possible only with the invention of the computer.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
T. Akaza, Poincaré theta series and singular sets of Schottky groups, Nagoya Math. J. 24 (1964), 43–65.
T. Akaza, Singular sets of some Kleinian groups, Nagoya Math. J. 29 (1967), 145–162.
T. Akaza and K. Inoue, Limit sets of a geometrically finite free Kleinian group, Tohoku Math. J. 36 (1984), 1–16.
A. F. Baker, Abel Theorem and the Allied Theory Including the Theory of Theta Functions, Cambrigde University Press, 1897.
E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.
A. B. Bogatyrëv, On evaluation of Chebyshev polynomials on several segments, Math. Sbornik 190 (1999) no.11, 15–50, LMS translations, 1571–1605.
A. B. Bogatyrëv, Representations of the moduli spaces and computation of extremal polynomials, Math. Sbornik 194 (2003) no.4, 3–28, LMS translations, 469–494.
A. B. Bogatyrëv, Effective solution of a problem about optimal stability polynomials, Math Sbornik 196 (2005) no.7, 27–50, LMS translations, 959–981.
A. B. Bogatyrëv, Extremal Polynomials and Riemann Surfaces (in Russian), Moscow, MCCME publishers, 2005.
—, Effective variational formula for Schottky-Klein prime factor, to appear.
W. Burnside, On a class of authomorphic functions, Proc. London Math. Soc. 23 (1892), 49–88.
W. Burnside, Further note on authomorphic functions, Proc. London Math. Soc. 23 (1892), 281–295.
D. Crowdy, Genus N algebraic reductions of the Benney hierarchy within a Schottky model, J. Phys. A: Math. and General 38 (2005), 10917–10934.
D. Crowdy and J. S. Marshall, On the construction of multiply connected quadrature domains, SIAM J. Appl. Math. 64 (2004), 1334–1359.
B. Deconinck, M. Heil, A. I. Bobenko et al. Computing Riemann theta functions, Math. Comp. 73 (2004), 1417–1442.
B. Deconinck and M. Van Hoeij, Computing Riemann matrices of algebraic curves, Physica D. 152 (2001), 28–46.
B. A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Surv. 36 (1981) no.2, 11–92.
J. Fay, Theta Functions on Riemann Surfaces, Springer, Lect. Notes in Math. 352, 1973.
J. Harris and I. Morrison, Moduli of Curves, Springer, 1998.
D. A. Hejhal, Sur les parameters accessoires pour l’uniformization de Schottky, C.R. Acad. Sc. Paris Serie A 279 (1974), 713–716.
R. A. Hidalgo and J. Figueroa, Numerical Schottky uniformizations, Geometriae Dedicata 111 (2005), 125–157.
I. M. Krichever and S. P. Novikov, Periodic and almost periodic potentials in the inverse problems, ArXive: math-ph/003004 (2000).
P. J. Myrberg, Zur Theorie der Konvergenz der Poincaré schen Reihen, Ann. Acad. Sci. Fenn. (A) 9 (1916) no.4, 1–75; 11 (1917) no.4, 1–29.
S. M. Natanzon, Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs, AMS, 2004.
J. Nuttall Sets of minimum capacity, Padé approximants and the bubble problem, in: C. Bardos and D. Bessis (eds.), Bifurcation Phenomena in Math. Physics and Related Topics, D. Reidel Publ. Company, 1980, 185–201.
M. Schmies, Computational methods for riemann surfaces and helicoids with handles, PhD thesis, TU Berlin, 2005.
F. Schottky, Uber eine spezielle Function welche bei einer bestimmten linearen Transformation ihres Argumenten unverändert bleibt, J. Reine Angew. Math. 101 (1887), 227–272.
Y. Wang, X. Gu, K. M. Hayashi et al., Brain Surface Parametrization Using Riemann Surfaces, in: J. Duncan and G. Gerig (eds.), MICCAI2005, Springer, Berlin-Heidelberg, (2005), 657–665.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by RFBR grants 05-01-01027, 05-01-00582 and grant MD 2488.2005.1.
Rights and permissions
About this article
Cite this article
Bogatyrëv, A.B. Computations in Moduli Spaces. Comput. Methods Funct. Theory 7, 309–324 (2007). https://doi.org/10.1007/BF03321647
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321647