Advertisement

Book Reviews: Riemann Surfaces of Infinite Genus by J. Feldman, H. Knörrer, and E. Trubowitz. CRM Monograph series. Vol. 20, Amer. Math. Soc.

Chapter
  • 490 Downloads
Part of the Contemporary Mathematicians book series (CM)

Abstract

One of the loveliest parts of mathematics is the subject of projective curves, developed, notably, by Jacobi, Abel, Riemann, and Poincaré over most of the nineteenth century.

Keywords

Riemann Surface Projective Curve Jacobi Variety Projective Curf Classical Story 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    R. D. M. Accola. Some classical theorems on open Riemann surfaces. Bull. Amer. Math. Soc., 73:13–26, 1967.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. V. Ahlfors. Normalintegrale auf offenen Riemannschen Flächen. Ann. Acad. Sci. Fenn. Math., 1947(35):1–24, 1947.MathSciNetGoogle Scholar
  3. [3]
    N. I. Akhiezer. Continuous analogues of orthogonal polynomials on a system of intervals. Dokl. Akad. Nauk SSSR, 141:263–266, 1961.MathSciNetGoogle Scholar
  4. [4]
    A. Andreotti. On a theorem of Torelli. Amer. J. Math., 80:801–828, 1958.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. Arbarello and C. De Concini. On a set of equations characterizing Riemann matrices. Ann. of Math. (2), 120(1):119–140, 1984.Google Scholar
  6. [6]
    H. F. Baker. Abelian Functions. Cambridge University Press, Cambridge, 1897. Reissued 1995.Google Scholar
  7. [7]
    H. F. Baker. An Introduction to the Theory of Multiply-Periodic Functions. Cambridge University Press, Cambridge, 1907.zbMATHGoogle Scholar
  8. [8]
    F. J. Dyson. Fredholm determinants and inverse scattering. Comm. Math. Phys., 47:171–183, 1976.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Ercolani and H. P. McKean. Geometry of KdV (4). Abel sums, Jacobi variety, and theta function in the scattering case. Invent. Math., 99(3):483–544, 1990.Google Scholar
  10. [10]
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Method for solving the Korteweg-de Vries equation. Phys. Rev. Letters, 19:1095–1097, 1967.CrossRefGoogle Scholar
  11. [11]
    M. Heins. Riemann surfaces of infinite genus. Ann. of Math. (2), 55:296–317, 1952.Google Scholar
  12. [12]
    A. R. Its and V. B. Matveev Hill operators with a finite number of lacunae. Funkcional. Anal. i Priložen., 9(1):69–70, 1975.MathSciNetCrossRefGoogle Scholar
  13. [13]
    C. G. J. Jacobi. Vorlesungen über analytische Mechanik. Deutsche Mathematiker Vereinigung, Freiburg; Fried. Vieweg & Sohn, Braunschweig, 1848.Google Scholar
  14. [14]
    G. Kempf. On the geometry of a theorem of Riemann. Ann. of Math. (2), 98:178–185, 1973.Google Scholar
  15. [15]
    S. Kowalevski. Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math., 12(1):177–232, 1889.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    P. Lax. Integrals of non-linear equations and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P. Lax. Periodic solutions of the Korteweg-de Vries equation. Comm. Pure Appl. Math., 28:141–188, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. P. McKean and P. van Moerbeke. The spectrum of Hill’s equation. Inv. Math., 30:217–274, 1975.CrossRefzbMATHGoogle Scholar
  19. [19]
    H. P. McKean and E. Trubowitz. Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math., 29:143–226, 1976.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. P. McKean and E. Trubowitz. Hill surfaces and their theta functions. Bull. Amer. Math. Soc., 84:1042–1085, 1978.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Merkl. A Riemann-Roch theorem for infinite genus Riemann surfaces. Invent. Math., 139(2):391–437, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. J. Myrberg. Über transzendente hyperelliptische Integrale erster Gattung. Ann. Acad. Sci. Fenn. Math., 14:1–32, 1943.zbMATHGoogle Scholar
  23. [23]
    P. J. Myrberg. Über analytische Funktionen auf transzendenten zweiblättrigen Riemannschen Flächen mit reellen Verzweigungspunkten. Acta Math., 76:185–224, 1945.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. J. Myrberg. Über analytische Funktionen auf transzendenten Riemannschen Flächen. In C. R. Dixième Congrès Math. Scandinaves 1946, pages 77–96. Jul. Gjellerups Forlag, Copenhagen, 1947.Google Scholar
  25. [25]
    C. Neumann. De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur. J. Reine Angew. Math., 56:46–63, 1859.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    R. Nevanlinna. Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit. Ann. Acad. Sci. Fenn. Math., 1941(1):1–34, 1941.MathSciNetGoogle Scholar
  27. [27]
    S. P. Novikov. The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl., 8:236–246, 1974.CrossRefzbMATHGoogle Scholar
  28. [28]
    M. U. Schmidt. Integrable systems and Riemann surfaces of infinite genus. Mem. Amer. Math. Soc., 122(581):viii+111, 1996.Google Scholar
  29. [29]
    T. Shiota. Characterization of Jacobian varieties in terms of soliton equations. Inv. Math., 83:333–382, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    S. Venakides. The infinite period limit of the inverse formalism for periodic potentials. Comm. Pure Appl. Math., 41:3–17, 1988.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    V. E. Zakharov and L. D. Faddeev. The Korteweg-de Vries equation is a fully integrable Hamiltonian system. Funkcional Anal. i Priložen, 5:18–27, 1974.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

Personalised recommendations