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Book Review

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Book Information:

Authors: J. Feldman, H. K\"norrer and E. Trubowitz
Title: Riemann surfaces of infinite genus
Additional book information: CRM Monograph series, vol. 20, Amer. Math. Soc., Providence, RI, 2003, 303 pp., ISBN 0-8218-3357-X, $79.00$

Robert D. M. Accola, Some classical theorems on open Riemann surfaces, Bull. Amer. Math. Soc. 73 (1967), 13–26. MR 207976, DOI 10.1090/S0002-9904-1967-11619-7

Lars V. Ahlfors, Normalintegrale auf offenen Riemannschen Flächen, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1947 (1947), no. 35, 24 (German). MR 25577

N. I. Ahiezer, Continuous analogues of orthogonal polynomials on a system of intervals, Dokl. Akad. Nauk SSSR 141 (1961), 263–266 (Russian). MR 0140970

Aldo Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. MR 102518, DOI 10.2307/2372835

Enrico Arbarello and Corrado De Concini, On a set of equations characterizing Riemann matrices, Ann. of Math. (2) 120 (1984), no. 1, 119–140. MR 750718, DOI 10.2307/2007073

H. F. Baker, Abelian functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Abel’s theorem and the allied theory of theta functions; Reprint of the 1897 original; With a foreword by Igor Krichever. MR 1386644

[1907]

-: An Introduction to the Theory of Multiply Periodic Functions. Cambridge U. Press, Cambridge, 1967.

Freeman J. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), no. 2, 171–183. MR 406201

V. E. Zaharov and L. D. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 18–27 (Russian). MR 0303132

[1967]

GARDINER, R., GREENE, J., KRUSKAL, M., and MIURA, R.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Letters 19 (1967) 1095-1097.

Maurice Heins, Riemann surfaces of infinite genus, Ann. of Math. (2) 55 (1952), 296–317. MR 45834, DOI 10.2307/1969780

A. R. Its and V. B. Matveev, Hill operators with a finite number of lacunae, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 69–70 (Russian). MR 0390355

[1884]

JACOBI, C.G.J.: Gesammelte Werke Bd.1 supplement: Vorlesungen über Dynamik, Reimer, Berlin, 1884.

George Kempf, On the geometry of a theorem of Riemann, Ann. of Math. (2) 98 (1973), 178–185. MR 349687, DOI 10.2307/1970910

Sophie Kowalevski, Sur le problème de la rotation d’un corps solide autour d’un point fixe, The Kowalevski property (Leeds, 2000) CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 315–372 (French). Reprinted from Acta Math. 12 (1889), 177–232. MR 1916790, DOI 10.1090/crmp/032/18

Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 235310, DOI 10.1002/cpa.3160210503

Peter D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28 (1975), 141–188. MR 369963, DOI 10.1002/cpa.3160280105

N. Ercolani and H. P. McKean, Geometry of KdV. IV. Abel sums, Jacobi variety, and theta function in the scattering case, Invent. Math. 99 (1990), no. 3, 483–544. MR 1032878, DOI 10.1007/BF01234429

H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567

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 (1976), no. 2, 143–226. MR 427731, DOI 10.1002/cpa.3160290203

H. P. McKean and E. Trubowitz, Hill’s surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1042–1085. MR 508448, DOI 10.1090/S0002-9904-1978-14542-X

Franz Merkl, A Riemann Roch theorem for infinite genus Riemann surfaces, Invent. Math. 139 (2000), no. 2, 391–437. MR 1738447, DOI 10.1007/s002229900031

P. J. Myrberg, Über transzendente hyperelliptische Integrale erster Gattung, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1943 (1943), no. 14, 32 (German). MR 15497

P. J. Myrberg, Über analytische Funktionen auf transzendenten zweiblättrigen Riemannschen Flächen mit reellen Verzweigungspunkten, Acta Math. 76 (1945), 185–224 (German). MR 12675, DOI 10.1007/BF02551576

P. J. Myrberg, Über analytische Funktionen auf transzendenten Riemannschen Flächen, C. R. Dixième Congrès Math. Scandinaves 1946, Jul. Gjellerups Forlag, Copenhagen, 1947, pp. 77–96 (German). MR 0020146

[1859]

NEUMANN, C.: De problemate quandum mechanica quod ad primam integralium meta-ellipticarem classem revocatur. J. Riene u. Angew. Math. 56 (1859) 54-66.

Rolf Nevanlinna, Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1941 (1941), no. 1, 34 (French). MR 15496

[1974]

NOVIKOFF, S.P.: The periodic problem for the Korteweg-de Vries equation. Funkt. Anal. Pril. 8 (1974) 59-66.

Martin U. Schmidt, Integrable systems and Riemann surfaces of infinite genus, Mem. Amer. Math. Soc. 122 (1996), no. 581, viii+111. MR 1329944, DOI 10.1090/memo/0581

Takahiro Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), no. 2, 333–382. MR 818357, DOI 10.1007/BF01388967

Stephanos Venakides, The infinite period limit of the inverse formalism for periodic potentials, Comm. Pure Appl. Math. 41 (1988), no. 1, 3–17. MR 917122, DOI 10.1002/cpa.3160410103

Review Information:

Reviewer: Henry P. McKean
Affiliation: Courant Institute of Mathematical Sciences
Email: mckean@cims.nyu.edu