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Riemann surfaces, theta functions and representation theory. (Italian) Zbl 0699.14055

This is a beautiful written exposition of the ideas which have been developed starting from the discovery of the connection between the Korteweg-de Vries equation and Riemann surfaces. Explained are the Kadomtsev-Petviashvili hierarchy, Sato’s infinite Grassmannian, the Novikov-Krichever-Dubrovin construction, Novikov’s conjecture on the characterisation of Jacobians among abelian varieties and the results of the author and C. De Concini [Duke Math. J. 54, 163-178 (1987; Zbl 0629.14022)] and T. Shiota [Invent. Math. 83, 333-382 (1986; Zbl 0621.35097)] on this subject. Discussed is also the relation between the infinite Grassmannian and the geometry of the moduli space of curves [cf. the author, C. De Concini, V. G. Kac and C. Procesi in Commun. Math. Phys. 117, No.1, 1-36 (1988; Zbl 0647.17010)].
Reviewer: A.Buium

MSC:

14K25 Theta functions and abelian varieties
14H25 Arithmetic ground fields for curves
30F30 Differentials on Riemann surfaces
14H52 Elliptic curves
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