Arbarello, Enrico Riemann surfaces, theta functions and representation theory. (Italian) Zbl 0699.14055 Boll. Unione Mat. Ital., VII. Ser., A 3, No. 1, 1-22 (1989). 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 Keywords:Korteweg-de Vries equation; Riemann surfaces; Kadomtsev-Petviashvili; hierarchy; Novikov’s conjecture Citations:Zbl 0629.14022; Zbl 0621.35097; Zbl 0647.17010 PDFBibTeX XMLCite \textit{E. Arbarello}, Boll. Unione Mat. Ital., VII. Ser., A 3, No. 1, 1--22 (1989; Zbl 0699.14055)