Introduction xvii Its various realizations based on the bilinear method, algebro-geometric inte- gration, integral equations, R-matrices, and Lagrangian mechanics were de- veloped in Hirota (1977a,b), Krichever (1978), Date-Jimbo-Miwa (1982-3), Quispel-Nijhoff-Capel-Van der Linden (1984), Faddeev-Takhtajan (1986), Moser-Veselov (1991) (here we give just a few representative references). An encyclopedic presentation of the Hamiltonian approach to the problem of integrable discretization is given in Suris (2003). The development of this field led to a progress in various branches of mathematics. Pairs of commuting difference operators were classified in Krichever-Novikov (2003). Laplace transformations of difference operators on regular lattices were constructed in Dynnikov-Novikov (1997) see also Dynnikov-Novikov (2003) for a related development of a discrete complex analysis on triangulated manifolds. A characterization of Jacobians of alge- braic curves based on algebro-geometric methods of integration of difference equations was given in Krichever (2006). From discrete to smooth. In differential geometry the original idea of an intelligent discretization was to find a simple explanation of sophisticated properties of smooth geometric objects. This was the main motivation for the early work in this field documented in Sauer (1937, 1970) and Wunder- lich (1951). The modern period began with the works by Bobenko-Pinkall (1996a,b) and by Doliwa-Santini (1997), where the relation to the theory of integrable systems was established. During the next decade this area wit- nessed a rapid development reflected in numerous publications. In particu- lar, joint efforts of the main contributors to this field resulted in the books Bobenko-Seiler (1999) and Bobenko-Schr¨ oder-Sullivan-Ziegler (2008). The present book gives a comprehensive presentation of the results of discrete differential geometry of parametrized surfaces and coordinate systems along with its relation to integrable systems. We leave the detailed bibliographical remarks to the notes at the end of individual chapters of the book. Discrete differential geometry deals with multidimensional discrete nets (i.e., maps from the regular cubic lattice Zm into RN or some other suitable space) specified by certain geometric properties. In this setting, discrete surfaces appear as two-dimensional layers of multidimensional discrete nets, and their transformations correspond to shifts in the transversal lattice direc- tions. A characteristic feature of the theory is that all lattice directions are considered on an equal footing with respect to the defining geometric prop- erties. Due to this symmetry, discrete surfaces and their transformations become indistinguishable. We associate such a situation with the multidi- mensional consistency (of geometric properties, and of the equations which serve for their analytic description). In each case, multidimensional con- sistency, and therefore the existence and construction of multidimensional
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