Introduction xix and conversely, many interesting integrable systems admit a differential- geometric interpretation. Having identified the roots of integrable differen- tial geometry in the multidimensional consistency of discrete nets, one is led to a new (geometric) understanding of integrability itself. First of all, we adopt the point of view that the central role in this theory is played by discrete integrable systems. In particular, a great variety of integrable differ- ential equations can be derived from several fundamental discrete systems by performing different continuous limits. Further, and more importantly, we arrive at the idea that the multidimensional consistency of discrete equa- tions may serve as a constructive and almost algorithmic definition of their integrability. This idea was introduced in Bobenko-Suris (2002a) (and inde- pendently in Nijhoff (2002)). This definition of integrability captures enough structure to guarantee such traditional attributes of integrable equations as zero curvature representations and acklund-Darboux transformations (which, in turn, serve as the basis for applying analytic methods such as in- verse scattering, finite gap integration, Riemann-Hilbert problems, etc.). A continuous counterpart (and consequence) of multidimensional consistency is the well-known fact that integrable systems never appear alone but are organized into hierarchies of commuting flows. This conceptual view of discrete differential geometry as the basis of the theory of surfaces and their transformations as well as of the theory of integrable systems is schematically represented in Figure 0.2. This general picture looks very natural, and there is a common belief that the smooth theories can be obtained in a limit from the corresponding discrete ones. This belief is supported by formal similarities of the cor- responding difference and differential equations. However one should not underestimate the difficulty of the convergence theorems required for a rig- orous justification of this philosophy. Solutions to similar problems are substantial in various areas of differen- tial geometry. Classical examples to be mentioned here are the fundamental results of Alexandrov and Pogorelov on the metric geometry of polyhedra and convex surfaces (see Alexandrov (2005) and Pogorelov (1973)). Alexan- drov’s theorem states that any abstract convex polyhedral metric is uniquely realized by a convex polyhedron in Euclidean 3-space. Pogorelov proved the corresponding existence and uniqueness result for convex Riemannian metrics by approximating smooth surfaces by polyhedra. Another example is Thurston’s approximation of conformal mappings by circle packings (see Thurston (1985)). The theory of circle packings (see the book by Stephenson (2005)) is treated as discrete complex analysis. At the core of this theory is the Koebe-Andreev-Thurston theorem which states that any simplicial decomposition of a sphere can be uniquely (up to obius transformations)
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