The subject of this paper is a special class of configurations, or patterns, of intersecting circles in constant curvature surfaces. The combinatorial aspect of such a pattern is described by a cellular decomposition of the surface. The faces of the cellular decomposition correspond to circles and the vertices correspond to points where circles intersect. (See figures 1 and 2.) In the most general case that we consider, the surface may have cone-like singularities in the centers of the circles and in the points of intersection. In oarticular, we treat...

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of...

Abstract. We discuss a new geometric approach to discrete integrability coming from discrete differential geometry. A d–dimensional equation is called consistent if it is valid for all d–dimensional sublattices of a (d + 1)–dimensional lattice. This algorithmically verifiable property implies analytical structures characteristic of integrability, such as the zero-curvature representation, and allows one to classify discrete integrable equations within certain natural classes. These ideas also apply to the noncommutative case. Theorems about the smooth limit of the theory are also presented. 1

We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the non-commutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general non-commutative ones.

Birational Yang-Baxter maps (‘set-theoretical solutions of the Yang-Baxter equation’) are considered. A birational map (x,y) ↦ → (u,v) is called quadrirational, if its graph is also a graph of a birational map (x,v) ↦ → (u,y). We obtain a classification of quadrirational maps on CP 1 × CP 1, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics. Keywords: Yang-Baxter map, Yang-Baxter equation, set-theoretical solution, 3D-consistency, quadrirational map

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Abstract. Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results. 1.

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quadgraphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the Bäcklund transformations of the nonlinear Schrödinger system and specific ferromagnetic models. 1

Abstract. We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi’s original Permutability Theorem and its generalisation by Dajczer– Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms. 1.

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this dierence scheme. These results are applied to hyperbolic systems of dierential{geometric origin, like the sine{Gordon equation describing the surfaces of the constant negative Gaussian curvature (K{surfaces). In particular, we prove the convergence of discrete K{surfaces and their Backlund transformations to their continuous counterparts. This puts on a rm basis the generally accepted belief (which however remained unproved untill this work) that the classical dierential geometry of integrable classes of surfaces and the classical theory of transformations of such surfaces may be obtained from a unifying multi{dimensional discrete theory by a re nement of the coordinate mesh{size in some of the directions.