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Splines and index theorem. (English) Zbl 1262.58014

This paper is focused on an intersection area of different mathematical topics – numerical analysis, index theory and symplectic geometry. The paper is divided into three main parts: Polytopes and splines, arithmetic and combinatorics and index theory. The Atiyah-Singer index theorem – one of key results in modern geometry – provides a fundamental link between differential geometry, partial differential equations, differential topology, operator algebras and many other fields. The theory of splines can be considered as a differentiable analogue of the arithmetic theory because many constructions with partition functions and differential equations were discovered as generalization of certain splines associated to a list of vectors. In the differentiable theory, the partition function is replaced by the multivariate spline.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
41A15 Spline approximation
19L10 Riemann-Roch theorems, Chern characters
65D07 Numerical computation using splines
51M20 Polyhedra and polytopes; regular figures, division of spaces
53D05 Symplectic manifolds (general theory)
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