Splines and index theorem (Q2908963)

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.