Geometric realisations of spline spaces on a simplicial complex

Nelly Villamizar (Swansea University)

Wednesday 31st October, 2018 16:00-17:00 Maths 311B


We consider the space of continuous splines (or piecewise polynomial functions) defined on a simplicial complex. Besides the practical applications of splines, including the solution of partial differential equations by the finite element method, and the approximation of shapes in geometric modeling, the space of continuous splines forms a ring, and one can study its algebraic structure. More precisely, the space of C^0-continuous splines is a quotient of the Stanley-Reisner ring of the corresponding simplicial complex, and the geometric realisation of the Stanley-Reisner ring reflects the structure of the simplicial complex. In the talk, we shall consider the generalised Stanley-Reisner rings associated to a simplicial complex, namely the ring of spline functions with higher order of global continuity on the simplicial complex, and give a description of their geometric realizations for particular instances of the dual graph of the complex. We will also discuss related open problems in this area.

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