Pure Maths Colloquium: Nellie Villazimar

Splines are piecewise polynomial functions defined over a real domain which are continuously differentiable to some order \(r\). For a fixed integer \(d\), the space of splines of degree at most \(d\) and smoothness \(r\) is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when \(r > 1\). It is particularly difficult to compute the dimension of splines on partitions in which a vertex is completely surrounded by tetrahedra – we call these domains vertex stars.

In the talk, I will present a lower bound formula on the dimension of splines on vertex stars and how that leads to prove a lower bound on the dimension of splines defined over general tetrahedral partitions in large degree. The proofs use apolarity, some results from rigidity theory, and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. I will show some examples and open problems in the area.