Hal Schenck: Combinatorics, Geometry, and…..Numerical Analysis????

Abstract: The main player in this talk is a two-dimensional simplicial complex, embedded in the plane. While a simplicial complex is a combinatorial object, embedding into the plane means that geometry enters the picture. As does applied math: a fundamental tool in numerical analysis is the finite element method, which uses {\em splines}: piecewise polynomial functions on the embedded simplicial complex. Even for a fixed planar triangulation, there are many open questions about splines: for a triangular mesh T and smoothness order one, the dimension of the vector space C^1_3(T) of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a formula for the dimension of the space C^1_2(T) in terms of the combinatorics and geometry of the mesh T, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology.