DMS Algebra Seminar

Speaker: Hal Schenck

Title: Numerical Analysis meets Topology

Abstract: One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM). A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, 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 topolog.