Numerical Analysis meets Topology

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 topology.