Abstract: A fundamental problem in mathematics is to approximate a given function on some region R with a nice function, such as a polynomial. In order to get a good approximation, the standard strategy is to subdivide R into smaller regions Δi, approximate f on those regions, and require compatibility conditions on Δi∩Δj. In the most studied case, the Δi are simplices, and the compatibility condition is Cr-smoothness. The set of piecewise polynomial functions of degree at most k and smoothness r on a triangulation Δ is a vector space, and even when Δ ⊆ R2, the dimension of Crk(Δ) is unknown. Work of Alfeld-Schumaker provides an answer if k≥3r+1, and Billera earned the Fulkerson prize for solving a conjecture of Strang for the case r=1 and a generic triangulation Δ. I will discuss recent progress on the dimension question using tools of algebraic geometry, when Δ is a polyhedral complex. I will also touch on a beautiful connection to toric geometry, provided by work of Payne on the equivariant Chow cohomology of toric varieties.