  This survey gives an overview of three central algebraic themes related to
the study of splines: duality, group actions, and homology. Splines are
piecewise polynomial functions of a prescribed order of smoothness on some
subdivided domain D in R^k, and appear in applications ranging from
approximation theory to geometric modeling to numerical analysis.
Alternatively, splines can be interpreted as a collection of polynomials
labeling the vertices of a (combinatorial) graph, with adjacent vertex-labels
differing by a power of an affine linear form attached to the edge. In most
cases of interest, the subdivided domain is essentially dual to the
combinatorial graph, and these two characterizations of splines coincide.
  Properties of splines depend on combinatorics, topology, geometry, and
symmetry of a simplicial or polyhedral subdivision of a region D in R^k, and
are often quite subtle. We describe how duality, group actions, and homology --
techniques which play a central role in many areas of both pure and applied
mathematics -- can be used to illuminate different questions about splines.
  Our target audience is nonspecialists: we provide a concrete introduction to
these methods, and illustrate them with many examples in the context of
splines. We also provide a tutorial on computational aspects: all of the
objects appearing in this note may be studied using open source computer
algebra software.
