Abstract:

For a ring R and a graph G with edges labeled by ideals in R, an algebraic spline on G is a labeling of the vertices with elements in R so that if two vertices are connected by an edge labeled I, then the difference of their labels is in I. Splines can be used to determine the cohomology of certain algebraic varieties, called GKM spaces for Goresky-Kottwitz-MacPherson. If two GKM spaces are nested, one inside the other, than the GKM graphs describing their cohomology should also be nested, one a subgraph of the other. Given a sequence of nested GKM spaces, can we use splines to find a universal basis for the cohomology? When can we find a universal basis for the splines on a given graph? This talk will describe a work in progress, joint with Nealy Bowden (University of New Hampshire).