Whitney Stratifications, Partitions and Permutations

The PI is investigating and laying fundamental work in the areas of algebraic, geometric and topo- logical combinatorics, including applications to other sciences. The PI's research is making major contributions to tree areas. First, the PI has a successful research program to study face incidences of polytopes via the cd-index, a non-commutative polynomial which removes all linear face redun- dancies for graded Eulerian posets. In new joint work with Goresky and Readdy, the PI shows the cd-index exists in the more general context of manifolds with Whitney stratified boundary. This extension allows for more straightforward cd-index computations which are topologically cognizant. Current research concerns proving linear inequalities among the coecients of the cd-index. Here the notion of shelling is relaxed and the essential question is to find shelling components having a non-negative contribution.