Explorations in Stratified, Topological and Geometric Combinatorics

Overview: The PI is investigating and laying fundamental work in the areas of algebraic, geomet- ric and topological combinatorics, including applications to other sciences. The PI proposes four research projects in these areas. Project I is a continuation of the PI''s successful research program to study face incidences of polytopes via the cd-index, a non-commutative polynomial which re- moves all linear face redundancies for graded Eulerian posets. In a joint work with Goresky and Readdy, the PI shows the cd-index exists in the more general context of manifolds with Whitney stratied boundary. This extension allows for more straightforward cd-index computations which are topologically cognizant. Project I concerns proving linear inequalities among the coecients of the cd-index. Here the notion of shelling is relaxed and the essential question is to nd shelling components having a non-negative contribution. Project II is to obtain linear inequalities for poly- topes by sharpening the results for proving the minimization inequalities for polytopes. Project III is continue to explore classes of inequalities motivated by the cohomology of Shimura varieties. Project IV is to continue the study of lters in the partition lattice and describe their topology. Results of Calderbank, Hanlon and Robinson; and Wachs have been extended by the PI, Jung and Hedmark. The projects are in various areas of enumerative combinatorics with connections to permutations, topology, Hopf algebras and graph theory.