The PI will study how combinatorial structure of partially ordered sets (posets) gives information about associated algebraic and topological structures. Specifically, she seeks ways to use the combinatorics of a face poset or closure poset together with limited topological information (e.g. information about codimension one cell incidences) to determine topological information such as rational homology or even in some cases homeomorphism type. A main goal is to prove that various stratified spaces whose closure posets are thin and shellable are in fact regular CW complexes. She also aims to construct boundary maps on important classes of posets such as geometric lattices, motivated by potential applications e.g. in theoretical computer science. She will also study representations of simple Lie algebras via an analysis of local structure within the poset that indicates which weight space generators appear with nonzero coefficient upon application of a raising or lowering operator to other weight space generators.
In computer science, there is a strong interest both in finding effective algorithms when this is possible, and also in understanding when this is not theoretically possible. It is often difficult to know how to even begin to obtain these impossibility results, but topology combined with combinatorics does have the potential to yield such results, and there have been some striking successes in the past. One of the proposed projects is somewhat in this vein -- to prove impossibility results for the question of which graphs (i.e. which models for networks of computers) admit data sorting algorithms by greedily swapping pieces of neighboring data that are out of order. In another direction, geometric structures such as the Schubert varieties of representation theory/algebraic geometry often have combinatorial data associated to them which is organized into what is called a partially ordered set. Much is known about how properties of the geometric structure force properties on the partially ordered set. The PI seeks a better understanding of how much can be said in the other direction, perhaps allowing some manageable extra data such as information about how edges are attached to 2-dimensional faces, how 2-dimensional faces are attached to 3-dimensional faces, and so on. This could have applications to finding new ways of figuring out what very complicated, high-dimensional geometric objects look like. In addition to continuing to mentor students and run seminars, the PI will also continue organizing conferences designed to foster fruitful new interactions and collaborations between those working in topological combinatorics and in related areas such as combinatorial representation theory and combinatorial commutative algebra.
