The proposers will collaborate on a suite of main projects in the combinatorics, geometry, and topology of Grassmannians, flag manifolds, and their Schubert varieties. These varieties are central to questions spanning many areas of classical mathematics, such as algebraic topology, enumerative geometry, and representation theory. They hope to resolve some thorny questions about classification of singularities and topology of Schubert varieties, e.g. the 30-year old problem of the fixed-point-property for Grassmannians. They also propose several additional projects, continuing some of the PI and co-PI's past work in algebraic combinatorics. These projects relate to the recently discovered "cyclic sieving phenomenon", to alternating subgroups of Coxeter groups, and to critical groups of graphs. Some of these projects are outgrowths of the PI's past (NSF-funded) REU's.
The appeal of this work is that counting problems, one of the staples of the field of combinatorics, which seem unrelated to algebra or geometry sometimes find the key to their solution and/or application in these other areas. This has been particularly true for the algebraic/geometric objects in this proposal: the Schubert varieties in flag manifolds and Grassmannians. These objects are the key to understanding how lines and planes can tilt and intersect at different angles in the two-dimensional plane, in three-dimensional space, and in higher dimensions.
