Combinatorial Intersection Theory

University Of Washington, Seattle WA

Investigators

Abstract

The investigator and colleagues study various objects involving the combinatorics of geometric intersections. These include the characteristic classes of oriented matroid bundles as defined by Gelfand and MacPherson, the configuration spaces of simplices first studied by Schubert and an application related to work of Sarkaria to extensions of Tverberg's theorem proposed by Kalai. Oriented matroid bundles are a combinatorial analog of vector bundles for which a classifying space is known. The cohomology of this classifying space is studied. The spaces of simplices are singular moduli spaces which arise in geometric ennumeration problems as well as in the study of generalized Schur modules. A combinatorially defined resolution of singularities is defined and its intersection properties studied. Tverberg's theorem has conjectured extensions by Kalai involving the dimensions of the Tverberg points. The relationship to the deleted join of simplices with the symmetric group action is studied. This work relates several previously studied approaches to the intersections of geometric objects to finite mathematical problems. This will provides a better understanding of both the finite objects in question, as for instance in the third case above, and of the geometric objects, as in the second case above. These finite objects arise frequently in both computer science and physics, and further understanding the relationships to geometric objects could have an impact on these fields.