This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Mitchell's project is an investigation of affine flag varieties from the point of view of topology, combinatorics, Lie theory and algebraic geometry. The focus is on Schubert, Birkhoff and Richardson varieties, especially the study of singular loci and equivariant cohomology. For example, in earlier joint work with Sara Billey, Mitchell completely determined the smooth and rationally smooth Schubert varieties in an affine Grassmannian; now the goal is to extend these results to more general affine flag varieties, and to refine them by determining the singular loci. In earlier joint work with Luke Gutzwiller, Mitchell completely determined the homotopy type of Birhkhoff varieties. However, many interesting problems concerning their equivariant cohomology and associated Richardson varieties remain unsolved, and will be addressed in the new research program.
Mitchell's project lies at the interface of geometry, topology and combinatorics. Topology, a cousin of geometry but of much more recent origin, is concerned with properties of geometric objects that are invariant under continuous deformations. In recent years it has found surprising applications to physics, engineering (e.g. robotics), computer science and biology (e.g. structure of DNA). Combinatorics, or ``finite math'', is the branch of mathematics most directly applicable to computer technology. This project is concerned with the pure mathematics that forms the foundation of these applications. The key buzzword attached to the project is ``Schubert variety'', a concept impossible to explain in a short paragraph but having a distinguished history of a century and half within mathematics itself, and with applications beginning to be found in fields such as computer graphics and computer vision. Thus the goal of Mitchell's project is to improve our understanding of the geometry, topology and combinatorics of these remarkable ``Schubert varieties''.
