Contemporary Schubert Calculus and Schubert Geometry (07w5112)

Objectives

Schubert calculus originally meant the calculus of enumerative geometry, which is the art of counting geometric figures determined by given incidence conditions. This was developed in the 19th century and presented in the classic treatise "Kälkul der abzählanden Geometrie" by Herman Cäser Hannibal Schubert in 1879. Schubert, Pieri, and Giambelli subsequently developed algorithms to solve enumerative geometric problems concerning linear subspaces of vector spaces, which we now understand to be computations in the cohomology ring of a Grassmannian. Their vision and technical skill exceeded the foundations of this subject, and Hilbert, in his 15th problem, asked for a rigorous foundation. This was largely completed by the middle of the 20th century, a centerpiece being the cohomology of Grassmannians.

By the 1950's it was discovered that the cohomology ring of the Grassmannian, with its natural geometric Schubert basis, was essentially the same as the algebra of symmetric functions, with its Schur basis. These are also essentially the same as the representation ring of the general linear group, with its basis of irreducible Weyl modules. In particular, the multiplication in the geometric Schubert basis of cohomology was governed by the combinatorial Littlewood-Richardson rule (which was only proved in the 1970's). These Schubert classes correspond to Schubert varieties in the Grassmannian, and Chevalley showed how there are natural Schubert varieties for any algebraic homogeneous space, thus pointing out a deep connection to algebraic groups. In the last 20 years, the Schubert calculus has come to refer to the study of the geometric, combinatoric, and algebraic aspects of the Schubert basis in various cohomology settings, and its relation to the rest of mathematics.

The past ten years have seen an explosion of progress in the subject. A major impetus for this resurgence was a meeting at Oberwolfach in 1997 at which some people from combinatorics and geometry met each other for the first time. Since then, there has been a flood of important work in this area by Buch, Eremenko-Gabrielov, Knutson-Tao, Vakil, Knutson-Miller-Shimozono and many, many others. For example, Coskun recently announced a solution to the long-standing Littlewood-Richardson problem for the quantum cohomology of a Grassmannian. This meeting at Banff in 2007, 10 years after the Oberwolfach meeting, is intended to assess this recent progress and chart the course for the next decade.