Equivariant Methods in Algebraic Geometry

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9870088 Graham The purpose of this project is to apply equivariant methods to some problems in algebraic geometry. These problems come from, and are motivated by, geometry, combinatorics, and representation theory. The PI plans to consider several questions related to Schubert varieties, in particular, a geometric positivity conjecture motivated by conjectures of Peterson and Billey, which is possibly related to work of Fulton and Lazarsfeld on positive polynomials in Chern classes. Another Schubert variety problem is to find a Pieri formula in the equivariant K-theory of the flag variety, generalizing work of Fulton and Lascoux for the general linear group. Besides Schubert varieties, the PI plans to investigate other directions where equivariant methods can be applied. One direction is to extend localization formulas in equivariant Chow groups to higher Chow groups, and use localization to investigate the Chow groups of interesting spaces. Another is to relate an equivariant Riemann-Roch theorem to results on the Chow groups of quotients; this is joint work with Dan Edidin. Finally, equivariant methods have been used to study multiplicities in representations; the PI would like to make these results more explicit, especially for the general linear group. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.