Equivariant Methods in Algebraic Geometry

小行星9870088 这个项目的目的是将等变方法应用于代数几何中的一些问题。 这些问题来自于几何学、组合学和表示论，并受到它们的启发。PI计划考虑与舒伯特品种有关的几个问题，特别是一个几何正性猜想的动机彼得森和比利，这可能是与工作的富尔顿和Lazarsfeld在陈类的正多项式。 另一个舒伯特品种的问题是找到一个皮耶里公式的等变K理论的旗帜品种，推广工作的富尔顿和Lascoux的一般线性群。 除了舒伯特品种，PI计划研究其他方向，其中可以应用等变方法。一个方向是将等变Chow群中的局部化公式推广到更高的Chow群，并利用局部化来研究有趣空间的Chow群。 另一个是涉及到一个等变黎曼-罗克定理的结果对周群的concurents;这是联合工作与丹Edidin。 最后，等变方法被用来研究表示中的多重性; PI希望使这些结果更明确，特别是对于一般线性群。 这是代数几何领域的研究。 代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。

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.