Problems in Equivariant Algebraic Geometry

本课题的目的是用等变方法研究具有群作用的代数簇的问题。这个项目的一部分涉及具有扭矩作用的变种的矩图。对于一个环面上有有限多个不动点和曲线的变种，可以定义一个组合对象，称为矩图。在用矩图计算上同调或交同调等拓扑不变量方面取得了很大的进展。研究人员建议将其扩展到K-理论。舒伯特变种是具有这种环面作用的最重要的变种之一；研究人员计划继续与BrianBoe就一个猜想进行合作，该猜想将简化舒伯特变种中某一点是否理性光滑的确定。此外，他计划将一些仅为舒伯特变种所知的事实扩展到具有这种环面作用的变种的更一般设置。此外，研究人员计划使用等变方法来研究某些有趣的变体：他计划计算Chern-Schwartz-MacPherson类的简并轨迹，并计算约化李群的幂零伴随轨道的有趣不变量(度，推进度量)。该项目属于被称为“代数几何”的数学领域。代数几何学通过将几何对象描述为多项式方程的解来研究几何对象--这样的对象被称为“代数簇”。幸运的是，尽管代数变体可能非常复杂，但它们中的许多都具有很大的对称性。数学家们一直在深入研究这种具有额外对称性的变种，原因有几个：这种额外对称性的存在使这些变种更容易研究，因此这些变种在开发研究所有变种的方法时是有价值的测试案例。此外，具有额外对称性的变种本身也很有趣：它们在数学的各个领域都扮演着重要的角色，包括数论、组合学和表示论。最近在开发理解这些品种的技术方面取得了相当大的进展；这个项目是关于推广这些技术，并利用它们来研究特定类别的品种。

The purpose of this project is to use equivariant methods to studyproblems concerning algebraic varieties with group actions. Partof this project involves moment graphs of varieties with torusactions. For a variety on which a torus acts with finitely manyfixed points and curves, one can define a combinatorial objectcalled a moment graph. There has been much recent progress incomputing topological invariants, such as cohomology orintersection homology, in terms of the moment graph. Theinvestigator proposes to extend this to K-theory. Schubertvarieties are among the most important varieties with this type oftorus action; the investigator plans to continue work with BrianBoe on a conjecture that would simplify determining if a point ina Schubert variety is rationally smooth. Moreover, he plans toextend some facts known only for Schubert varieties to the moregeneral setting of varieties with this type of torus action. Inaddition, the investigator plans to use equivariant methods tostudy certain interesting varieties: he plans to calculateChern-Schwartz-MacPherson classes of degeneracy loci, and tocalculate interesting invariants (degrees, push-forward measures)of nilpotent adjoint orbits of reductive Lie groups.This project is in the area of mathematics referred to as"algebraic geometry." Algebraic geometry studies geometricobjects by describing them as solutions to polynomial equations-- such objects are called "algebraic varieties." Fortunately,although algebraic varieties can be very complicated, many ofthem have a great deal of symmetry. Mathematicians have beenintensely investigating such varieties with extra symmetry forseveral reasons: The presence of such extra symmetry makesthese varieties easier to study, so that these varieties arevaluable test cases in developing methods to investigate allvarieties. Moreover, varieties with extra symmetry are of greatinterest in their own right: they play important roles invarious areas of mathematics, including number theory,combinatorics, and representation theory. There has beenconsiderable recent progress in developing techniques tounderstand these kinds of varieties; this project is aboutextending these techniques, and using them to study particularclasses of varieties.