The Geometry of Moduli Spaces of Sheaves

滑轮模空间的几何

• 批准号: 0700742
• 负责人: Alina Marian
• 金额: $ 12万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2008-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700742&HistoricalAwards=false
• 关键词: Geometry; Moduli; Spaces; Sheaves

The PI plans to explore the geometry of parameter spaces of sheaves onalgebraic curves and surfaces. She wants to calculate some of theintersection theory, in particular Verlinde numbers, on Gieseker modulispaces of semistable sheaves on a smooth surface. Theseintersection-theoretic calculations will help the PI probe whether thelevel-rank duality which is a feature of moduli spaces of bundles on acurve, can be associated with moduli spaces of sheaves on a surface aswell. The author also wishes to explore some of the representation theoryof classical and affine Lie algebras on the cohomology of Grothendieckquot schemes on curves, as well as on the cohomology of Gieseker modulispaces of sheaves on a surface, which would result in a betterunderstanding of the geometric structure of these moduli spaces.The study of vector bundles in algebraic geometry parallels that of gauge field theories in physics. The most familiar example of a gauge field is the electromagnetic field. Higher-rank vector bundles correspond to nonabelian gauge fields, which appear in high-energy physics, and mathematically are organized as the entries of a matrix. Studying the geometry and intersection theory on the moduli space of bundles, which is the goal of this project, leads in particular to understanding the structure of various integrals on the space of all possible gauge fields. These integrals are mathematical expressions for scattering amplitudes of various physical processes in high-energy physics.

PI计划探索代数曲线和曲面上的层的参数空间的几何。她想计算一些theintersection理论，特别是Verlinde数，对Gieseker modulispaces的半稳定层在一个光滑的表面。这些相交理论的计算将有助于PI探测是否thelevel-rank对偶，这是一个特点，模空间的丛上的曲线，可以与模空间的层上的表面以及。作者还希望探讨经典李代数和仿射李代数在曲线上的Grothendieckquot方案的上同调以及曲面上层的Gieseker模空间的上同调上的一些表示理论，这将有助于更好地理解这些模空间的几何结构。规范场最熟悉的例子是电磁场。高阶向量丛对应于非阿贝尔规范场，它出现在高能物理中，在数学上被组织为矩阵的元素。 研究几何和相交理论的模空间的丛，这是这个项目的目标，导致特别是理解各种积分的结构上的空间的所有可能的规范场。这些积分是高能物理中各种物理过程散射振幅的数学表达式。