The Geometry of Moduli Spaces of Sheaves

PI计划在代数曲线和曲面上探索轮轴参数空间的几何形状。她想计算光滑表面上半稳定轮轴的Gieseker模空间上的一些交点理论，特别是Verlinde数。这些相交理论计算将有助于PI探测作为曲线上束的模空间特征的水平-秩对偶性是否也可以与表面上束的模空间相关联。本文还探讨了经典李代数和仿射李代数关于曲线上Grothendieck格式的上同调的一些表示理论，以及关于曲面上束的Gieseker模空间的上同调的一些表示理论，从而更好地理解这些模空间的几何结构。代数几何中向量束的研究与物理学中规范场论的研究相似。规范场最常见的例子是电磁场。高阶向量束对应于高能物理中出现的非阿贝尔规范场，并且在数学上被组织为矩阵的条目。研究束模空间上的几何和交点理论是本课题的目标，它特别有助于理解所有可能规范域空间上各种积分的结构。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。

The PI plans to explore the geometry of parameter spaces of sheaves on algebraic curves and surfaces. She wants to calculate some of the intersection theory, in particular Verlinde numbers, on Gieseker moduli spaces of semistable sheaves on a smooth surface. These intersection-theoretic calculations will help the PI probe whether the level-rank duality which is a feature of moduli spaces of bundles on a curve, can be associated with moduli spaces of sheaves on a surface as well. The author also wishes to explore some of the representation theory of classical and affine Lie algebras on the cohomology of Grothendieck quot schemes on curves, as well as on the cohomology of Gieseker moduli spaces of sheaves on a surface, which would result in a better understanding 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.