The Geometry of Moduli Spaces of Sheaves

PI计划探索代数曲线和曲面上的滑轮参数空间的几何。她想要计算光滑表面上半稳定滑轮的Gieseker模空间上的一些交集理论，特别是Verlinde数。这些交论计算将有助于PI探索作为曲线上丛的模空间的特征的水平秩对偶是否也可以与曲面上的滑轮的模空间相关联。作者还希望探讨经典李代数和仿射李代数关于曲线上Grothendieck Quot格式的上同调的一些表示理论，以及关于曲面上的层的Gieseker模空间的上同调，这将有助于更好地理解这些模空间的几何结构。代数几何中的矢丛的研究与物理学中规范场理论的研究是平行的。最常见的规范场的例子是电磁场。高阶向量丛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。研究丛的模空间上的几何和交集理论，这是本项目的目标，特别是有助于理解所有可能的规范场空间上的各种积分的结构。这些积分是高能物理中各种物理过程的散射幅度的数学表达式。

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.