Aspects of the moduli theory of sheaves and varieties

PI建议研究曲线和曲面上的各种轴的空间，通常与曲线的模空间和某些曲面的模空间的几何有关。特别是，PI将研究在g属光滑曲线的模空间上的Verlinde轴的几何形状如何反映在kappa环的结构中。PI还计划进一步了解目前对表面上的轴的模空间上的行列式线束的几何形状的理解。对于阿贝尔曲面或K3曲面，当允许曲面变化时，关于极化阿贝尔曲面或K3曲面模空间的上同调性问题引起了人们的关注。在另一个方向上，PI将从几何上解释二维杨-米尔斯理论的q-变形。在对重要几何对象的数学研究中，以几何方式描述某一类型的所有对象的集合本身具有相当大的兴趣。这样的集合称为模空间。特别是变轴上的模空间在数学和理论物理中都具有重要的意义。而在物理学中，矢量束的模空间是在规范理论的基础研究中自然出现的。PI计划在不同的设置中分析轮轴的模空间，有可能揭示一系列有趣的几何形状。

The PI proposes to study various spaces of sheaves on curves and surfaces, often in relation to the geometry of the moduli space of curves and of some moduli spaces of surfaces. In particular, the PI will investigate how the geometry of the Verlinde sheaves over the moduli space of smooth curves of genus g is reflected in the structure of the kappa ring. The PI also plans to further the current understanding of the geometry of determinant line bundles over moduli spaces of sheaves on a surface. For abelian or K3 surfaces, when the surface is allowed to vary, interesting questions arise about the cohomology of the moduli space of polarized abelian or K3 surfaces. In a different direction, the PI will interpret geometrically a q-defomation of two-dimensional Yang-Mills theory. In the mathematical study of important geometric objects, there is considerable interest in describing geometrically the set itself of all objects of a certain type. Such a set is called a moduli space. In particular, moduli spaces of sheaves on varieties are important in both mathematics and theoretical physics. Spaces of sheaves over a variety often reflect geometric properties of the underlying variety, while in physics, moduli spaces of vector bundles arise naturally in the fundamental study of gauge theories. The PI plans to analyze moduli spaces of sheaves in various settings, with the potential to shed light on a range of interesting geometries.