Universal Series, Chow Rings, and Dualities in the Moduli Theory of Sheaves

这个项目是在代数几何领域，可以描述为研究由多项式方程组的解集切割出的几何空间。这门学科有很强的古典传统。目前，在一个重要的方面，该领域的研究工作是为早期宇宙的物理学建立数学基础。在这一努力中，模量理论发挥了特别重要的作用，它涉及同一类型的重要几何对象的分类和变形特性。目前的项目将研究层的模量理论，层是物理学中与规范场（如电磁场）相对应的对象。与层的模空间相关的数学量对应于粒子物理学中散射过程的振幅。因此，研究这些量的结构关系到几何学和高能物理学中的重要问题。因此，在对几何学做出贡献的同时，该项目还试图提高我们对理论物理学的理解。在一个方向上，PI将研究表面上的点的希尔伯特方案上的中心不变量系列-空间已经成为几何和表示论中的基本对象。 她还将试图把严格的几何基础上的q变形的二维杨米尔斯理论，通过解释它在四维的背景下，并将其与最近的工作模块化的不变量的模空间的稳定层的表面。在另一个不同的方向，PI的努力将集中在理解曲线和K3曲面上稳定层的模空间的模有理等价的循环组，尤其是为了确认这些空间的霍奇猜想。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is in the field of algebraic geometry which can be described as the study of geometric spaces cut out by solution sets of systems of polynomial equations. The subject has a strong classical tradition. At present, in one significant strand, research efforts in the field go toward establishing mathematical foundations for the physics of the early universe. A particularly important role in this effort is played by moduli theory, which deals with the classification and deformation properties of important geometric objects of the same type. The current project will investigate moduli theory of sheaves, objects corresponding to gauge fields (such as the electromagnetic field) in physics. Mathematical quantities associated with moduli spaces of sheaves correspond to amplitudes of scattering processes in particle physics. Studying the structure of these quantities therefore bears on important questions in both geometry and high-energy physics. Thus, while contributing to geometry, the project also seeks to improve our understanding of theoretical physics.In one direction, the PI will investigate central series of invariants on Hilbert schemes of points on surfaces -- spaces which have emerged as essential objects in both geometry and representation theory. She will also try to put on rigorous geometric footing a q-deformation of two dimensional Yang-Mills theory, by interpreting it in a four-dimensional context, and connecting it with recent work on modularity of invariants for moduli spaces of stable sheaves on surfaces. In a different direction, the PI's efforts will be concentrated on understanding the groups of cycles modulo rational equivalence of moduli spaces of stable sheaves on curves and K3 surfaces, not least with a view toward confirming the Hodge conjecture for these spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.