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

这个项目属于代数几何领域，它可以被描述为对多项式方程系统解集所切割的几何空间的研究。这门学科有很强的古典传统。目前，该领域的一个重要研究方向是为早期宇宙的物理学建立数学基础。模理论在这项工作中扮演了一个特别重要的角色，它处理相同类型的重要几何对象的分类和变形特性。目前的项目将研究与物理规范场（如电磁场）相对应的物体的模理论。与轴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，研究这些量的结构涉及几何和高能物理的重要问题。因此，在为几何学做出贡献的同时，该项目也试图提高我们对理论物理的理解。在一个方向上，PI将研究表面上点的希尔伯特方案的中心不变量序列——这些空间在几何和表示理论中都是必不可少的对象。她还将尝试把二维杨-米尔斯理论的q-变形建立在严格的几何基础上，通过在四维环境中解释它，并将其与最近关于表面上稳定轴的模空间的不变量的模性的研究联系起来。在不同的方向上，PI的努力将集中在理解曲线和K3曲面上稳定轴的模空间的模有理等价的循环群，尤其是为了证实这些空间的Hodge猜想。该奖项反映了美国国家科学基金会的法定使命，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.