CAREER: Higher Enumerative Geometry via Representation Theory and Mathematical Physics

这个项目将通过研究几何本身的基本对称性来加强对枚举不变量的理解，枚举不变量是在各种几何的数学研究中出现的重要数字，也是量子物理理论中的基本物理量。计数几何与数学和物理的许多其他部分相联系，在从纽结理论到组合学的各个领域都有具体的应用。因此，来自不同背景的研究生对这个领域感兴趣，为此，PI将每年组织与该项目相关的主题暑期班。将学习几何的空间分为两个主要班级之一。首先，由图的组合数据构建的箭图变种。箭图簇的对称性与Kac-Moody李代数、延安和量子群有关。其次，曲面和三叠层上的滑层的模空间，特别是在Hilbert格式的重要例子中，将全局对象参数化，它们的研究涉及不同的技术。这两类空间的交集由局部几何组成，如平面或环面，其中许多物理量都可以显式计算。在这一背景下，感兴趣的计数不变量是通过在空间上积分上同调类，利用线丛的欧拉特征，或者在更高的水平上甚至研究空间本身的派生范畴来获得的。本项目将研究所有这三种方法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will enhance the understanding of enumerative invariants, which are important numbers that arise both in the mathematical study of various geometries, as well as fundamental physical quantities in quantum physical theories, through the study of the underlying symmetries of the geometry itself. Enumerative geometry is connected with many other parts of mathematics and physics, with concrete applications to fields from knot theory to combinatorics. As such, the field is of interest to graduate students from a variety of backgrounds, and to this end the PI will organize yearly summer schools on topics connected with the project.The spaces whose geometry to be studied fall into one of two major classes. First, quiver varieties that are built from the combinatorial datum of a graph. Symmetries of quiver varieties are connected to Kac-Moody Lie algebras, Yangians and quantum groups. Second, moduli spaces of sheaves on surfaces and threefolds, especially in the important example of Hilbert schemes, parametrize global objects, and their study involves different techniques. The intersection of the two classes of spaces consists of local geometries such as the plane or toric surfaces, wherein many physical quantities can be explicitly computed. In this context, the enumerative invariants of interest are obtained by integrating cohomology classes over the space, taking the Euler characteristic of line bundles, or at a higher level even studying the derived categories of the spaces themselves. This project will study all three approaches.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.