CAREER: Higher Enumerative Geometry via Representation Theory and Mathematical Physics

职业：通过表示论和数学物理进行高等枚举几何

• 批准号: 1845034
• 负责人: Andrei Negut
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1845034&HistoricalAwards=false
• 关键词: CAREER; Higher; Enumerative; Geometry; via

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.

该项目将增强对列举不变性的理解，这是对各种几何形状的数学研究中出现的重要数字，以及量子物理理论的基本物理量，通过研究几何本身的基本对称性。枚举几何形状与数学和物理学的许多其他部分有关，并在结理论到组合学的领域进行了具体应用。因此，该领域对来自各种背景的研究生感兴趣，为此，PI将在与该项目相关的主题上组织年度暑期学校。将研究的几何形状属于两个主要类别之一。首先，由图的组合基准构建的Quiver品种。箭量品种的对称性与Kac-Moody Lie代数，Yangians和量子组有关。其次，表面和三倍的滑轮的模量空间，尤其是在希尔伯特方案的重要示例中，参数化全局对象及其研究涉及不同的技术。两个类别的空间的相交由局部几何形状组成，例如平面或感谢您的表面，其中可以明确计算许多物理数量。在这种情况下，通过在空间上集成了共同体学类别，采用线条捆绑包的欧拉特征，或者甚至在研究空间本身的派生类别，从而获得了枚举的枚举不变性。该项目将研究所有三种方法。该奖项反映了NSF的法定任务，并被认为是使用基金会的知识分子优点和更广泛的影响评论标准的评估值得支持的。