Quantum Algebras, Quiver Varieties, and Applications

量子代数、箭袋种类和应用

• 批准号: 1600375
• 负责人: Andrei Negut
• 金额: $ 18.12万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600375&HistoricalAwards=false
• 关键词: Quantum; Algebras; Quiver; Varieties; Applications

Representation theory is the study of symmetries in mathematics, while algebraic geometry is the study of spaces that can be described by algebraic equations. The interface between these two fields is a rich subject, which has had recent applications to various branches of mathematics, theoretical physics, and combinatorics. In this project the principal investigator will study a particular class of spaces called quiver varieties, which are determined by certain graphs, from which they inherit many interesting symmetries. Abstracting the properties of quiver varieties allows mathematicians to discover many fascinating formulas, whose applications range between such distant fields as string theory and the study of knots. The approach of the principal investigator is two-fold: first, to study the general properties of quiver varieties (cohomology, K-theory, derived categories) by using a technical tool called the shuffle algebra, and second, to apply these techniques in important particular cases in order to solve concrete problems. For example, the study of flag Hilbert schemes leads to a geometric realization of knot invariants. Similarly, the study of moduli spaces of higher rank sheaves leads to a mathematical understanding of the relations between gauge theory and conformal field theory. These and other applications will be pursued by the principal investigator and his coauthors.

表示论是研究数学中的对称性，而代数几何是研究可以用代数方程描述的空间。这两个领域之间的接口是一个丰富的主题，最近已应用于数学，理论物理和组合学的各个分支。在这个项目中，首席研究员将研究一类特殊的空间，称为“类簇”，它们由某些图决定，它们继承了许多有趣的对称性。抽象的性质，使数学家发现许多迷人的公式，其应用范围之间的弦理论和研究结等遥远的领域。 主要研究者的方法是双重的：第一，通过使用称为洗牌代数的技术工具来研究多个簇（上同调，K-理论，导出范畴）的一般性质，第二，将这些技术应用于重要的特殊情况，以解决具体问题。例如，旗希尔伯特方案的研究导致结不变量的几何实现。同样地，高阶层的模空间的研究也导致了对规范理论和共形场论之间关系的数学理解。这些和其他应用程序将由主要研究员及其合著者进行。