Formulas for Quiver Varieties and Quantum Schubert Calculus

箭袋品种和量子舒伯特微积分的公式

• 批准号: 0603822
• 负责人: Anders Buch
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603822&HistoricalAwards=false
• 关键词: Formulas; Quiver; Varieties; Quantum; Schubert

The investigator will study topics in the broad area of enumerativegeometry. Quiver varieties form a general type of degeneracy locus,which includes many important varieties such as determinantalvarieties and Schubert varieties. The investigator will attempt togeneralize known geometric and combinatorial formulas andconstructions concerning equioriented quiver varieties of type A towork for more general quiver varieties of Dynkin type. He will alsostudy various cohomology theories of generalized flag manifolds withan emphasis on the multiplication of Schubert classes. In particular,he will attempt to understand the quantum cohomology of isotropicGrassmannians (with A. Kresch and H. Tamvakis). He also hopes todetermine if the Gromov-Witten invariants on Grassmannians and otherhomogeneous spaces are determined by positivity properties. As aseparate goal in his research, the investigator will develop computerprograms for computing the studied formulas and invariants.Much development in algebraic geometry has been motivated byenumerative geometric problems, that is, problems that ask for thenumber of geometric objects of a specified type that satisfy certainconditions. For example, quantum cohomology is a theory for countingthe number of curves of given degree in a variety that meet a numberof fixed subvarieties. The invention of quantum cohomology wasmotivated by physics, where the ability to count curves hasapplications to mirror symmetry. In general, enumerative problems canbe attacked by constructing a moduli space that has one point for eachgeometric object that could be counted, such that the conditions onthe objects correspond to subvarieties of the moduli space. Thisreduces an enumerative problem to understanding the intersection ofsubvarieties of a larger space. The relevant subvarieties can oftenbe constructed as special cases of quiver varieties, so this type ofvarieties provide a powerful tool in enumerative geometry. Inaddition, the study of quiver varieties has given rise to many newrelations between algebraic geometry and combinatorics, as well aspowerful methods and constructions in both fields.

研究者将研究枚举植物计量学广泛领域的课题。颤抖型变种是退化位点的一种一般类型，它包括许多重要的变种，如决定性变种和舒伯特变种。研究者将尝试推广已知的关于A型等取向颤振的几何和组合公式和结构，以适用于更一般的Dynkin型颤振。他还将研究广义旗流形的各种上同调理论，重点是舒伯特类的乘法。特别是，他将尝试理解各向同性格拉斯曼人的量子上同源性（与A. Kresch和H. Tamvakis）。他还希望确定格拉斯曼空间和其他齐次空间上的Gromov-Witten不变量是否由正性决定。作为他研究的另一个目标，研究者将开发计算机程序来计算所研究的公式和不变量。代数几何的许多发展都是由枚举几何问题推动的，即要求满足某些条件的特定类型的几何对象的数量的问题。例如，量子上同调是一种计算某一种类中满足若干固定子种类的给定度曲线数目的理论。量子上同调的发明是由物理学激发的，在物理学中，计算曲线的能力可以应用于镜像对称。一般来说，枚举问题可以通过构造一个模空间来解决，该模空间对每个可计数的几何对象都有一个点，这样对象上的条件就对应于模空间的子变体。这将枚举问题简化为理解更大空间的子变种的交集。相关的子变种通常可以构造为颤振变种的特殊情况，因此这类变种在枚举几何中提供了一个强有力的工具。此外，颤振变型的研究在代数几何和组合学之间产生了许多新的关系，并在这两个领域中产生了强有力的方法和构造。