FRG: Collaborative Research: Quantum Cohomology, Quantized Algebraic Varieties, and Representation Theory

FRG：合作研究：量子上同调、量化代数簇和表示论

• 批准号: 0854792
• 负责人: Valerio Toledano Laredo
• 金额: $ 25.82万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854792&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Quantum; Cohomology

Recent results of the PI's and the co-PI's suggest a strong connection between the following mathematical objects and constructions: localization theory in representation theory in zero and positive characteristic; derived categories of coherent sheaves on algebraic symplectic varieties; small equivariant quantum cohomology; Casimir-type connections and their monodromy.The goal of the project is to gain a deeper and more detailed understanding of the links between these objects and develop new methods for enumerative algebraic geometry and representation theory based on those links.Representation theory is a branch of mathematics based on the fact that surprisingly rich information about a mathematical or physical object is often hidden in the structure of its symmetries. Throughout some 100 years of its history, a major source of motivation and methods in representation theory has been the interaction with neighboring fields, such as the physics of elementary particles, number theory and geometry. The idea of the present project comes from a new connection of this sort, this time with recent constructions in algebraic geometry motivated by high energy physics. At present this connection has only been observed in particular, though impressive, examples. The aim of the project is to gain a better understanding of the nature of this connection and use this understanding to develop new methods for attacking current problems in several areas of mathematics.

PI 和 co-PI 的最新结果表明以下数学对象和构造之间存在密切联系：零和正特征表示论中的定位理论；代数辛簇上相干滑轮的派生范畴；小等变量子上同调；卡西米尔型联系及其单性。该项目的目标是更深入、更详细地理解这些对象之间的联系，并基于这些联系开发枚举代数几何和表示论的新方法。表示论是数学基于这样一个事实：关于数学或物理对象的令人惊讶的丰富信息通常隐藏在其对称结构中。纵观其一百多年的历史，表示论的动机和方法的一个主要来源是与邻近领域的相互作用，例如基本粒子物理学、数论和几何学。当前项目的想法来自于这种新的联系，这次是与高能物理推动的代数几何的最新构造。目前，这种联系仅在一些令人印象深刻的例子中被观察到。该项目的目的是更好地理解这种联系的本质，并利用这种理解来开发新方法来解决多个数学领域中当前的问题。