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

PI‘s和co-PI’s的最新结果表明下列数学对象和结构之间有很强的联系：零点和正特征表示理论中的局部化理论；代数辛簇上的连贯层的派生范畴；小等变量子上同调；Casimir型连接及其单列。该项目的目标是更深入和详细地了解这些对象之间的联系，并在这些联系的基础上发展计数代数几何和表示理论的新方法。表示论是数学的一个分支，基于这样一个事实，即关于数学或物理对象的惊人丰富的信息往往隐藏在其对称性结构中。在其大约100年的历史中，表象理论的一个主要动机和方法一直是与邻近领域的相互作用，如基本粒子物理学、数论和几何。本项目的想法来自于这类新的联系，这一次是与高能物理推动的代数几何的最新结构。目前，这种联系只观察到了特别的例子，尽管令人印象深刻。该项目的目的是更好地了解这种联系的性质，并利用这种理解来开发新的方法来解决当前数学领域中的几个问题。

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.