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

PI和co-PI的最新结果表明，以下数学对象和结构之间存在着很强的联系：表征中的局部化理论、零和正特征中的局部化理论；代数辛变簇上相干束的派生范畴小等变量子上同调；卡西米尔型连接及其单一性。该项目的目标是更深入和更详细地了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理论的新方法。表征理论是数学的一个分支，它基于这样一个事实，即关于数学或物理对象的丰富信息往往隐藏在其对称结构中。在大约100年的历史中，表征理论的动力和方法的主要来源是与邻近领域的相互作用，如基本粒子物理学、数论和几何。当前项目的想法来自于这类新的联系，这一次是与最近由高能物理驱动的代数几何结构有关。目前，这种联系虽然令人印象深刻，但只被特别观察到。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的方法来解决几个数学领域的当前问题。

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.