Geometry of Langlands Duality

该项目旨在构建有望成为数学（朗兰兹计划）和物理学（规范理论和弦理论）中各种对偶现象的基础和解释的思想。在每一种情况下，“二元性”意味着一种现象有两种同样有效的描述。然而，在目前的理解水平上，这些似乎无关紧要。对同一物体的两种不同的描述表明，一个更全面的框架将有助于对所观察到的二元性作出概念性解释。本项目的目的是建立这样一个框架。这些发展的数学起源是所谓的朗兰兹纲领，这是经典数论的现代观点。朗兰兹纲领是数学（表示论、代数几何、同伦理论）和物理（规范理论、弦理论）之间的桥梁。研究生将参与该研究。用更专业的术语来说，该项目将几何朗兰兹纲领的制定从一维扩展到任意维。要超越一维，需要重新表述几何朗兰兹纲领，使其更好地匹配更高维的物理理论。这个项目的基础是一个代数上的“代数几何的内上同调”的建设，包括非交换系数。这里，“内”意味着这个上同调的值将是导出代数几何中的对象，而不是简单的集合。因此，任意维的几何类场论对于这种内上同调都应该是Poincare对偶，几何Langlands程序应该是Poincare对偶到非交换上同调的推广.一个经典的问题是，系数在非阿贝尔群中的上同调在二次以上没有定义，并且在一次以上没有太多的内容。该项目的观点是，在约化群中系数的上同调可以定义为在环面中系数的上同调的局部分布。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project is directed towards building ideas that are expected to underlie and explain various duality phenomena in mathematics (the Langlands program) and physics (gauge theory and string theory). In each case, "duality" means that one phenomenon has two equally valid descriptions. However, at the current level of understanding these seem unrelated. Two distinct descriptions of the same object suggest that a more comprehensive framework would serve to give a conceptual explanation of the observed dualities. The aim of this project is to develop such a framework. The mathematical origin of these developments is the so called Langlands program, which is the modern view on classical number theory. The Langlands program functions as a bridge between mathematics (representation theory, algebraic geometry, homotopy theory) and physics (gauge theory, string theory). Graduate students will be involved in the research.In more technical terms, this project will extend the formulation of the geometric Langlands program from one dimension to arbitrary dimensions. Going beyond dimension one requires a reformulation of the geometric Langlands program that better matches physical theories in higher dimensions. The basis of this project is the construction of a conjectural "inner cohomology in algebraic geometry," including noncommutative coefficients. Here, "inner" means that the values of this cohomology would be objects in derived algebraic geometry rather than simply sets. As a consequence, geometric class field theory in arbitrary dimension should be Poincare duality for this inner cohomology and the geometric Langlands program should be an extension of Poincare duality to non-commutative cohomology. A classical problem is that cohomology with coefficients in a non-abelian group is not defined beyond degree two and does not have much content beyond degree one. The point of view taken in this project is that higher cohomology with coefficients in a reductive group can be defined as certain local distributions on cohomology with coefficients in tori.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.