Lie algebroids and groupoids, supermanifolds, and noncommutative geometry

本项目将研究可微堆栈和代数双模范畴上的群结构，重点研究泊松和量子环面的例子。该项目还将研究复李代数，以深入了解接触流形上Cauchy-Riemann结构的相对指标问题，并构建与广义复流形相关的全纯对象。最后，类群将应用于广义相对论和其他群不够灵活，无法编码系统的自然对称性的领域。在经典几何中，“空间”的概念通常等同于光滑流形或其他具有附加结构的集合。但是，对全局和局部商对象的研究，有时是由数学、物理和工程中的具体问题引起的，往往超出了光滑流形的世界，甚至超出了集合的世界，而被称为（可微）堆栈的对象。在这个新的世界里，在代数的对偶世界里，几何对象可以被识别为函数代数，堆栈成为一个新的范畴里的对象，在这个范畴里的同构就是通常所说的森田等价。该项目致力于理解这些新世界中相应的对称概念。

This project will investigate group structures on differentiable stacks and in bimodule categories of algebras, concentrating on the example of Poisson and quantum tori. The project will also study complex Lie algebroids to gain insight into the relative index problem for Cauchy-Riemann structures on contact manifolds, and to construct holomorphic objects associated with generalized complex manifolds. Finally, applications of groupoids will be made to general relativity and other areas where groups are not flexible enough to encode the natural symmetries of a system.In classical geometry, the notion of "space" is commonly identified with that of smooth manifold or some other kind of set with additional structure. But the study of global and local quotient objects, sometimes arising from concrete problems in mathematics, physics, and engineering, often leads beyond the world of smooth manifolds, and even beyond the world of sets, to objects known as (differentiable) stacks. In this new world, and the dual world of algebra, where geometric objects may be identified with algebras of functions, stacks become objects in a new category in which the isomorphisms are what is generally known as Morita equivalences. The project is devoted to understanding the corresponding notions of symmetry in these new worlds.