Lie algebroids and groupoids, supermanifolds, and noncommutative geometry

这个项目将研究代数的可微堆叠和双模范畴上的群结构，集中在Poisson和量子Tori的例子上。该项目还将研究复李代数群，以深入了解接触流形上Cauchy-Riemann结构的相对指标问题，并构造与广义复流形相关的全纯对象。在经典几何中，“空间”的概念通常等同于光滑流形或具有附加结构的其他类型的集合。但对全局和局部商对象的研究，有时源于数学、物理和工程中的具体问题，往往会导致超越光滑流形的世界，甚至超越集合的世界，通向被称为(可微)栈的对象。在这个新的世界中，在代数的对偶世界中，几何对象可以用函数的代数来标识，堆栈成为一个新范畴中的对象，在这个新范畴中，同构是通常所说的Morita等价。该项目致力于理解这些新世界中相应的对称性概念。

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.