Singular Foliations: Desingularization and the Baum-Connes Conjecture

奇异叶是动力系统的一个例子，它们出现在大量的几何情况中，如李群的作用和泊松几何。事实上，泊松结构完全由其相关的奇异叶理（辛）决定。由于奇异叶的特殊性，研究奇异叶并理解它们，就需要开发新的工具。奇异叶理的动力学被编码在它的完整群拟和相关群拟C*-代数中。要理解这些，我们必须理解他们的k理论，最常见的是通过Baum-Connes猜想。要做到这一点，第一个大的开放问题是预期答案的构建：奇异叶理适当作用的分类空间。我们建议使用高阶（高分类）方法来实现这一目标。这将作为奇异叶理的完全去象形化（通过适当的分辨率）来完成。也就是说，问题是找到一个具有足够可微结构的空间，作为叶空间的模型。最后的目标是应用这些方法计算沿奇异面理的Schrödinger型算符的谱。

Singular foliations are examples of dynamical systems and they appear in an abundance of geometric situations, such as actions of Lie groups and Poisson geometry. In fact, Poisson structures are completely determined by their associated singular foliation (symplectic). To work with singular foliations and to understand them requires precisely because of their singular nature the development of new tools. The dynamics of a (singular foliation) is encoded in its holonomy groupoid and the associated groupoid C*-algebra. To understand those, one must understand their K-theory, most commonly via a Baum-Connes conjecture.The first big open question to do this is the construction of the expectedanswer: the classifying space for proper action of thesingular foliation. We propose to achieve this using higher order (higher Liecategory) methods. This will be done as a completedesingularization of singular foliation (via suitable resolutions). Namely, the problem is to find a space withenough differentiable structure, which acts as a model for the leaf space.A final goal then is the application of these methods for the calculation ofthe spectrum of Schrödinger type operators along the singular foliation.