Singular Foliations: Desingularization and the Baum-Connes Conjecture

奇异叶状结构：去奇异化和鲍姆-康尼斯猜想

• 批准号: 272988935
• 负责人: Professor Dr. Thomas Schick
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/272988935?language=en
• 关键词: Singular; Foliations; Desingularization; Baum; Connes

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.

奇异叶是动力系统的例子，它们出现在大量的几何情形中，例如李群和泊松几何的作用。事实上，泊松结构完全由它们相关的奇异叶理(辛)决定。要处理奇异的叶面并理解它们，正是因为它们的独特性质，所以需要开发新的工具。一个(奇异叶)的动力学被编码在它的完整群胚及其相关的群胚C*-代数中。要理解这些，我们必须理解他们的K理论，最常见的是通过鲍姆-康奈斯猜想。要做到这一点，第一个悬而未决的大问题是构建预期的答案：单叶的适当行动的分类空间。我们建议使用更高阶(更高Liecategory)方法来实现这一点。这将作为单一叶理的完全单一化(通过适当的决议)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。然后，最终目标是应用这些方法来计算沿奇异叶系的薛定谔算子的谱。

项目成果

Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves

StefanâSussmann 奇异叶状结构、奇异子代数体及其相关滑轮

• DOI: 10.1142/s0219887816410012
• 发表时间: 2016
• 期刊: International Journal of Geometric Methods in Modern Physics
• 影响因子: 1.8
• 作者: I. Androulidakis;M. Zambon
• 通讯作者: M. Zambon

Almost regular Poisson manifolds and their holonomy groupoids

几乎正则泊松流形及其完整群群

• DOI: 10.1007/s00029-017-0319-5
• 发表时间: 2017
• 期刊: Selecta Mathematica
• 作者: I. Androulidakis;M. Zambon
• 通讯作者: M. Zambon