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.
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