The smooth algebra of a one-dimensional singular foliation

The analytic index of elliptic pseudodifferential operators on a singular foliation

In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic foliation" TF on $M \times \R$, which we introduce here.

Pseudodifferential calculus on a singular foliation

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation F on a compact manifold M, and we show that it can be naturally understood as a positive, unbounded, self-adjoint operator on L2(M).

Blow-up groupoid of singular foliations

We introduce a blow-up construction of a smooth manifold along the singular leaves of an arbitrary singular foliation in the sense of Stefan and Sussmann, as well as a blow-up construction of the holonomy groupoid defined by Androulidakis and Skandalis. Our construction gives a locally compact locally Hausdorff groupoid, which can be regarded as a desingularisation of the singular foliation. We show that it retains some smooth structure.

关于与给定超曲面相切的奇异叶状结构

On singular foliations tangent to a given hypersurface

本文研究一类与给定超曲面相切的奇异叶状结构，通过引入横截阶参数并利用环路诱导的jet层次完整不变量，实现了此类叶状结构的完全分类及其群胚和代数的具体描述。