The idea of assigning weights to local coordinate functions appears in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted deformation spaces. As an application, I will discuss the osculating tangent bundle for Lie filtrations, and the corresponding tangent groupoid of Choi-Ponge, van Erp-Yuncken, and Haj-Higson. (Based on joint work with Yiannis Loizides.)

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