Poincaré inequalities and geometry

Poincaré inequalities and Loewner conditions are intricately connected to the study of functions on the space and the geometry of the space. However, their characterization has been challenging. My work in this area started from the study of the infinitesimal geometry of RNP-differentiability spaces, which lead to a new way of characterizing spaces satisfying Poincaré inequalities. Remarkably, a covering by subsets of spaces with Poincaré inequalities can be characterized by a strong form of Cheeger’s differential structure. Further, the new characterizations showed that essentially all “weak” notions of Poincaré inequalities imply real Poincaré inequalities. Say Orlicz-Poincaré, non-homogenous Poincaré, Measure Contraction Property, or existence of “thick families of nearly connecting curves/geodesics”.

Later, this morphed into other conditions and applications. Specifically, we were able to show the existence of a Semmes family with a short and self-contained proof. These tools, and the short proof with it seem to shed light on other problems where the existence of many curves imposes structure on the underlying space.

My current focus is on applying this “curve-wise” philosophy further with the aim of giving a more concrete and direct understanding of these inequalities and the structures, such as Cheeger’s differential structure, that they impose.


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