Abstract

A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs…) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs.

A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead. In this talk, we will motivate the (asymptotic) notion of infrasup umbel convexity (introduced in collaboration with Chris Gartland (UC San Diego)) and discuss the value of this invariant for Heisenberg groups. This (asymptotic) invariant is inspired by the profound work of Lee, Mendel, Naor, and Peres on the (local) notion of Markov convexity. If time permits we will discuss the notion of bicone convexity, a new asymptotic invariant, inspired by the work of Eskenazis, Mendel, and Naor on the (local) notion of diamond convexity.

All these metric invariants share the common feature of being derived from point-configuration inequalities which generalize curvature inequalities.