Geometric Analysis and Topology Seminar

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Shortcut Laakso spaces, pure PI unrectifiability, and differentiability of Lipschitz functions

Speaker: David Bate, University of Warwick

Location: Online

Videoconference link: https://nyu.zoom.us/s/96390398010

Date: Friday, March 6, 2026, 11 a.m.

Synopsis:

A Lipschitz differentiability space (LDS) is a metric measure space that
satisfies Cheeger's generalisation of Rademacher's theorem. In seminal
work, Cheeger proved that any doubling space that satisfies a Poincaré
inequality is a LDS. It has since been an open question whether a LDS
must necessarily posses a Poincaré inequality.

After an overview of this history of this problem, this talk will
present a general procedure to construct LDS that do not possess a
Poincaré inequality by contracting the distance of a given metric space
X. In fact, we provide a general investigation into the geometry of the
resulting "shortcut" metric space and characterise when such spaces are
PI rectifiable, and when they are Lipschitz differentiability spaces. We
then specialise to X a Laakso space, show that our characterisations
yield a new family of purely PI unrectifiable LDS, and investigate the
possible Banach space targets for which Lipschitz differentiability holds.

Our constructions highlight that uniform convexity of the target plays a
crucial role for Lipschitz differentiability in the absence of a
Poincaré inequality.
 

Notes:

This talk will be on Zoom.