Introductory description

Analysis on metric spaces emerged in the 90’s when interest in extending various results from the classical smooth to non-smooth settings arose. Motivations for this came from quasiconformal analysis, Ricci limit spaces, sub-Riemannian geometry as well as geometric measure theory. A first order calculus was developed for spaces with enough analytical structure, namely spaces supporting a Poincare inequality (PI spaces), culminating with Cheeger’s introduction of a weak differentiable structure in PI spaces. Although PI-spaces enjoy a rich analytical structure, they can exhibit highly non-manifold-like behaviour, which makes Cheeger’s result all the more remarkable. All of
hese developments rely on a theory of Sobolev spaces over metric spaces equipped with a measure; Sobolev spaces,
n turn, are based on a suitable substitute for weak derivatives in the absence of a coordinate structure.

A notion that came to underlie most of the theory is that of upper gradients, which control the oscillation of a function f along curves. Moreover, curves turn out to be central in non-smooth analysis. For example the Poincare inequality should be seen as a quantitative way to turn curwise control, given by the upper gradient, into control over the average oscillation in the sense of the measure.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Possible topics comprise:

• Curves in metric spaces, measure theory, upper gradients, Sobolev spaces, Poincare inequality, differentiability in metric spaces

Learning outcomes

By the end of the module, students should be able to:

• Define a Sobolev over a metric measure space
• Understand the Poincare inequality and some of its basic geometric implications
• Understand Cheeger's differentiable structure and the proof of its existence on PI-spaces.

Subject specific skills

Curves in metric spaces, modulus of curve families, upper gradients, Sobolev spaces on metric spaces, Poincare inequalities

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)