Ttile:
New invariants of metric spaces: magnitude and maximum entropy.
Abstract:
This is the story of some of the geometrical fruits of a large-scale categorical programme to investigate invariants of size. One such fruit is magnitude, a (newish) real invariant of compact metric spaces, whose asymptotic behaviour determines classical invariants such as volume, surface area, dimension, etc. Another is a suite of new measures of entropy, generalizing classical quantities from information theory and closely related to measures of biological diversity. We’ll see that every compact metric space carries a canonical probability measure, which maximizes entropy in infinitely many senses at once (a result joint with Emily Roff). I will explain all this, starting from the beginning.