Magnitude

Magnitude is an invariant that originated in category theory. It is defined in the very wide generality of enriched categories, and specializes in several significant directions, thus producing numerical invariants of ordinary categories, linear categories, ordered sets, graphs, and - most significantly - metric spaces. Far from being of purely categorical interest, magnitude has found application in metric geometry (using some sophisticated analytic techniques), graph theory, and certain parts of mathematical biology (via the related concept of maximum diversity).

Subsequently, magnitude was shown to be the shadow of a new graded homology theory (magnitude homology), in the sense that magnitude is the Euler characteristic of magnitude homology. At first, magnitude homology was only defined for graphs (by Hepworth and Willerton). Work of Shulman and Leinster later extended the definition to arbitrary enriched categories (including metric spaces), but aspects of the framework remain to be clarified improved.

This project has two parts. First, the student will work on extending the maximum diversity theorem from finite spaces to arbitrary compact spaces.
This task will make use of some of her expertise in analysis. Second, she will work to clarify and extend the framework of magnitude homology for enriched categories, with the ultimate aim of showing that magnitude can be recovered as the Euler characteristic of magnitude homology not only for finite spaces, but for compact spaces in general.