The convex magnitude conjecture


Venue   British Mathematical Colloquium 2013, University of Sheffield, 28 March 2013.

Abstract   (Joint work with Simon Willerton)

Magnitude is a real-valued invariant of metric spaces, springing from a category-theoretic study of size. Unlike most invariants of metric spaces, it changes unpredictably as the space is scaled up or down. It therefore assigns to each space a real-valued function on the positive real line. Roughly, the Convex Magnitude Conjecture states that for convex subsets of Rn, this function is a polynomial encoding all the most important quantities associated with convex sets: dimension, volume, surface area, perimeter, and so on.

I will explain where magnitude comes from, how it is defined, and what makes the conjecture interesting. I will also explain the conjecture's unusual status: while there is compelling evidence in its favour, not a single nontrivial example is known.

Slides   In this pdf file. The final page contains clickable links to references.

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