Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

January 10, 2018

On the Magnitude Function of Domains in Euclidean Space, I

Posted by Simon Willerton

Guest post by Heiko Gimperlein and Magnus Goffeng.

The magnitude of a metric space was born, nearly ten years ago, on this blog, although it went by the name of cardinality back then. There has been much development since (for instance, see Tom Leinster and Mark Meckes’ survey of what was known in 2016). Basic geometric questions about magnitude, however, remain open even for compact subsets of n\mathbb{R}^n: Tom Leinster and Simon Willerton suggested that magnitude could be computed from intrinsic volumes, and the algebraic origin of magnitude created hopes for an inclusion-exclusion principle.

In this sequence of three posts we would like to discuss our recent article, which is about asymptotic geometric content in the magnitude function and also how it relates to scattering theory.

For “nice” compact domains in n\mathbb{R}^n we prove an asymptotic variant of Leinster and Willerton’s conjecture, as well as an asymptotic inclusion-exclusion principle. Starting from ideas by Juan Antonio Barceló and Tony Carbery, our approach connects the magnitude function with ideas from spectral geometry, heat kernels and the Atiyah-Singer index theorem.

We will also address the location of the poles in the complex plane of the magnitude function. For example, here is a plot of the poles and zeros of the magnitude function of the 2121-dimensional ball.

poles and zeros of the magnitude function of the 21-dim ball

We thank Simon for inviting us to write this post and also for his paper on the magnitude of odd balls as the computations in it rescued us from some tedious combinatorics.

The plan for the three café posts is as follows:

  1. State the recent results on the asymptotic behaviour as a metric space is scaled up and on the meromorphic extension of the magnitude function.

  2. Discuss the proof in the toy case of a compact domain XX\subseteq \mathbb{R} and indicate how it generalizes to arbitrary odd dimension.

  3. Consider the relationship of the methods to geometric analysis and potential ramifications; also state some open problems that could be interesting.

Posted at 5:11 PM UTC | Permalink | Followups (13)