### Instantaneous Dimension of Finite Metric Spaces via Magnitude and Spread

#### Posted by Simon Willerton

In June I went to the following conference.

This was held at the Będlewo Conference Centre which is run by the Polish Academy of Sciences’ Institute of Mathematics. Like Oberwolfach it is kind of in the middle of nowhere, being about half an hour’s bus ride from Poznan. (As our excursion guide told us, Poznan is 300km from anywhere: 300 km from Warsaw, 300 km from Berlin, 300 km from the sea and 300 km from the mountains.) You get to eat and drink in the palace pictured below; the seminar rooms and accommodation are in a somewhat less grand building out of shot of the photo.

I gave a 20-minute long, magnitude-related talk. You can download the slides below. Do try the BuzzFeed-like quiz at the end. How many of the ten spaces can just identify just from their dimension profile?

To watch the animation I think that you will have to use acrobat reader. If you don’t want to use that then there’s a movie-free version.

Here’s the abstract.

Some spaces seem to have different dimensions at different scales. A long thin strip might appear one-dimensional at a distance, then two-dimensional when zoomed in on, but when zoomed in on even closer it is seen to be made of a finite array of points, so at that scale it seems zero-dimensional. I will present a way of quantifying this phenomenon.

The main idea is to think of dimension as corresponding to growth rate of size: when you double distances, a line will double in size and a square will quadruple in size. You then just need some good notions of size of metric spaces. One such notion is ‘magnitude’ which was introduced by Leinster, using category theoretic ideas. but was found to have links to many other areas of maths such as biodiversity and potential theory. There’s a closely related, but computationally more tractable, family of notions of size called ‘spreads’ which I introduced following connections with biodiversity.

Meckes showed that the

asymptoticgrowth rate of the magnitude of a metric space is the Minkowski dimension (i.e. the usual dimension for squares and lines and the usual fractal dimension for things like Cantor sets). But this is zero for finite metric spaces. However, by considering growth ratenon-asymptotically you get interesting looking results for finite metric spaces, such as the phenomenon described in the first parargraph.

I have blogged about instantaneous dimension before at this post. One connection with applied topology is that as in for persistent homology, one is considering what is happens to a metric space as you scale the metric.

The talk was in the smallest room of three parallel talks, so I had a reasonably small audience. However, it was very nice that almost everyone who was in the talk came up and spoke to me about it afterwards; some even told me how I could calculate magnitude of large metric spaces much faster! For instance Brad Nelson showed me how you can use iterative methods, such as the Krylov subspace method, for solving large linear systems numerically. This is much faster than just naively asking Maple to solve the linear system.

Anyway, do say below how well you did in the quiz!

## Re: Instantaneous Dimension of Finite Metric Spaces via Magnitude and Spread

Meanwhile, over in Sapporo, Tom Leinster taught us some stuff about your work on instantaneous dimension. Nice!