The n-Category Café

In a sense, there’s nothing new here: I’ve probably written about all the mathematical content at least once before on this blog. But in another sense, it was a really new talk. I had to think very hard about how to present this material for a mixed group of ecologists, botanists, epidemiologists, mathematical modellers, and so on, all of whom are active professional scientists but some of whom haven’t studied mathematics since high school. That’s why I began the talk with an explanation of how pure mathematics looks these days.

I presented two pieces of evidence that diversity is intimately connected to ancient, fundamental mathematical concepts.

The first piece of evidence is a connection at one remove, and schematically looks like this:

maximum diversity $\leftrightarrow$ magnitude $\leftrightarrow$ intrinsic volumes

The left leg is a theorem asserting that when you have a collection of species and some notion of inter-species distance (e.g. genetic distance), the maximum diversity over all possible abundance distributions is closely related to the magnitude of the metric space that the species form.

The right leg is a conjecture by Simon Willerton and me. It states that for convex subsets of $\mathbb{R}^n$, magnitude is closely related to perimeter, volume, surface area, and so on. When I mentioned “quantities that have been studied continuously since the time of Euclid”, that’s what I had in mind. The full-strength conjecture requires you to know about “intrinsic volumes”, which are the higher-dimensional versions of these quantities. But the 2-dimensional conjecture is very elementary, and described here.

The second piece of evidence was a very brief account of a theorem of Mark Meckes, concerning fractional dimension of subsets $X$ of $\mathbb{R}^n$ (slide 15, and Corollary 7.4 here). One of the standard notions of fractional dimension is Minkowski dimension (also known by other names such as Kolmogorov or box-counting dimension). On the other hand, the rate of growth of the magnitude function $t \mapsto \left| t X \right|$ is also a decent notion of dimension. Mark showed that they are, in fact, the same. Thus, for any compact $X \subseteq \mathbb{R}^n$ with a well-defined Minkowski dimension $dim X$, there are positive constants $c$ and $C$ such that

$$c t^{dim X} \leq \left| t X \right| \leq C t^{dim X}$$

for all $t \gg 0$.

One remarkable feature of the proof is that it makes essential use of the concept of maximum diversity, where diversity is measured in precisely the way that Christina Cobbold and I came up with for use in ecology.

So, work on diversity has already got to the stage where application-driven problems are enabling advances in pure mathematics. This is a familiar dynamic in older fields of application such as physics, but I think the fact that this is already happening in the relatively new field of diversity theory is a promising sign. It suggests that aside from all the applications, the mathematics of diversity has a lot to give pure mathematics itself.

Next April, John Baez and friends are running a three-day investigative workshop on Entropy and information in biological systems at the National Institute for Mathematical and Biological Synthesis in Knoxville, Tennessee. I hope this will provide a good opportunity for deepening our understanding of the interplay between mathematics and diversity (which is closely related to entropy and information). If you’re interested in coming, you can apply online.