## July 20, 2014

### The Place of Diversity in Pure Mathematics

#### Posted by Tom Leinster

Nope, this isn’t about gender or social balance in math departments, important as those are. On Friday, Glasgow’s interdisciplinary Boyd Orr Centre for Population and Ecosystem Health — named after the whirlwind of Nobel-Peace-Prize-winning scientific energy that was John Boyd Orr — held a day of conference on diversity in multiple biological senses, from the large scale of rainforest ecosystems right down to the microscopic scale of pathogens in your blood.

I used my talk (slides here) to argue that the concept of diversity is fundamentally a mathematical one, and that, moreover, it is closely related to core mathematical quantities that have been studied continuously since the time of Euclid.

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.

Posted at July 20, 2014 2:10 PM UTC

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### Re: The Place of Diversity in Pure Mathematics

Like John, I had the urge to rewrite my blog entry. I think it’s better now.

Posted by: Tom Leinster on July 20, 2014 11:14 PM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

Thanks for mentioning that Banff workshop on information and entropy in biological systems!

Anyone who wants more details can check out my blog article on Azimuth. With a little luck Tobias Fritz, Tom Leinster and Christina Cobbold will be there, as well as other people whose names aren’t so familiar on the n-Café: biologists and information theorists.

Posted by: John Baez on July 21, 2014 7:45 AM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

Did you really mean Banff, or were you mixing up the Knoxville workshop with this one?

Posted by: Mark Meckes on July 21, 2014 8:06 AM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

It was a nice talk. For those of you who are slightly worried about Tom’s use of genus rather than Euler characteristic on slide 6, he did say, probably just for my benefit, that he really meant Euler characteristic, but that was harder to draw and there was probably only two people in the room who appreciated the difference :-)

I think that the generic biologist finds our desire for precision and process of abstraction very foreign. Afterwards, at dinner, the biologist without a maths degree that I was with said at one point to me “You speak a lot like Tom.” He was referring to my use of language, perhaps one might say, to my pedantry in choice and interpretation of words.

Posted by: Simon Willerton on July 21, 2014 6:15 PM | Permalink | Reply to this

### Diversity in Pure Mathematics

Interesting work. This metatheorem also applied to dimensionality problems in Physics, Complex Systems and Logical and Physical Power Systems.

Posted by: Jonah Lissner on August 20, 2014 6:47 PM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

Tom, have you looked into notions of diversity in finance (eg the notion of a diversified portfolio)? I wonder whether mathematicians can learn anything about diversity from financiers/finance.

Posted by: trent on August 21, 2014 10:50 PM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

Someone mentioned that to me before (maybe you?), but I can’t say I’ve looked into it properly. On the other hand, I have learned some things about diversity from economists.

Posted by: Tom Leinster on August 23, 2014 3:22 PM | Permalink | Reply to this

### Re: The Place of Diversity in Pure Mathematics

Wasn’t me. Curious to hear what you’ve learned from economists though.

Posted by: trent on August 23, 2014 9:43 PM | Permalink | Reply to this

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