The n-Category Café

One notable feature: of the 38 papers listed, 100% of them are on the arXiv.

Splendid thanks! I was just trying to cobble something together yesterday for my PhD student from various Café posts.

Nice idea! I hadn’t been aware of all of these.

Have you considered annotating them in any way? It’s not always the case that a paper’s most important contribution (in retrospect) is apparent from its abstract, so a sentence here or there about what an article establishes might be helpful. This thread could be a good place to gather suggestions for that, too.

(Unfortunately for me, I’ve only ever published/posted things about the diversity measures, rather than magnitude proper, so I can’t use this as an opportunity for self-promotion.)

I just added a few more recent papers on magnitude to the bibliography. In an ideal world I’d write an explanatory blog post about each, but in the actual world, I’ll just list them here:

• Kiyonori Gomi, Magnitude homology of geodesic space. arXiv:1902.07044, 2019.

• Yasuhiko Asao, Magnitude homology of geodesic metric spaces with an upper curvature bound. arXiv:1903.11794, 2019.

• Mark Meckes, On the magnitude and intrinsic volumes of a convex body in Euclidean space. arXiv:1904.08923, 2019.

The one I understand best is the last, by Mark Meckes. The way I’d explain it is with reference to a false conjecture that Simon Willerton and I made long ago, which has nevertheless turned out to have many true aspects.

Simon and I guessed that for convex compact subsets $K$ of Euclidean space $\mathbb{R}^n$, the magnitude function $t \mapsto |t K|$ is a polynomial:

$$|t K| = \sum_{i = 0}^n \frac{1}{i!\omega_i} V_i(K) t^i,$$

where $V_i$ is the $i$-dimensional intrinsic volume and $\omega_i$ is the volume of the unit $i$-dimensional ball. For example, if $n = 2$, the conjecture says that the magnitude function $t \mapsto |t K|$ of a planar convex set $K$ is a quadratic whose constant term is $1$ (or $0$ if the set is empty), whose coefficient of $t^1$ is $1/4$ times the perimeter, and whose coefficient of $t^2$ is $1/2\pi$ times the area.

This turned out to be false. The first counterexample, provided by Juan Antonio Barceló and Tony Carbery, was the 5-dimensional Euclidean ball, whose magnitude function turns out to be a rational function of $t$ that isn’t a polynomial. But as I said a couple of paragraphs ago, there are some respects in which the conjecture makes correct predictions.

Mark has provided a couple more. The first is that, as he proves,

$$|t K| \leq \sum_{i = 0}^n \frac{\omega_i}{4^i} V_i(K) t^i$$

for convex sets $K$ in $\mathbb{R}^n$. We already knew that the magnitude function wasn’t a polynomial. And in fact, the example of the 5-ball showed that the conjectured polynomial was too small. But Mark has provided a formally similar, explicit, polynomial that is an upper bound.

That’s the first of the main two results in Mark’s paper. The second is to do with the “$t^1$” term in the magnitude function of balls. It doesn’t really make sense to speak of the $t^1$ term, since the magnitude function is not a polynomial. But Mark proves that for odd-dimensional Euclidean balls $B$,

$$\lim_{t \to 0} \frac{|t B| - 1}{t} = V_1(B)/2.$$

Up to a known constant, the 1-dimensional intrinsic volume of a Euclidean ball is just its radius. If $|t B|$ was a polynomial in $t$, this result would say that its coefficient of $t^1$ was the radius of $B$ (up to a known constant). That’s because we know that $\lim_{t \to 0} |t B|$ (which would be the constant term if it was a polynomial) is $1$.

This result was conjectured by Simon Willerton, and Mark makes the further natural conjecture that the same result holds for all compact convex sets, not just odd-dimensional balls.

Results like this are complementary to recent theorems by Heiko Gimperlein and Magnus Goffeng, blogged about recently here. They’ve found out lots about the top terms of the asymptotic expansion of $|t K|$, showing that, for instance, the first two terms give the volume and surface area (or its higher-dimensional analogue) of $K$. Mark’s second result here concerns the other end of the expansion.

I’ve just compiled a list of all the Café posts I could find on magnitude, diversity, and closely related ideas. This is mainly for my own reference in case I want to dig up an old discussion (there are lots of interesting ideas buried in there, not all of which have been thoroughly explored!), but in case anyone else would like to refer to such a thing, here it is. Maybe I’ll get around to formatting it more nicely at some point, or maybe not.