The n-Category Café

Semyon Alesker kicked off. He’s giving one of the two courses that form the heart of the week’s activities. It’s an hour and a half every day for five days, which was enough to put the fear of God into me. I’d also had enough contact with the subject to know that he’s a total hotshot, so I sat down in the lecture room with some trepidation, suspecting I’d be rapidly lost. My fears weren’t calmed when he began by explaining that he appreciated what a diverse audience we were: he comes from convex geometry, whereas many of us come from integral geometry. But it was a really nice lecture, paced slowly enough that I could follow right the way through, and strikingly well organized—in short, a pleasure.

The CRM also has the great procedure of getting its lecturers to produce notes beforehand, so that when you sit down to hear Lecture 1, you already have a printed set of notes before you.

The other course is being given by Joseph Fu. I’d previously been reading a different set of lecture notes by him, on ‘Blaschkean integral geometry’, so it was good to see him in action. He talked a lot about how Alesker has revolutionized the subject (while being rather modest, I suspect, about his own contributions).

Among other things, Fu spoke about what he calls the template method. This is, essentially, what I described as ‘the grit around which the pearl forms’ back here, when I was explaining a nice solution to the Buffon needle problem. The general situation is as follows. In integral geometry, you often end up with an equation of the form ‘$f(A) = g(A)$ for all $A$’ involving some constants. You might not know what those constants are, but you know that there are some constants for which the equation holds. The template method is to stick in some particularly convenient value or values of $A$ in order to discover those constants. But it’s a bit of an ad hoc method, and can lead to tricky calculations… so I think he’s going to suggest something better later in the week.

In the afternoon, Luis Cruz-Orive gave a talk about computations on subsets of $\mathbb{R}^3$, from an engineering perspective. We can argue endlessly about what constitutes concrete or abstract mathematics, or whether the word ‘abstract’ actually means anything. But this talk was about granular cementing: if that’s not concrete, what is?

Partway through the talk, Cruz-Orive stopped, turned dramatically to the audience, and said something along the following lines. “You all prove general theorems, and that’s good. But I urge you, I urge you, take a break, take a sabbatical year, and spend a year applying your theorems in dimension three.” So, $n$-categorists: spend a year applying your theorems to tricategories. It’s not such inappropriate advice.

Eva Vedel Jensen gave the last talk, on rotational Crofton formulas. Now, Crofton’s formula is a really nice thing to explain, and I’d kind of like to explain it—it’s a generalization of that solution to Buffon’s needle problem that I explained before and mentioned above. It’s a staple of integral geometry. Unfortunately I don’t have time to tell the story now… but if anyone reading wants to have a go, please do!

Incidentally, Crofton published his work in an unusual place: the Encyclopaedia Britannica. More specifically, it was the entry on Probability in the 9th edition, 1885. In their book Introduction to Geometric Probability, Klain and Rota describe it as

Crofton’s article in the ninth edition of the Encyclopaedia Britannica, an article that created the subject from scratch and that is still worth reading today.

I once looked for it online, thinking that it must be out of copyright and available, but couldn’t find it. If anyone else can find it, I’d love to know.