The n-Category Café

2 very minor comments:

Arnol’d’s facility with the English language permitted
a rapid fire delivery surpassed (if at all) only by Atiyah.

Arnol’d was quite a walker. When visiting Chapel Hill, he thought nothing of walking to not-sonearby Hillsborough.

The link to Lecture 2 should be to arnlect2, not arn-lect2.

I only got to see one Arnol’d lecture. I was sad that such a huge fraction of it was spent on priority disputes between Russian and Western mathematicians (“what you call the So-and-so theorem from 1950, which we call the So-and-sofsky theorem from 1948”).

Regarding the scare quotes around “functorial”, I’d read it as suggesting that we should expect the constructions to be X, for some X with a similar flavour to functoriality. So X might be weaker than functoriality in some ways as you suggest, but might well also be stronger in others (carry extra structure, etc.).

I only saw Arnol’d in action once, at Gelfand’s eightieth birthday conference. The overriding impression I got was of someone who could be very forceful and intimidating. After a certain French mathematician gave a talk, Arnol’d stood up and began grilling him (I took it that he didn’t think highly of some of the abstract formalisms). I was also impressed by the great sangfroid of the speaker, a young fellow, in his response.

Arnol’d was well known as a strong detractor of Bourbaki and of what he saw as Bourbaki’s very deleterious influence on French education.

I’d posted the Arnold obits on Facebook the day of death, and also this, related to Arnold’s aesthetics:

Autumn 2009
A Mindful Beauty
What poetry and applied mathematics have in common
By Joel E. Cohen
http://www.theamericanscholar.org/a-mindful-beauty/

“…The same continuum runs in the visual arts from journalistic photography at the extreme of pointing to purely abstract art at the extreme of patterning. Between those extremes lies most of the world of art, mixing apples and oranges, mixing meanings and patterns, along with poetry and applied mathematics.”

“The differences between poetry and applied mathematics coexist with shared strategies for symbolizing experiences. Understanding those commonalities makes poetry a point of entry into understanding the heart of applied mathematics, and makes applied mathematics a point of entry into understanding the heart of poetry. With this understanding, both poetry and applied mathematics become points of entry into understanding others and ourselves as animals who make and use symbols.”

Some reminiscences by his student Oleg Karpenkov.

The Los Angeles Times reports that Arnol’d has an asteroid named after him: Vladarnolda!

I wonder if it moves in a chaotic or completely integrable trajectory?

I found my way back to this post via ten years’ worth of obituaries, so I was in a rather wistful and contemplative mood. And I remembered being particularly struck by one of the trinities that Arnold suggested: “the triangles which are sold in the stationery shops” having angles

$$(60, 60, 60); (45, 45, 90); (30, 60, 90).$$

The reason I have occasionally (and fruitlessly) wondered about this has to do with the polygonal numbers:

$$P_k(n) = n + (k-2) \frac{n(n-1)}{2}.$$

For $k = 3$, these are the triangular numbers $n(n+1)/2$; for $k = 4$, they are the square numbers, etc. They count the number of real degrees of freedom in an $n \times n$ “self-adjoint matrix” defined over a field with $k-3$ imaginary units. Of course, having that many imaginary units only really makes sense for $k = 3$, $k = 4$, $k = 6$ and $k = 10$; otherwise, we can’t have a normed division algebra. (5 is a quirky, prickly and uncooperative number, and Hamilton was never able to multiply triplets.) And for a lot of things we’d like to do with matrices, we should rule out $k = 10$ because we prefer our multiplication to be associative. So, we have self-adjoint $n \times n$ matrices over $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$, and counting up the dimensions of the spaces those define, we get the regular polygons that tile the plane. And when we break those polygons down into triangles the way the formula suggests, the triangles that we get are the ones sold in stationery shops, in just the order they should be.

I was never able to make anything of that; it probably is a coincidence of small numbers, but it’s a hard one to shake fully.