### Vladimir Arnold, 12 June 1937 - 3 June 2010

#### Posted by David Corfield

Perhaps people have memories they’d like to share of Vladimir Arnold who died last week.

Arnold had very strong views on education, which he was not reluctant to impart, as in this Interview in the Notices of the AMS (April 1997), and in The antiscientifical revolution and mathematics:

In the middle of the twentieth century a strong mafia of left-brained mathematicians succeeded in eliminating all geometry from the mathematical education (first in France and later in most other countries), replacing the study of all content in mathematics by the training in formal proofs and the manipulation of abstract notions. Of course, all the geometry, and, consequently, all relations with the real world and other sciences have been eliminated from the mathematics teaching.

The writings I’ve enjoyed most, though, are the *Toronto Lectures* and *Polymathematics: is mathematics a single science or a set of arts?* found on this page of lectures.

Image: www.kremlin.ru

In Lecture 2: Symplectization, Complexification and Mathematical Trinities, Arnold argues for a family relation between different geometries. He begins with the finite-dimensional geometries as given by Coxeter groups. So that there is an $A$ geometry and its sisters $B$, $C$ and $D$. He then discusses the infinite-dimensional case, where 6 family members can be found: differential, volume-preserving, symplectic, contact, complex, and a variant. (According to Bryant’s lectures (p. 110), volume-preserving and symplectic each have an extension, geometries in which preservation is up to a constant multiple.) Arnold then looks for versions of theorems and constructions in differential geometry and topology for the two sisters – symplectic and complex geometry.

Finally, he moves on to describe various trinities, starting from $(\mathbb{R}, \mathbb{C}, \mathbb{H})$. He writes

The main dream (or conjecture) is that all these trinities are united by some rectangular “commutative diagrams”. I mean the existence of some “functorial” constructions connecting different trinities. (Arnold 1997: 10)

I always find fascinating the idea of there being larger systems of which we have only glimpsed some rocky outcrops. I wonder what the scare quotes are meant to imply – that we shouldn’t expected fully fledged functors?

## Re: Vladimir Arnold, 12 June 1937 - 3 June 2010

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.