Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

June 8, 2010

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 AA geometry and its sisters BB, CC and DD. 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?

Posted at June 8, 2010 11:52 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2224

16 Comments & 0 Trackbacks

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.

Posted by: jim stasheff on June 8, 2010 1:45 PM | Permalink | Reply to this

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

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”).

Posted by: Allen Knutson on June 8, 2010 2:49 PM | Permalink | Reply to this

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

Typo fixed, thanks.

Re priority, he clearly thought people weren’t properly consulting the literature. He gave a long answer to how this should be done to a question put to him by Milnor at the end of Toronto Lecture 1.

Regarding the transfer of constructions across different geometries, I take it that the Euler-Arnold equation described by Tao in this post is an example. Here, it is geodesic flows for the sister geometries.

Posted by: David Corfield on June 8, 2010 3:16 PM | Permalink | Reply to this

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

It’s hard to consult the literature if you don’t know it exists. Recently I’ve belatedly discovered work on the arXiv on infty-MC equations from years ago which were listed only on hep-th and yet clearly deserved a cross listing to e.g quantum algebra. I wonder how much good stuff here and on the n-lab never makes it out to the wider world??

Posted by: jim stasheff on June 9, 2010 12:19 PM | Permalink | Reply to this

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

At least we have search engines and central databases now…knowledge management is lagging behind at bit, though. Knowledge management is about

  • identifing important information,

  • making it available, for example by writing about it on Wikipedia or the nLab,

  • and adding a structure that makes it easy to find it,

Example: The ideas around the semantic web are about defining data structures that search engines, for example, could use to avoid listing clothing shops when you search for “strings”.

If Arnol’d was a little bit bitter that many results are attributed to western mathematicians instead of those who lived in the Soviet Union, he would have my sympathy. (Did you know that my advisor discovered Wilson loops several months before Wilson? But he published his ideas in German only…).

Posted by: Tim van Beek on June 9, 2010 1:55 PM | Permalink | Reply to this

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

I wonder how much good stuff here and on the n-lab never makes it out to the wider world??

Probably a lot, but it’s much, much easier to search the nLab than it is this blog. Not long ago I was trying to find a nugget which I knew was embedded somewhere within hundreds of detailed comments to a Café post. I sifted for about ten or fifteen minutes before giving up.

The reputation of the nLab seems to be growing, and one can find information and details there that is hard to find anywhere else online without a subscription.

Posted by: Todd Trimble on June 9, 2010 2:01 PM | Permalink | Reply to this

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

…Not long ago I was trying to find a nugget which I knew was embedded somewhere within hundreds of detailed comments to a Café post. I sifted for about ten or fifteen minutes before giving up.

You could try typing something like this into Google’s search box:

+site:http://golem.ph.utexas.edu/category/ +“phrase you’re looking for”

Posted by: Eugene Lerman on June 9, 2010 4:11 PM | Permalink | Reply to this

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

Thanks, but as far as I can tell, that just sends me to the top of the post when I click on a search result, not to the comment itself.

Posted by: Todd Trimble on June 9, 2010 4:50 PM | Permalink | Reply to this

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

Sorry for shooting off the hip. But if it gets you to the right post, you could then use the search feature of your browser to find the exact comment. For instance in Firefox it’s Ctrl F (also in the edit manue). Apologies for going off topic on this subthread.

Posted by: Eugene Lerman on June 9, 2010 5:05 PM | Permalink | Reply to this

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

Although it makes me look ignorant for not knowing about ctrl-F, thanks for telling me that! As far as I’m concerned, you’ve done a really good deed for the day. Thanks a bunch, Eugene.

Posted by: Todd Trimble on June 9, 2010 5:44 PM | Permalink | Reply to this

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

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.).

Posted by: Peter LeFanu Lumsdaine on June 8, 2010 4:35 PM | Permalink | Reply to this

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

I’d be interested to know what might be done to capture the sense of analogy used here. In The geometry of infinite-dimensional groups there is some working out of the real-complex correspondence without a functor or category in sight. E.g.,

The correspondence between the real and complex cases leads to somewhat surprising analogies between notions in differential topology (such as orientation, boundary, and the Stokes theorem) and those in complex algebraic geometry (a meromorphic differential form, its divisor of poles, and the Cauchy-Stokes formula). These analogies are formalzed in the notion of polar homology, and their applications include the construction of a holomorphic linking number for a pair of complex curves in a complex threefold. The definition of the latter is closely related to a holomorphic version of the Chern-Simons functional. (p. 5)

Were such constructions to be addressed functorially, might we not say that the analogy had been replaced by a general theory. This is the thrust of Kevin Buzzard’s response to John’s talk of analogy between the rationals and ration complex functions in TWF 218. Where, say, the pp-adic integers and [[za]]\mathbb{C}[[z - a]] are subsumed as complete discrete valuation rings, the relation between rational fields and imaginary quadratic fields is a proper analogy

because no-one has (as far as I know) a clue how to do this more generally.

Posted by: David Corfield on June 8, 2010 5:04 PM | Permalink | Reply to this

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

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.

Posted by: Todd Trimble on June 9, 2010 2:14 PM | Permalink | Reply to this

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

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.”

Posted by: Jonathan Vos Post on June 10, 2010 6:20 AM | Permalink | Reply to this

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

Some reminiscences by his student Oleg Karpenkov.

Posted by: David Corfield on July 6, 2010 5:02 PM | Permalink | Reply to this

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

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?

Posted by: John Baez on July 8, 2010 4:22 PM | Permalink | Reply to this

Post a New Comment