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Topic: This Week's Finds in Mathematical Physics (Week 140)
Replies: 10   Last Post: Oct 27, 1999 9:53 PM

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Christopher Hillman

Posts: 388
Registered: 12/12/04
Re: This Week's Finds in Mathematical Physics (Week 140)
Posted: Oct 21, 1999 10:09 PM
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On 16 Oct 1999, John Baez wrote:

> Hans Bethe,

I was once in the passenger seat of a car driving on an icy two-lane road
in the hills above Ithaca. Suddenly we hit a patch of black ice and went
into a skid, right into the path of an oncoming car. The driver of the
car I riding in turned into the skid, recovered traction, and we got back
on our side of the road just as I glanced over and saw Bethe whizzing by
at the wheel of his ancient grey car, grinning from ear to ear. This was
pretty much his normal expression, but still, I was impressed. We, in the
other car, were pretty shaken up, since this was a narrow road with no
shoulder, a steep rock wall on one side and a cliff on the other, and
there would have been no possible way to avoid a head on collision if we
hadn't recovered from the skid in the nick of time.

> no dope himself,

And no coward :-)

> said of von Neumann that "I always thought his brain indicated that he
> belonged to a new species, an evolution beyond man".

Von Neumann frequently gave all night wild parties in his bungalow at Los
Alamos during the war, in which all the drinks and even the food were not
just alchoholic but -strongly- alchoholic. According to legend, he was
usually the only one still standing by morning. The saying at Los Alamos
was "Johnny is really a god, but he has made a close study of humans and
has learned to imitate them to perfection." Both according to my memory
of what I read in Richard Rhodes' (excellent) book on the Manhattan
project several years ago.

> The mathematician Polya said "Johnny was the only student I was ever
> afraid of."

The story I heard is that once during a class, Polya mentioned an open
problem which he considered very difficult and then passed on to other
things. After five minutes, von Neumann raised his hand, came to the
board, and without saying a word, wrote out a short but complete proof of
the conjecture. Polya recalled many years later: "I was never afraid of my
students, but after that, I was afraid of von Neumann".

Another story concerns a skeptical mathematician, hoping to test von
Nuemann's legendary ability to think on his feet, who started to tell him
about a hard problem to which there was a very clever "shortcut" solution
which someone else had discovered. The "obvious" approach to solving this
particular problem, however, involved summing a very difficult infinite
series which would require several minutes of hard computation and
ingenious manipulation by a good mathematician. Von Neumann interrupted
almost immediately to give the correct answer. The other man exclaimed
"Oh, you heard about the short proof!" Von Neumann replied: "What short
proof? I just summed the series!".

> figuring out how to turn a sphere inside out without any crinkles,

People keep improving on the original construction, and there are two
different movies, one depicting a construction by Thurston and a more
recent one depicting an even simpler "sphere eversion". You can get still
shots from the Geometry Center website.

> cooking up strange attactors using the horseshoe map,

A small plug for symbolic dynamics: one way of thinking about this
construction is that it is a trick for finding symbolic dynamical systems
hiding inside "natural" ones. The symbolic ones are much easier to study,
and horseshoes generate shifts of finite type, which are very well
understood (in one dimension). See for instance my posting "symbolic
dynamics inside the Mandlebrot set" on

for a one dimensional construction which is somewhat similar in spirit.
Other papers on the page (not by me) offer short expositions of symbolic
dynamics. Another very interesting development which can probably be
traced back to Smale's work is the discovery of a specific smooth
dynamical system (even the term "dynamical system" is due to Smale) in
which there are repelling closed orbits representing -every- knot type, up
to isotopy. This is the "universal knot template" of Ghrist. (I don't
think I've stated it right, but you get the flavor of the result.)

Chris Hillman

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