One way to increase the blogshare-value of this site might be to

give readers more of what they want. In fact, there is an excellent

guide for those who really want to increase traffic on their site

called 26

Steps to 15k a Day. A somewhat sobering suggestion is rule S :

“Think about what people want. They

aren't coming to your site to view “your content”,

they are coming to your site looking for “their

content”.”

But how do we know what

people want? Well, by paying attention to Google-referrals according

to rule U :

“The search engines will

tell you exactly what they want to be fed – listen closely, there is

gold in referral logs, it's just a matter of panning for

it.”

And what do these Google-referrals

show over the last couple of days? Well, here are the top recent

key-words given to Google to get here :

13 :

carolyn dean jacobian conjecture

11 : carolyn dean jacobian

9 : brauer severi varieties

7 : latexrender

7 : brauer severi

7 : spinor bundles

7 : ingalls

azumaya

6 : [Unparseable or potentially dangerous latex

formula Error 6 ]

6 : jacobian conjecture carolyn dean

See a pattern? People love to hear right now about

the solution of the Jacobian conjecture in the plane by Carolyn Dean.

Fortunately, there are a couple of things more I can say about this

and it may take a while before you know why there is a photo of Tracy

Chapman next to this post…

First, it seems I only got

part of the Melvin Hochster

email. Here is the final part I was unaware of (thanks to not even wrong)

Earlier papers established the following: if

there is

a counterexample, the leading forms of $f$ and $g$

may

be assumed to have the form $(x^a y^b)^J$ and $(x^a

y^b)^K$,

where $a$ and $b$ are relatively prime and neither

$J$

nor $K$ divides the other (Abhyankar, 1977). It is known

that

$a$ and $b$ cannot both be $1$ (Lang, 1991) and that one

may

assume that $C[f,g]$ does not contain a degree one

polynomial

in $x, y$ (Formanek, 1994).

Let $D_x$ and $D_y$ indicate partial differentiation with respect

to $x$ and $y$, respectively. A difficult result of Bass (1989)

asserts that if $D$ is a non-zero operator that is a polynomial

over $C$ in $x D_x$ and $y D_y$, $G$ is in $C[x,y]$ and $D(G)$

is in $C[f,g]$, then $G$ is in $C[f,g]$.

The proof

proceeds by starting with $f$ and $g$ that give

a

counterexample, and recursively constructing sequences of

elements and derivations with remarkable, intricate and

surprising relationships. Ultimately, a contradiction is

obtained by studying a sequence of positive integers associated

with the degrees of the elements constructed. One delicate

argument shows that the sequence is bounded. Another delicate

argument shows that it is not. Assuming the results described

above, the proof, while complicated, is remarkably self-contained

and can be understood with minimal background in algebra.

Speaking about the Jacobian

conjecture-post at not even wrong and

the discussion in the comments to it : there were a few instances I

really wanted to join in but I'll do it here. To begin, I was a

bit surprised of the implicit attack in the post

Dean hasn't published any papers in almost 15 years and is

nominally a lecturer in mathematics education at Michigan.

But this was immediately addressed and retracted in

the comments :

Just curious. What exactly did

you mean by “nominally a lecturer”?

Posted by mm

at November 10, 2004 10:54 PM

I don't know

anything about Carolyn Dean personally, just that one place on the

Michigan web-site refers to her as a “lecturer”, another

as a “visiting lecturer”. As I'm quite well aware from

personal experience, these kinds of titles can refer to all sorts of

different kinds of actual positions. So the title doesn't tell you

much, which is what I was awkwardly expressing.

Posted by Peter

at November 10, 2004 11:05 PM

Well, I know a few things

about Carolyn Dean personally, the most relevant being that she is a

very careful mathematician. I met her a while back (fall of 1985) at

UCSD where she was finishing (or had finished) her Ph.D. If Lance

Small's description of me would have been more reassuring, we

might even have ended up sharing an apartment (quod non). Instead I

ended up with Claudio

Procesi… Anyway, it was a very enjoyable month with a group

of young starting mathematicians and I fondly remember some

dinner-parties we organized. The last news I heard about Carolyn was

10 to 15 years ago in Oberwolfach when it was rumoured that she had

solved the Jacobian conjecture in the plane… As far as I recall,

the method sketched by Hochster in his email was also the one back

then. Unfortunately, at the time she still didn't have all pieces

in place and a gap was found (was it by Toby Stafford? or was it

Hochster?, I forgot). Anyway, she promptly acknowledged that there was

a gap.

At the time I was dubious about the approach (mostly

because I was secretly trying to solve it myself) but today my gut

feeling is that she really did solve it. In recent years there have

been significant advances in polynomial automorphisms (in particular

the tame-wild problem) and in the study of the Hilbert scheme of

points in the plane (which I always thought might lead to a proof) so

perhaps some of these recent results did give Carolyn clues to finish

off her old approach? I haven't seen one letter of the proof so

I'm merely speculating here. Anyway, Hochster's assurance that

the proof is correct is good enough for me right now.

Another

discussion in the NotEvenWrong-comments was on the issue that several

old problems were recently solved by people who devoted themselves for

several years solely to that problem and didn't join the parade of

dedicated follower of fashion-mathematicians.

It is remarkable that the last decade has seen great progress in

math (Wiles proving Fermat's Last Theorem, Perelman proving the

Poincare Conjecture, now Dean the Jacobian Conjecture), all achieved

by people willing to spend 7 years or more focusing on a single

problem. That's not the way academic research is generally

structured, if you want grants, etc. you should be working on much

shorter term projects. It's also remarkable that two out of three

of these people didn't have a regular tenured position.

I think particle theory should learn from this. If

some of the smarter people in the field would actually spend 7 years

concentrating on one problem, the field might actually go somewhere

instead of being dead in the water

Posted by Peter at November

13, 2004 08:56 AM

Here we come close to a major problem of

today's mathematics. I have the feeling that far too few

mathematicians dedicate themselves to problems in which they have a

personal interest, independent of what the rest of the world might

think about these problems. Far too many resort to doing trendy,

technical mathematics merely because it is approved by so called

'better' mathematicians. Mind you, I admit that I did fall in

that trap myself several times but lately I feel quite relieved to be

doing just the things I like to do no matter what the rest may think

about it. Here is a little bit of advice to some colleagues : get

yourself an iPod and take

some time to listen to songs like this one :

Don't be tempted by the shiny apple

Don't you eat

of a bitter fruit

Hunger only for a taste of justice

Hunger only for a world of truth

'Cause all that you have

is your soul

from Tracy Chapman's All

that you have is your soul