Proof of the Jacobian Conjecture

The Jacobian Conjecture is one of the most well-known open problems in algebraic geometry. It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables (for more variables, many people believe it is not even true). For more information about this, see Graham Leuschke’s weblog.

Dean hasn’t published any papers in almost 15 years and is nominally a lecturer in mathematics education at Michigan. There have been many false proofs of this conjecture over the years, and if this one holds up it will be quite a story. The paper doesn’t seem to be publicly available yet, but Dean will be lecturing on the proof at Michigan next month. One of the experts in the field, Mel Hochster, has gone over it carefully and is convinced it is correct. The rumor I hear is that it has been submitted for publication to the Journal of the American Mathematical Society.

Update: There’s an announcement of Dean’s talks posted on sci.math.research.

Update: Someone wrote in with a comment to another post pointing out that Dean has found a hole in her proof. For some more information about this, go here.

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13 Responses to Proof of the Jacobian Conjecture

  1. Chris W. says:

    The contrasting examples of string theory and the Jacobian Conjecture (and the other notable recent achievements mentioned below) illustrate why physics is different, and arguably harder, than mathematics. Once a mathematical proof has been carefully reviewed and published it stands forever. One may argue about its significance, but one doesn’t have to worry that it will be contradicted by existing or future observations.

    The other crucial point is that, as Shiing-shen Chern observed in an essay published during the Einstein Centennial year, mathematicians’ problems, while often very difficult to solve, come to them more or less clearly formulated. In the natural sciences one must sometimes struggle for years to properly formulate the problem that needs to be solved.

    I would suggest that the epistemological pathologies of string theory stem from an inclination to evade those painful realities, knowing the enormous and often lonely dedication that went into the theory’s early development. Also, one should consider the example of Einstein’s lonely pursuit of a unified field theory in the latter part of his career, and the disdain with which he was treated by many younger researchers engrossed in the development of quantum mechanics and quantum field theory, with the support of an ample supply of experimental input. No one relishes being in Einstein’s position for very long.

  2. Chris Oakley says:

    Did not Superstring theory come from a similar multi-year, fashion-resisting, career-endangering focus?

    It did, which only goes to show that when rebels become the establishment, they are even more reactionary than those they supplanted.

  3. Arun says:

    Did not Superstring theory come from a similar multi-year, fashion-resisting, career-endangering focus?

  4. mm says:

    One correction to the original post. There is no department of Math Education at Michigan. There is Math in LSA and there is the School of Education. Carolyn is a lecturer in the Math Department. Her research interests are listed as being in math education. Bass and Ball both have joint appointments in Math and Education. After she was denied tenure, she founded and initially was the director for MMSS, a summer program for high school students. I believe she has recently taught, among other things, honors calculus and math for future (elementary?) teachers.

    Carolyn had a postdoc at University of Chicago. She probably went to Michigan in 1989. She was definitely there in 1992.

    I agree with Graham’s comments about tenure. It is perhaps bittersweet for Carolyn that she was aided in solving the Jacobian conjecture by being a lecturer at Michigan given that I am pretty sure that she was working on it while she was working towards tenure. Of course, it could just have been the passage of time. In any event, she could have chosen a research agenda with a shorter time horizon. Instead, she gambled and lost (the tenure game, at least).

  5. Graham says:

    As far as I know, Bass hasn’t done any research mathematics in several years — he’s been thinking about mathematics education, much of it with Deborah Ball. I would seriously doubt that he even knows Dean was working on JC.

    About Dean: I’m told that she has been at at Michigan since the mid-nineties, when she had a tenure-track position and was denied tenure. Her husband is also on faculty there (Stafford), which I imagine is part of the reason she’s still there. I would be surprised if Michigan now hired her with tenure; it would be tantamount to admitting that they’d made a mistake. However, she is clearly a “serious mathematician”, as one of your other commenters put it. The fact that she’s not tenured does not detract in the slightest from her talent or skill or intelligence, only from her paycheck and job security.

    It is interesting that recent quantum leaps in mathematics have been due to people who showed exceptional ability to focus on one problem for years at a time, but it’s certainly not surprising. In fact, this is in part what the tenure system is designed to encourage — having nothing to lose. If there were no such thing as tenure, Wiles would never have been able to devote years to working on the solution to an incredibly difficult problem — he would have had to show some progress on something in order to keep his job every year. Dean is in the same position for a different reason; she has nothing to lose because she’s not regular faculty. Perelman, too — his position in St. Petersburg wasn’t going to be taken away from him. Doing away with tenure isn’t going to bring about this kind of freedom for everyone — for most of us, it would mean we’d have to work on simplistic problems that we knew we could publish on before the year’s end.

  6. yuhan says:

    good to hear from you too.

    Actually I know little about Dean beyond some googling. And yes, I was wondering about the “tenure question” too which I’m sure must be a delicate one. But I didn’t know that she had been thinking about the jacobian conjecture for several years. Tenacity!

    If I’m not mistaken, in some parts of europe there’s more of math culture of keeping postdocs on practically indefinitely (at just slightly above a grad students level) and giving them a huge block of time to create a significant work. Is this correct?

    I must correct myself in my last post: Hyman is *both* at math and education at Michigan.

  7. Chris Oakley says:

    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.

    Of the two functions of a University, namely teaching and research, the former IMHO tends to be done a lot better than the latter, possibly because it is easier to apply correction mechanisms when things start to go wrong. Research, OTOH, tends to be islands of excellence amid a sea of mediocrity, and the mediocre seems to be highly resistant to any attempt to reform it. I think that something needs to be done, but I am not quite sure what. My proposal was to abolish tenured positions. Then at least people who are useless will only allowed to be useless for a limited time. One of the most frustrating things possible for a young researcher with a new idea is people of this kind being in a position to block their every move. And these are tenured staff who more often than not would not be shortlisted if they were forced to reapply for their jobs. I do not believe that my suggestion would ever become a reality, but am interested to know whether those reading this blog (who I suspect may well be more critical of “the establishment” than average) either (a) think that there is a problem with academic research and (b) have any ideas as to what to do about it.

  8. Peter says:

    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

  9. The Great Gazoo says:

    Does it seem strange that several conjectures have fallen in recent years due to mathematicians that have spent nearly a decade solely focusing on a single problem? This is an oddity isn’t it? I mean don’t mathematicians generally not pour all of their energy into one thing that may or may not pan out?

  10. Peter says:

    Hi Yuhan,
    Good to hear from you, and I’m glad you’ve been enjoying the weblog.

    I really don’t know anything about Carolyn Dean, it sounds like you know more than I do. So I don’t know whether Hyman was in any way involved in her work on the Jacobian Conjecture. It is an amazing story and a very impressive achievement. It will be interesting to see whether academia rewards her the way it should. Will Michigan give her a tenured position?

  11. yuhan says:

    Dear Peter,

    I’ve been a lurker, enjoying reading your blog, for some time now. But I am blown away by the news that Carolyn Dean proved the jacobian conjecture (not that I know her personally). I am wondering how she will be regarded — after all conventional wisdom has it that lecturers can’t do research and here she has clearly outdone “serious mathematicians”. She sure has a good brain and a good heart too considering her passion in running math summer camps for high school kids.

    Interesting thing is that our ex-columbian Hyman Bass, a key contributor to the jacobian conjecture, is also at Michigan but in the education dept. An internet search shows that Dean and Bass had contact in education forums. Any connection?

  12. Peter says:

    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.

  13. mm says:

    Just curious. What exactly did you mean by “nominally a lecturer”?

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