Friday, February 17, 2012

People solve math problems for the prize money! NOT!

Why do people or organizations offer Prize Money for mathematics?
  1. Paul Erdos: He offered money to solve problems that he found interesting. I assume he wanted them solved but he also wanted to encourage a line of research beyond the problem. He had a (well deserved) reputation as a brilliant mathematician, so if he couldn't solve a problem it was hard. People would somtimes not cash the check and frame it. I've heard that with color copiers people now copy it, frame the copy, and cash the check. Did he insist that it appear in a journal or just need to be convinced? I don't know but I would think just need to be convinced.
  2. Bill Gasarch: He offered $289 dollars for one problem, which, as you know, was recently solved by Steinbach and Posthoff (see here). While Gasarch has nowhere near the reputation of Erdos and his problem was not a deep math problem, this problem caught on as a matter of luck and timing. The blog helped, and Brian Hayes picking up on it helped. Gasarch wanted to get this problem solved, but did not quite know if it would inspire a line of research. It did (according to the solvers) present a problem just on the edge of what is possible to solve of this type. Gasarch used paypal. Hence, alas, Steinbach and Postoff won't be able to frame a check or its copy.
  3. Scott Aaronson: His 100,000 offer (see here) for ...demonstration, convincing to me, that scalable quantum computing is impossible in the physical world This is different than most prize offers in seveal ways: (1) He gets to decide, not a ``refereed journal''. (SIDE NOTE-here is an idea: a prize that pays out only if the article appears in a non-elsevier journal.) (2) He does not expect to pay out (but he happily will if someone really convinces him). I believe him on this, though 100,000 is a lot of money. He wants to inspire people to think about these questions. The only thing analagous I can think of is prizes for REAL parapsychology- they don't expect to pay out but would be happy to since the world is more interesting if parapsychology is true.
  4. Millienium prizes: I believe these one million dollar prizes are the most ever offered for solving particular math problems by an order of magnitude (if that is not correct, please leave a polite comment correcting me). Clearly the Clay Instuite wants to encourage research in these areas. Why so much money? I assume to REALLY put these problems on the map. There is no mathematician of the stature of Hilbert nowadays who could state problems of importance in a way people would listen. Smale tried (see here) but those problems never got the status of either Hilbert's problems or the Millenium problems.
  5. Godel Prize: Best paper in theory published in the last 14 years (used to be 7). I wonder- if someone posted a solution to P vs NP on arXiv and it was correct, would they really not get the Godel prize? I suppose not. A bit awkward in that if your publish in a period of time when many good papers come out you could be out of luck. Why did they extend the window from 7 years to 14 years? Speculation: people are getting worse at getting papers out into journals so they had to extend it. Enablers? Given once a year.
  6. Turing Award: I am not quite sure if this is for one paper, a body of work around one idea, or a career. It can go to people who never proved a theorem since its open to all computer scientists. The prize money has gone from $2000 to $250,000. Given once a year.
  7. Fields Medal: Given for a body of work. About $15,000. High Prestige, low dollars. How come the Turing Award was able to increase its money value but the Fields medal was not? I honestly want to know. Given once every 4 years to a set (group? locus?) of people.
  8. King Faisal prize: I blogged about this here so I'll be brief: High dollars ($400,000), but low prestige. I assume the origin was to try to give glory and prestige to Saudi Arabia who gives out the prize. I don't think it worked. Aside from its origins it also has the problem of being unfocused in that they have awards for Sciene (which is sometimes math) and also for Muslim scholarship, and other areas.
  9. Here is a list of other prize. Some thoughts
    1. Some are for solving a particular problem, some are for a body of work in a particular area, some are for a body of work and the area can be anything within mathematics.
    2. Some are restricted to a subset of people, some are not. Thats a tautology!
    3. People do not solve problems for the money. Most of the prizes are too small for that and those that are large are for really hard problems.
    4. There are many of them, more than I thought. I still doubt I'll win one. The closest I ever came was being linked to on the Wikipedia page on the Godel Prize (see this Blog Entry about why that happened.)


  1. Celebrated among algebraic geometers is the prize that the University of Alaska's Elizabeth Allman has offered for a proof of the Salmon Conjecture in projective geometry As an aside, the Salmon Conjecture is named after its prize, namely a (delicious) wild Alaska salmon. This same class of manifolds is naturally equipped with the complex, metric, and symplectic structures that locally determine quantum dynamical trajectories; yet being non-Hilbert globally, these manifolds may someday yield plausible candidates for (what might be called) the Aaronson Quantum Prize. In any case, these manifolds are fun to study, and they are very useful in practical calculations too.

  2. The Godel Prize changed from 7 to 14 years but the start date for these calculations changed so they are not completely comparable. The 7 years was from the time of journal publication. The 14 years starts the clock as soon as a version of the paper appears in a conference or journal.

    The prize was originally started to encourage people with the very best work to actually get it published in a refereed journal. I remember discussions at that time that, particularly for people in industrial research labs, there was essentially no reward for publishing work in refereed journals if it had already appeared in a conference. There were entire research topics, e.g. communication complexity, where many of the major papers only were available in conference versions that did not include some key proofs. (Since then the incentive for journal publication for those in academia has actually been reduced.)

    I think that the change from 7 to 14 years was meant to encourage "timely" conversion to journal paper and to lengthen the actual window a bit. I have never been on a prize committee but I suspect that every year the committee is particularly cognizant of work that is about to "time out". I have no knowledge of why 14 was chosen for the switch. There were a few winners under the 7 year rule that would be counted as 10-11 years under the new rule. When the change was made, I wouldn't be surprised if there was one or more candidate paper in danger of timing out with the switch based on any shorter rule. (I think of the 14 as 4+10 - 4 years as a generous time to get things from conference to journal and then another 10 of eligibility.) The last two years are the only prizes where the gap from conference version to award has been more than 10-11 years. Some papers have won it in roughly the first year of eligibility: quantum factoring, Primes in P, decidability of DPDA equivalence, undirected connectivity in log space.

    Unless that P v NP resolution got published in a refereed journal it would NOT get the Godel Prize. It is well known that Cook's original paper on NP-completeness of SAT would not have been eligible because it never appeared in journal form. On the other hand, Levin's related dissertation work was only fully published in a refereed journal in 2010 (Annals of Pure and Applied Logic - Elsevier) but since there was an initial partial publication many years ago the paper is not eligible under the 14 year rule, though it would have been eligible under the previous 7 year rule!

  3. On #4: There was at one point a $1,000,000 prize offered for the proof of the Goldbach conjecture. This is somewhat of a weird case though. The prize wasn't being offered to promote the conjecture (or mathematics at all) so much as to promote a novel whose plot revolved around it (Doxiadis' "Uncle Petros and Goldbach's Conjecture"). Furthermore, the prize had a strict time had to prove the conjecture by 2002 to win it.

  4. Clearly the Clay Instuite wants to encourage research in these areas. Why so much money? I assume to REALLY put these problems on the map.

    This would not be a cost-effective way to encourage research (compared with giving out grants or funding workshops). The polite theory is that Clay did it to help get good publicity for mathematics, since for example newspapers are more likely to report on a million-dollar prize than just on the solution of a problem their readers have never heard of and can't really understand.

    A more cynical theory is that Clay did it to make himself famous, by ensuring that nobody will ever talk about these problems again without mentioning him and his prize.

  5. This would not be a cost-effective way to encourage research (compared with giving out grants or funding workshops).

    Keep in mind one does not actually need the money to offer an award. The Goldbach $1 million dollar prize mentioned in the comments above was provided by insurance (for a much lower rate than the full million). I don't know the details about the Clay but I imagine they also were not expecting to have to pay all $7 million at once.

  6. I don't know the details about the Clay but I imagine they also were not expecting to have to pay all $7 million at once.

    That's true, but the extra incentive from the prize is nearly meaningless. Spending a few hundred thousand dollars on grants or workshops would do more to speed up the process of solving these problems than offering seven million dollars of prizes.

    1. typed like a naive fool

    2. If you think winning a million dollar prize is a serious motivation for solving a Clay problem, then I doubt you've ever talked with anyone who has even the slightest chance of solving one. If you're smart and care about money, it's not that hard to earn a million dollars, and far more mathematicians have made fortunes in their spare time than will ever solve a Clay problem. Revolutionizing the field and winning eternal fame is a much bigger reward.

      There are two of the problems (the Yang-Mills mass gap and the Hodge conjecture) where the Clay prize probably has made the problem a little more famous. However, if you prove P!=NP or the Riemann hypothesis, then winning the Clay prize will be just a footnote, and even for Navier-Stokes or Birch-Swinnerton-Dyer it's really not such a big deal.

    3. spending a few hundred thousand dollars a small number of times is a million dollars and done as grants could lead to nothing and there's no reason to think it would

    4. I'm talking about spending it once. Of course the effect would be tiny, possibly some minor progress and likely nothing, but the extra incentive from the Clay prizes is even closer to zero.

  7. The money given to these prizes are not to motivate mathematicians but to generate publicity for the prizes and the fields they represent. The Clay Math Millennium prizes gave a huge boon to the P v NP problem, even though the money is unlikely to make a proof come any faster.

    1. Indeed, the fastest way to convey to a layperson that the problem is hard and important is to tell them there's a million dollar prize for solving it.

  8. Just a remark to your analogy under (3): Parapsychology--the discipline--is certainly real. The controversy lies in its subject "Does psi exist?". Curiously, some advocates of a positive answer speculate about a connection with quantum theory (see e.g., D. Radin, Entangled minds). While a negative answer could well be just in terms of normal science (debunking of alleged psi effects), the game changing positive answer would certainly be worth a couple of Nobel prizes. So there's already more than enough money at stake for that problem. But it may be harder to get than any of the Clay prizes (a tautology, of course, for the non-believers in psi).