The Abel prize is a new yearly prize in mathematics, intended to function somewhat like a Nobel Prize for mathematics. The first one was awarded last year to Jean-Pierre Serre and this year’s has gone to Sir Michael Atiyah and Isadore Singer, specifically for their development of the Atiyah-Singer index theorem.
Atiyah and Singer are great heroes of mine, especially Atiyah. They have both taken a great interest in the relation of mathematics and quantum field theory and are responsible for much of the fruitful exchange of ideas between the two subjects over the last 25 years. I consider much of my mathematical education to have come from a lot of time spent reading through Atiyah’s five-volume collected works. His interests have ranged over a wide swath of modern mathematics and his writing is always a model of clarity. A good case could be made that Atiyah has been the most important figure in mathematics during the second half of the twentieth century.
The Atiyah-Singer index theorem is perhaps the single most important theorem of the last half century. It links together analysis, topology, geometry and representation theory in a fundamental and surprising way, one that is dear to any physicist since it involves the Dirac operator. Very roughly, what Atiyah and Singer discovered was that the dimension of the space of solutions of certain PDEs on compact manifolds was a topological invariant, one that they could explicitly compute in terms of more well-understood topological invariants: the cohomology classes of the manifold.
They did this by noticing that the general case could be reduced to the case of the Dirac operator, twisted by the various possible vector bundles on the manifold. They actually rediscovered the Dirac operator for themselves in the course of their research. The natural abstract framework for these new topological invariants is K-theory, where classes are represented by vector bundles, much as cohomology classes are represented by differential forms.
Things get even more interesting if there is a symmetry group acting on the whole set-up. Then the space of solutions carries not just an integer, the dimension, but is a representation of the group. You can actually use the Atiyah-Singer index theorem to classify and construct geometrically the representations of many classes of Lie groups, or going the other way, use representation theory to get more powerful topological invariants. The explicit cohomological formulas you get in these cases often have versions which localize to fixed points of the group action; so you can find your answer just by locating the fixed points and looking at what happens in a neighborhood about them.
I hope Atiyah and Singer enjoy their shared $875,000!
Hi Peter,
Could you elaborate on Grothendieck’s new ideas of “space”? Do you mean change of element (e.g. Pluecker) or something totally new?
I can hardly believe I’d never heard of this person!
-danny
Hi Erin,
I can’t make any definite statements about LQG because I don’t know enough about it. My understanding is – it makes more physical sense than strings but there are fewer “results”. Neither is finished in the sense of having a definite schemata and methods. IMO string theory is ab initio preposterous, while LQG is excessively formal. Having some detailed knowledge of the history of physics, progress just does not come that way.
I don’t want to spam Peter’s blog with details, but if you are interested in what I did I’d be happy to explain it elsewhere. In contrast to LQG or strings, it’s a complete scheme with sensible elements and very little arbitrariness, and does not purport to be quantum gravity up front (although I am quite confident that strict locality in this theory makes it a much better candidate for quantization than GR). It is a shame to have to work in isolation because I could really use input from field theorists with physical grounding at this point. This after all is why collaborative environments in universities were invented.
Grothendieck’s work and life are truly amazing. Someone should write a biography and make a movie; it would make John Nash’s story (“A Beautiful Mind”) look pretty prosaic.
I had looked at that web-site last year, but there is a lot more material there now. It has a pdf of “Recoltes et Semailles” the ravings of Grothendieck’s later years, scanned by a student here at Columbia (Max Lipyansky) from a copy I had in my office. For physicists, I recommend reading the article by Pierre Cartier about Grothendieck’s work, subtitled “The evolution of concepts of space and symmetry”. Grothendieck’s mathematical concerns were far from physics, but he developed fantastic new ideas about how to think about a “space”. Since physicists are now so sure that the standard geometrical concepts of space and time need to be generalized, they might get some inspiration from Grothendieck’s ideas.
Hi Peter,
Yes, it would be very interesting if the Abel Prize were to be awarded to – or attempted to be awarded to – Grothendieck. I believe he also refused to travel to Moscow in 1966 to collect his Fields Medal. Have you ever seen the following website? I recommend it:
http://www.grothendieck-circle.org
Grothendieck is/was a fascinating mathematician! His work is astonishing, in my opinion. (I’m a bit of a fan of his, even if most of his major works remain untranslated into English.)
Hi Erin,
I think the whole idea of setting up the Abel prize was to have something more or less equivalent to a Nobel prize in mathematics. One reason for this certainly is to raise the public profile of mathematics. We’ll see if it works.
The Fields medal until now has been the most well-known prize in mathematics (although there are some others, including the Wolf prize). It is somewhat different in that it is given only once every four years (to 2-4 people), is restricted to mathematicians under the age of forty, and carries little or no money with it. So there are relatively few Fields medalists compared to, say Physics Nobelists, and quite a few very good mathematicians don’t have one since either they didn’t do their best work or it wasn’t recognized before they were 40. An example is Andrew Wiles, who was a bit over 40 by the time he found a proof of Fermat’s last theorem.
It’s kind of a strange situation, to start up a new prize like this. One has all the best living mathematicians to choose from; many of them have Fields medals, some don’t. There was some speculation the Abel prize committee might decide to first award prizes to those who didn’t get a Fields medal (e.g. Wiles), but that isn’t what they are doing. I’d guess a lot of mathematicians would list as the top three living mathematicians of the last half century Serre, Atiyah and Grothendieck. It will be interesting to see what the Abel committee does about Grothendieck. He’s still alive, but stopped doing mathematics and hid himself away in the mountains in France. The last time someone tried to give him a prize (the Crafoord Prize), he very publicly rejected it. I’d assume he’d do the same thing if the Abel committee tries to award him the prize.
Hi Peter,
It’s wonderful to see great mathematicians of our era such as these recognized and honoured like this. (I realize I should have posted these comments earlier, but I’ll do so anyway now, disconnected as they are from the preceding posts.)
I’ve often wondered if the existence of a Nobel Prize in Mathematics would have improved the global public perception of mathematics, and especially made mathematics be considered on a par with the other sciences for which Nobel Prizes exist (which surely it is, isn’t it?). Perhaps more people would study mathematics if they knew there were Nobel Prizes awarded for it. I myself think Nobel did mathematics a bit of diservice by excluding it.
So if the Fields Medal is the `equivalent’ of the Nobel Prize in mathematics (despite being awarded only every four years, restricted to those younger then 40 years of age and having a smaller prize fund), then where does that leave the Abel Prize? Doesn’t the Abel Prize perhaps go some way to being the Nobel Prize in mathematics (awarded annually, Scandanavian, large prize fund, likely to be awarded to those over 40 years of age… perhaps always awarded to fewer than three people?)?
Another thing: both Serre and Atiyah already have Fields Medals, but Singer does not. How many other future Abel Prize winners will also be Fields Medallists or have worked with one?
Erin
P.S. Danny: isn’t the theory of loop quantum gravity equivalent to background-independent – presumably synonymous with “background-free” – Yang-Mills gauge theory? If it is, then there are already at least some physicists working on this theory.
Hi Chris,
I’ve seen the academic system (at least its US version) at work from closeup in two different fields, particle physics and math, and the differences are interesting. There are problems with how it works in math that are somewhat the same as those in physics: it is tough for young people who want to try to do something really ambitious, and the safest bet for one’s career is to work on a not too ambitious topic in a hot area. But things are very different in math for several reasons:
1. The ratio of good young people to jobs is nowhere near as bad as in particle theory. Very roughly I’d guess the situation in math is that there are perhaps twice as many people getting Ph.D.s in math as there are permanent teaching jobs that might allow you time to do research. In particle theory the situation is closer to ten times as many people as permanent jobs. The game of musical chairs for young people in particle theory is a lot more brutal than in math.
2. Particle theory has always been a much trendier subject than math, partly for the good reason that , historically, as unexpected new experimental results became available, the smart thing to do was generally to be thinking about what they meant. The lack of any unexpected experimental results for nearly 30 years has been hugely problematic for particle theory.
So, while we agree that the particle theory academic system is quite unhealthy these days, I don’t think that moving the power base to the young would help much (besides being extremely unlikely to happen; the old do not give up power so easily!). If older people as well as younger ones were all in the same situation of way too few jobs, with the only way to survive to be working on the latest trend, things might not be any better.
My own general ideas about how to make particle theory a healthier business are something like the following:
1. First of all, the hype has to be brought under control. As long as ideas that don’t work are being sold as huge successes, the field is going to be really unhealthy no matter how it is organized.
2. Bring the ratio of good young people to jobs under control, either by finding a way to fund more positions (this may happen to some extent naturally as the rate of retirements picks up) or by limiting the size and number of particle theory Ph.D. programs.
3. Find ways of encouraging people to work on less trendy things. This would require a real change in the whole sociology of how particle theory is organized. There are a lot of ways you could imagine encouraging such a change.
I must admit though, that prospects for any such changes look pretty dismal.
Hi Danny,
Nice to hear an assenting voice. Or any voice at all, for that matter. My beliefs on QFT [anti-renormalization] and academic research in general [anti-gerontocracy] have thus far provoked a deafening silence. Maybe there are too many vested interests. If Dirac had lived 50 years later, I might have survived better. I particularly liked his quote, when talking about QED: “Just because a theory agrees with experiment, it does not mean that it is correct.” Thank-you, Paul. As usual concise and to the point.
I have to say, though, that in many ways I believe that I am only stating the obvious. In the first case it is the obvious fact that you can get any answer you want by subtracting infinity from infinity and in the second case, I, like everyone else, was more creative and mentally agile at the age of 22 than I am now at 44. One should accept these things, and it is grossly unfair that someone my age now who has an academic post can put obstacles in the way of a young researcher with new ideas just because they are not *my* ideas.
I am not answering your question … I am *sort of* still working on QFT in my spare time, but on a kind of QFT text book (sans renormalization, obviously) rather than research. I will post it on my site in the not-too-distant future.
In the meantime, in particular as regards the komissar, do not forget the old adage:
Non Illegitemi Carborundum
You just have to stick to the ideas and hope the world gets better. At least you are still working.
I remember your work, it was interesting. Are you still at it?
If it’s any consolation, I actually managed to solve Einstein’s problem and get it published, only to be blacklisted from the archive by its komissar. No one in the entire world seems to be interested in background-free electrodynamics (and coming soon, background-free Yang-Mills!) This to me is the deepest mystery of all.
Keep at it, all you can do is – all you can do.
-danny
This is neither an ad for sexual-performance-enhancing chemicals or an assessment of whether or not Michael Atiyah deserved $437,500 for his index theorem, being rather some general observations about academic research which have already been made on my web site, but will be expanded on when I can be bothered. It follows on from Peter’s post “My (Not So) Brilliant Career”.
As far as I can tell, the research situation now is much the same as it was when I was a graduate student/research fellow (1982-1987). The basic fact is that an institution (pure academic research) that has no clearly-defined objective will by definition lack a self-correction mechanism. This is not the case in financial software, which is what I work in now, as measures of success and failure are often brutally apparent, even to those who have never written a line of code in their lives.
If the academic system worked as it should, every graduate student entering the field would be told something along the lines of the following: “You are where you are because have exceptional ability in mathematics and science. It may well be, although it does not necessarily follow, that you have valuable contributions to make to your subject. It does not necessarily follow because what you are now embarked upon is essentially different to what you have done up till now. Instead of jumping through hoops that others have set up, you will now be deciding the nature and shape of the hoops themselves. If the task of passing exams were to be likened to being an accountant, the task of doing research is like being an entrepreneur. Read and absorb everything you can get your hands on. Do not rely on others to tell you what is important as these assessments have to come from within yourself. If having done this, you decide that, in all honesty, all you can hope to contribute is a few dotted i’s and crossed t’s then get out and do something useful instead. You will almost certainly be better paid if you do.”
No-one said this to me in 1981. What it was about was not doing what you thought was right, but getting noticed. You had to attach yourself to some research project, however little you believed in it, and then hope that the “big shots” in this field liked you enough to offer you employment. Fine if you believed in it, but if there was *nothing* that they were doing that you believed in, and you wanted to start your own line of enquiry, then there was nowhere to go except out. The herding tendency was embarrassing. “Safety in numbers” was and is the principle, and I am amazed at the extent to which researchers were and are prepared to sacrifice integrity, enjoyment and quality of life for the sake of the distant prospect of a poorly-paid tenured academic job. If the situation in particle physics was bad 20 years ago, then it is ten times worse today.
The model of researchers as inquisitive children rather than flocks of sheep would work much better for me. How might one get there? The obvious step would be to move the power base to younger people, by (a) only granting research posts on a rolling 5-year basis, i.e. NO permanent posts and (b) giving equal votes in hiring, etc. to ALL involved in research, including graduate students. These steps would perhaps not solve *all* the problems, but they would at least make academic research more interesting.