There’s a fascinating interview with Atiyah and Singer now on-line. It was conducted in May at the time they were awarded the Abel prize. The interview and Atiyah and Singer’s acceptance speeches are also available in video form.
The whole interview is very much worth reading and both Atiyah and Singer make extensive comments about the relation of mathematics and physics. Atiyah makes the provocative prediction that ideas from quantum theory will ultimately have a revolutionary effect on number theory, helping to understand why the Riemann hypothesis or Langlands conjectures are true. He notes that Wiles says this is nonsense. He also predicts that new progress in theoretical physics will come from a better understanding of classical four-dimensional geometry. By this I think he has in mind something like twistor methods. Singer’s comments about string theory are probably typical of the attitude of many mathematicians. He says that, because of the Landscape “you cannot expect to make predictions from string theory. Its inital promise has not been fulfilled”, but he still is an “enthusiastic supporter of superstring theory”, largely because of the interesting mathematics it leads to.
Singer also makes the following sociological comment about mathematics, but I think what he has to say is also very true in physics:
“I observe a trend towards early specialization driven by economic considerations. You must show early promise to get good letters of recommendations to get good first jobs. You can’t afford to branch out until you have established yourself and have a secure position. The realities of life force a narrowness in perspective that is not inherent to mathematics. We can counter too much specialization with new resources that would give young people more freedom than they presently have, freedom to explore mathematics more broadly, or to explore connections with other subjects, like biology these day where there is lots to be discovered.
When I was young the job market was good. It was important to be at a major university but you could still prosper at a smaller one. I am distressed by the coercive effect of today’s job market. Young mathematicians should have the freedom of choice we had when we were young.”
Chris W.,
I appreciate the link below you gave On the numbers.
You know what is interesting to me, is how any number system could have began?
If probabilistics determinations rules our lives, then what said that the Pinball drop could may have defined how life could have manifested, in the number sequence of this flower, and the number of it’s petals?:)
Pascal’s Triangle
If Ramanujan modulars functions can well serve to explain the string’s world sheet, then how much more abstract are we going to get, if we wanted to apply some other kind of math to this function. Etc. Etc. Etc:)
Quite early in my playing around with numbers, I was quite surprise to see how the Ancients used these numbers, as told by Manjul Bhargava. I seem to have a certainty affinity to rythmns, as well as the sequences describe by Manjul, may also be found in Pascal’s triangle.
The movie PI has some weird ideas here, but may not be so weird when considered in context of what rythmns are found, as patterns in life?
To be caution for sure the slight’s given to the Indian influence that Lubos warn’s, John Baez, also gives us this in the link following the pinball at is source.
Speaking of Andrew Wiles (mentioned briefly in this post), a couple of days ago NPR did a profile of one of his recent advisees, now a full professor at Princeton at age 28.
Sorry, this is the link I meant you to have as well
Then I am sure you would like to see the issues on
cosmic clumping and what is being done here in the latest research with Max Tegmark.
The pics are direct links.
The ultimate geometry would have been Martin Rees snake biting it’s tail:), in the unification of the small with the very large, that we have psychologically induced reform. The big gumball( that’s what my wife calls it) that you find in the links by Tegmark, as very revealling.
I joined Andrey Kravtsov’s models, to Tegmarks.
This is so much fun I thought I’d mention it here:
http://www.hep.upenn.edu/~max/toe.pdf
You might call this “hospitality theory” since its basic rules are 1) we need self-aware systems, so be kind to them 2) all consistent systems that are compatible with SASes are physically real!
I just love Fig. 7, although I disagree with his green zone injunction 🙂
-drl
Klein`s Ordering of the Geometries
“A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean
geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.”
http://www.ensc.sfu.ca/people/grad/brassard/personal/THESIS/node21.html
Peter,
Of course some of Klein’s work (theory of the top, automorphic functions, icoashedron) relates number theory (implicitly) to spinors – and this was ages ago. See for example
http://store.yahoo.net/doverpublications/0486495280.html
-drl
“Mathematics is always a continuum, linked to its history, the past – nothing comes out of zero” Atiyah
I found this very statement revealing.
One could not of denied any mathematical interpretation that would have arisen in theory, that could have postulated some emergent property out of string theory? Is this statement valid?
I thought I would challenged any mathematician then to discount the validation of string theory, if it did not emerge from some mathematical interpretaion, how it could not have been considered?
Between 1870 and 1970, the number of freshly minted physics PhDs in the US rose from 1 to 10,000 anually. Today we have a steady-state situation, where each advisor can expect only one of her students to become an advisor, on the average. This is nothing to compain about. Exponential growth cannot be sustained forever, and the number of positions in academia has probably saturated.
Hi Lubos,
One sociological fact that Singer was referring to was that in the late fifties and early sixties, due to the huge expansion in the size of American universities, there were lots of jobs to go around. Since the competition for jobs was much less stiff, people could get away with being less focused on getting results quickly, so could take the time to learn about different fields and not stay so specialized.
I’ll go out on a limb and make a more specific conjecture along the lines of Atiyah’s comments about number theory. Maybe this is the kind of thing Atiyah had in mind, maybe not. From one point of view, the central object in number theory is the absolute Galois group of the field of rational numbers, and the study of its representations. Langlands theory relates these to other sorts of representations (“automorphic representations”). One point of view on particle theory is that central objects are the gauge and diffeomorphism groups and their representations. There are tantalizing analogies between these groups and the ones that appear in number theory. Perhaps new ideas about these representations coming out of a QFT framework may give new ideas about how to study the representations that occur in number theory.
I’m guessing Brian was referring to the “large extra dimensions” scenarios where there are effects that in principle could show up at the LHC when it starts collecting data in 2007-8. Here the idea is that some of the extra dimensions besides the four we know about are not all curled up and unobservably small, but instead we live on a 4d subspace of some higher dimensional space. One might see effects of this as energy disappearing from our 4d world as it moves into the other dimensions.
Most string theorists don’t actually seem to think that these scenarios are at all likely, but they do pull them out when they want to hold out hope that there will some day be experimental results relevant to string theory.
I find resonance with Atyiah’s vision about connections between number theory and physics.
Topological Geometrodynamics allows a formulation as what might be called a generalized number theory. p-Adic number fields and the requirement that real theory allows algebraic continuation to various p-adic number fields is an extremely powerful constraint.
The notion of infinite primes is second powerful notion and very physical: their construction is structurally equivalent to a repeated second quantization of a super-symmetric arithemetic quantum field theory with states labelled by primes.
TGD has inspired also a proposal for a proof of Riemann hypothesis based on a very simple conformally invariant dynamical system having zeros of Zeta as conformal weights. Rieman Zeta defines overlaps for general coherent states labelled by conformal weights z which are zeros z=1/2+iy of Riemann Zeta. Rieman hypothesis follows from the absence of state with negative norm (which corresponds to z=0).
M. Pitkanen (2002), A Strategy for Proving Riemann Hypothesis, math\@arXiv.org/0111262.
M. Pitkanen (2003), A Strategy for Proving Riemann Hypothesis, Acta Math. Univ. Comeniae, vol. 72.
The zeros of Zeta and their certain combinations appear also as conformal weights of super-canonical algebra playing together with Super Kac-Moody algebra a key role in TGD. Hermiticity requirement implies what I call conformal confinement: net conformal weights are real for physical states. For instance, quarks and gluons could have complex conformal weights and color confinement could reduce to conformal confinement.
For details see for instance
http://wwww.physics.helsinki.fi/~matpitka/tgd.html#mless .
http://wwww.physics.helsinki.fi/~matpitka/tgd.html#number .
http://wwww.physics.helsinki.fi/~matpitka/padtgd.html#mass1 .
Matti Pitkanen
Your colleague Brian Greene came to Portland to talk about his new book. It was an interesting talk which he finished with a couple of statements. One of them was that there was going to be an experiment in the near future that could substantiate string theory because it predicts that under this experiment “energy would disappear.” No one asked him if String Theory was even still valid.
Do you know any more details about this experiment?
Thanks!
That’s very interesting.
Well, I don’t quite see the particular connections between various fields of mathematics and physics that Atiyah envisions, but he certainly has some reasons to talk about them. 😉 I might propose similar, but different connections, and no one else would understand me. 🙂
I would expect Atiyah to have more visions about higher-dimensional (e.g. seven-dimensional) geometry etc.
Concerning overspecialization, it’s of course a wrong tendency, but I don’t think that the change is due to a different social environment. People are just deciding differently today than they were deciding when Singer was getting started.
In my opinion it has two main (and related) reasons:
* there is a general (and probably true) feeling that the obvious “big questions” have been solved, and therefore people must attack smaller, more specialized questions
* people in average are less ambitious today than they were a few decades ago, and a bigger fraction of the talented youth with big goals is eaten by the commercial sector and similar enterprises
It’s not quite clear to me whether it is better if the good people try to concentrate at the top universities, or they spread all over the world (or at least over the country). Concentration may help communication, but of course the “second class” universities suffer.