A commenter on the last posting pointed to the new video available at the IAS site of Witten’s recent public talk there on Knots and Quantum Theory. The talk is aimed at a general audience, including supporters of the IAS, so it’s rather non-technical. For the technical details behind what Witten is talking about (his recent work on Khovanov homology and QFT), see this survey for mathematicians, a survey for physicists at Strings 2011, and this paper.
For me an interesting part of Witten’s talk was how he described the evolution of his ideas about this topic, and the relationship to geometric Langlands. He also had interesting comments about number theory and the Langlands program, denying any real knowledge of the subject, but arguing that sooner or later (probably later, after his career is over), there would be some convergence of number theory Langlands and physics. He finds the coincidence of geometric Langlands showing up in QFT so remarkable as to indicate that there are deep connections there still to be explored. I suspect that he sees the likely path of information going more from physics to math, with QFT ideas giving insight into number theory. While I agree with him about the existence of deep connections, I suspect the influence might go the other way, with the powerful ideas behind the Langlands program in number theory someday providing some clues about QFT useful to physicists.
Also on the Langlands program topic, this semester we’re having a wonderful series of lectures on the topic by Dick Gross. He’s a fantastically gifted lecturer, and this series is pitched at just the right level for me, explicating many of the parts of the subject I’ve been trying to learn in recent years but have found quite confusing. It’s a beautiful, very deep, but rather intricate subject, bringing together a range of remarkable ideas about mathematics. In the end though, the Langlands program is really mostly about new ideas in representation theory, and since I’m convinced that deeper understanding of QFT will require new ideas about how to handle symmetries, which is the same thing as representation theory, perhaps finding connections between the subjects won’t have to await Witten’s retirement.
Pingback: Witten and Knots « viXra log
Of course there is a big difference between Geometric Langlands Program and Langlands Program in the sense that Langlands uses the term. When you say that Witten expects a connection between number theory and physics does he mean Geometric Langlands or number theory?
martin,
Witten has already done extensive work on the connection between geometric Langlands and physics (and there was such a connection, to conformal field theory, there from the beginning). What he was speculating about was the possibility of a connection to the number theory case.
Peter Woit,
Thakns, so he did mean number theory. I’ve just finished watching the talk, and I saw the part where he said that about the Langlands program, but it was only one sentence in passing. Has he explained somewhere why he thinks so? Or is ti really just a speculation, he did say that he was not very familiar with number theory.
martin,
As far as I know he doesn’t have anything specific in mind, this is just speculation. Philip Gibbs at vixra log transcribed the relevant part of his comments:
“I had in mind something a little bit more ambitious like whether physics could affect number theory at a really serious structural level like shedding light on the Langlands program. I’m only going to give you a physicists answer but personally I think it is unlikely that it is an accident that Geometric Langlands has a natural description in terms of quantum physics, and I am confident that that description is natural even though Ithink it mught take a long time for the math world to properly understand it. So I think there is a very large gap between these fields of maths and physics. I think if anything the gap is larger than most people appreciate and therefore I think that the pieces we actually see are only fragements of a much bigger totality.”
I really enjoyed Gross’s lectures on undergraduate abstract algebra. He is a great speaker.
Knot even wrong?
On the topic of possible applications of the Langlands program to physics, there is a famous conjecture of Atle Selberg about the lowest nonzero eigenvalue of the Laplace-Beltrami operator on 2-dimensional cusped hyperbolic manifolds obtained by quotienting 2-dimensional hyperbolic space, with sectional curvature -1, by a k principal congruence subgroup of SL(2,Z), k >= 3, which has no torsion. Selberg conjectured that the lowest nonzero eigenvalue is 1/4, and proved it is >= 3/16. Selberg’s lower bound was improved to 171/784 = 0.2181… by Luo, Rudnick, and Sarnak, and has been increased to 975/4096 = 0.238… by Kim and Sarnak. The analogous conjecture for the lowest non-zero eigenvalue on cusped principal congruence quotients of n-dimensional hyperbolic space H^n is that it is (n – 1)^2/4, which is where the spectrum starts on H^n. Sarnak has obtained a lower bound of 21/25 = 0.84 for a family of cusped hyperbolic 3-manifolds called Bianchi manifolds.
I understand from page 5 of this article by Peter Sarnak that Selberg’s conjecture, and from pages 14 to 16, possibly also the above generalization to cusped principal congruence quotients of H^n, are now understood to be part of Generalized Ramanujan Conjectures that are part of the (number theory) Langlands program.
If I have got that correct, (and I have to admit that there is a lot in Sarnak’s article that I don’t understand), does Ngo’s proof of the Fundamental Lemma of the Langlands program mean that Selberg’s conjecture, and its generalization to higher dimensional cusped hyperbolic manifolds as above, are now proved?
Chris,
The proof of the Fundamental lemma only proves a small piece of the conjectures that make up the Langlands program, and as far as I know doesn’t tell you anything about these Generalized Ramanujan conjectures.
For an interesting connection between physics and number theory that Sarnak has worked on, see his lectures about “Arithmetic Quantum Chaos” here
http://publications.ias.edu/sites/default/files/Arithmetic%20Quantum%20Chaos.pdf
There are evidently examples of chaotic dynamical systems such that the problem of quantizing them and finding their spectrum is related to problems like the one you mention. This however I think is very different than what Witten has in mind, which involves not just QM, but quantum field theory.
Thanks, Peter, I’ll take a look.
I guess I should educate you physics types 🙂 The history of the Selberg conjecture and related issues is fascinating, McKean in the early 70s published a “proof” of the fact that the first eigenvalue was > 1/4 for ANY surface, thing was there was a mistake in the proof. Burt Randol (who taught me complex and real analysis) showed this by coming up with examples of surfaces with first eigenvalue going to zero (Randol is both a great mathematician and the best teacher I ever had, he just retired). The reason that 1/4 is so important is that it is the bottom of the L^2 spectrum of the Laplacian for the hyperbolic plane, and is related to the zeros of the zeta function (that is in fact how Selberg got his 3/16 result, he used Weil’s theorem on zeta functions over finite fields). This sort of thing is why I ended up in geometric analysis, somehow everything seems to come together in strange and interesting ways…
Thanks, Jeff. Randol’s article appears as ref [26] of this review by Peter Sarnak. The examples with 1st eigenvalue going to 0 on the 2nd page here use a thin neck such that the length of a closed geodesic going round it goes to 0. You can’t have examples like that in n > 2 dimensions because you can’t put a hyperbolic metric on the “neck” of a connected sum for n > 2, and furthermore a finite volume hyperbolic structure has no shape moduli for n > 2 by Mostow rigidity. Which raises the question of whether the result claimed by McKean, (which I had not previously heard of), might be valid at least for n >= 4, with 1/4 replaced by (n – 1)^2/4. For n = 3 you could perhaps look for examples with large Dehn surgery coefficients in Thurston’s construction.
Chris,
Yes, n=2 is a special case. Basically for hyperbolic manifolds, there are 3 different cases, n=2, n=3 and n>3. In dimension two you can have continuous deformation, in dimension 3 you can still shrink a geodesic to length 0 to get a cusp, but the deformation isn’t continuous, and in dimension 4 and higher you can’t even do that, there is a direct relation between the length of the shortest geodesic and the volume. My thesis was the n=3 case for the Laplacian on differential forms, and I studied the accumulation of eigenvalues near the bottom of the essential spectrum as the length of the cusp goes to infinity. For functions the bottom of the essential spectrum is (n-1)^2/4, but there can be isolated eigenvalues below that. The relevant theorem is due to Buser, Colbois, and Dodziuk, and says that for n>2 with pinched negative curvature, the number of eigenvalues below the bottom of the essential spectrum is a constant times the volume, and the constant depends only on n and the curvature bound. I have a paper with Ruth Gornet which discusses the bounded curvature case for forms, it’s in Contemporary Math 237, and if I remember right it lists at least most of the relevant references in case you’re interested.
Peter – hope it’s OK to mention a paper of mine here, it was the best place I could think of for a reasonably complete list of interesting references for Chris.
Thanks a lot Jeff, I’m reading your article, “Small eigenvalues of the Hodge Laplacian for three-manifolds with pinched negative curvature,” in Contemporary Mathematics 237, now.
To find examples with small eigenvalues for all n >= 2, section 2.8.C in Gromov – Piatetski-Shapiro gives examples for all n >= 2 of finite-volume hyperbolic n-manifolds X, both compact and cusped, that contain a 2-sided non-separating embedded closed hypersurface S. If we take 2N copies of X, cut each along S, and join side b of copy 1 to side a of copy 2, side b of copy 2 to side a of copy 3, … , side b of copy 2N to side a of copy 1, we get a hyperbolic n-manifold that is a 2N-fold cover of X, and if we choose the function f on page 2 of the Sarnak review to be 1 on copies 1 to N and -1 on copies N + 1 to 2N, with smooth transitions across the two copies of S where f changes value, we get an upper bound on the first non-zero eigenvalue that is reduced by a factor 1/N, so can be made arbitrarily small by choosing N large.
Peter, I’m sorry if this has got off-topic.
Off-topic but highly interesting: a recent paper, with lots of apparently serious people backing it, claims the wavefunction has no statistical interpretation, it is real: see this Nature description…
Honestly, I have watched this presentation and was disappointed.
He failed according to Feynman’s criterion, “You do not really understand something unless you can explain it to your grandmother”, and a lot of Witten’s classic papers are classic precisely because they succeed in this respect, in the sense that they are remarkably clear in their arguments, both in stating the conclusion, supporting it, and elucidating the consequences.
I understand of course that this is a public talk on a specialized subject, but I failed to get even an inkling of what he was trying to accomplish, and why this worthwhile to do, and this is what the talk should have been about.
Witten’s old paper on Free Fermions on an Algebraic Curve has some connection to number field Langlands, as well as geometric Langlands (that it predates).
hello!,
could you tell us which paper. Thanks.
mbn,
The paper “Free Fermions on an Algebraic Curve” is in
The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 329-344. 16. Garland, H. and Zuckerman, G.: R
A more extensive version is
Quantum Field theory, Grassmannians, and algebraic curves,
Comm. Math. Phys. 113 (1988), 529-600.
For a more recent paper along similar lines, see Leon Takhtajan’s
http://arxiv.org/abs/0812.0169
I think it’s fair to say that ideas from the geometric Langlands program have recently started having an impact in number theory. The starting point is Ngo’s celebrated work on the Fundamental Lemma, which doesn’t explicitly use geometric Langlands but uses ideas that are very much part of that world. Since then this interaction has expanded significantly, primarily (in my not completely informed view) thanks to work of Ngo, Zhiwei Yun (MIT) and Xinwen Zhu (Harvard). In a spectacular recent preprint on his website, following up on ideas of Frenkel-Gross and his prior joint work on Kloosterman sheaves with Heinloth and Ngo, Yun used geometric Langlands ideas to construct motives with exceptional groups as Galois groups (in particular solving the inverse Galois problem for a large family of exceptional finite groups of Lie type). Other exciting developments in this direction include Zhu’s resolution of a conjecture of Pappas and Rapoport on Shimura varieties and a variety of works deepening our understanding of various fundamental lemmas and their geometric origin. Behind all of this at some basic level is the realization of the motivic or geometric nature of many quantities of arithmetic interest, such as the constituents of the trace formula (see eg Nadler’s BAMS article on the Fundamental Lemma), which enables arithmetic questions to be deduced out of more structured questions that can be approached in a function field setting using geometric ideas.
(Let me note that some of this work, eg Yun’s work on global Springer theory, EXPLICITLY acknowledges a debt to physics: the influence of Kapustin-Witten on our understanding of the geometric Langlands program shouldn’t be underestimated.)
I should also mention the fascinating work of Frenkel-Langlands-Ngo, proposing a strategy to understand Langlands functoriality based again on ideas of geometric (and physical) origin.
David,
Thanks for your comments. I should note that, following up on Dick Gross this fall, Edward Frenkel will be the Eilenberg lecturer here this spring. I’m hoping we’ll be hearing more about some of the topics you mention from him.
Gross’ Eilenberg lecture series is being made available:
http://www.math.columbia.edu/~staff/EilenbergVideos/Gross/index.html
kudos to Columbia for doing this.
Kudos to Columbia???? – what is SliverLite?
Generic problem: science for the rest of us should not be dependent
on on the fact that (a) not everyone uses Windows, (b) not everyone
has access to a university library – e.g. it costs ~$30 to read an article
published in Nature or a Springer or IOP journal.
Lee and Ignatz,
The videos were produced, with no help from the university or department, for free by some of our grad students (especially Alex Waldron and Ioan Filip, maybe others I don’t know about). They definitely deserve thanks from the whole math community for doing this on their own. One of our staff members, Nathan Schweer, helped put them up on the web server.
As for the format, it works fine on Macs (I just tried it out). Yes, you do need to install a “Silverlight” browser plugin from Microsoft. If you’re allergic to them, in this day of Apple and Google dominating the internet, you should try and get over it and get outraged about something more up-to-date. Under Linux, there seems to be a Silverlight firefox addon, which, as usual in these things, doesn’t work with the version of firefox I have, so I gave up. If we were paying somebody for this video, I suppose I’d look into demanding that they use a Linux-friendly format, but, you get what you pay for in the free software and content world….
As for the Nature/Springer/IOP problems, yes, in many cases it’s outrageous that content produced by scientists that should be in the public domain isn’t. In some cases though, I’m linking to material produced by professional science journalists, and someone has to pay them…
Well I confess I did not notice the dependency when I watched using Chrome on whatever my netbook was booted as at the time, of course barriers are offensive just as it’s offensive that there are millions who have no access to the Internet at all.
As one of the fortunate it astonishes me that material like this is there (thanks to grad students in this case); if I had been told 25 years ago when I was a student that material like this and some of the open courseware, lecture notes and also videos of the quality found on SCGP,KITP and PIRSA and elsewhere would be “available” it would have seemed to me the stuff of science fiction.