- For an Oxford conference last week, Langlands contributed a one-hour video talk, filmed in his office. One hour was not enough, so hours two and three are also available, as well as a separate text, and some additional comments.
- The latest AMS Notices has a long section of excellent articles about Friedrich Hirzebruch and his mathematical work.
- Also in the AMS notices is a long defense of the NSA, written by a mathematician who worked there for 41 years. About the main recent controversy here, the Snowden revelation of an NSA backdoor in an NIST standard, all the author has to say is:

I have never heard of any proven weakness in a cryptographic

algorithm that’s linked to NSA; just innuendo.This seems to me to exemplify pretty well the disturbing tactic of the US security establishment of claiming there is no problem while refusing to discuss anything problematic since it is classified.

- Bhargava, Skinner and my colleague Wei Zhang have a new paper out proving that better than 66% of elliptic curves satisfy the BSD conjecture. It seems not implausible that they or others might in the not too distant future get to 100%. One should note though that showing 100% of elliptic curves satisfy BSD wouldn’t be the same thing as showing all elliptic curves satisfy BSD, so wouldn’t be eligible for the $1 million Millennium prize.
- With the ICM less than a month away, I find it outrageous that no one has yet leaked to me the names of the Fields Medal winners. All I’ve heard is speculation, and the only name I’d bet any money on is Bhargava.

**
Update**: For something both ICM and Langlands related, Michael Harris on his web site has his ICM contribution Automorphic Galois representations and the cohomology of Shimura varieties. Many of the ICM 2014 proceedings contributions are already available on arXiv, via this search.

For the Fields medal winners, my money is on Bhargava, Lurie, Avila and, perhaps, Scholze.

Peter, can you explain how proving that 100% of elliptic curves satisfy BSD is not the same thing as proving that all elliptic curves satisfy BSD? Is this some mathematician’s definition of 100% that does not mean “all”, or is the particular prize question in terms of a single general proof rather than breaking the result into subcategories? (Bhargava gave the single most impressive math talk I’ve ever seen, and he used only a blank transparency and a felt-tip marker.)

Douglas Natelson,

100% of numbers in the interval [0,1] are not 1/2 but not all numbers in [0,1] are not 1/2.

@Douglas,

for example another statement: 100% of integers are not prime.

To expand on the remark of Cartan, the standard conjecture I believe is that if you order elliptic curves by height, in the limit as you go off to infinity, 50% will have rank 0, and 50% will have rank 1. Of course there are plenty of elliptic curves of higher rank, and to prove BSD you need to prove it for them. In the limit though, these do make up 0% of the curves.

Got it. It’s an issue of measures. Thanks!

Speculation is that there will be a female Fields medalist.

NSA mathematician Richard George protests that talk of the NSA deliberately weakening cryptographic algorithms is “innuendo”. But over at the

n-Category Café, user bgg points out another piece of evidence to the contrary: the NSA’s own 2013 budget request, which says the NSA will use funds to:and

Peter,

Not that it matters now, but perhaps better than 100% is “almost all.” Then again, I guess someone will ask what that actually means. How about “except for a set of measure 0.” I’m hoping no one asks for a definition of “measure.”

Bill,

I’ll raise you: I’m hearing speculation of TWO female Fields medalists.

Tom,

That “innuendo” business is just outrageous. Does anyone believe the NIST issued a warning not to use that standard based on “innuendo”???? Someone should write into the Notices about this. Makes me agree with Beilinson’s point of view…

It cannot be emphasized too strongly how much the relatively short paper of Bhargava, Skinner, and Zhang rests on tremendous amounts of substantial prior work of the 3 authors separately (e.g., item [15] in the bibliography).

Also, though it will be spectacular to know some day that “0%” of elliptic curves over Q have higher rank (and that “100%” satisfy the rank-BSD conjecture), the “0%” case can of course occupy a substantial % of the proof of a big theorem: recall the crucial role of a higher-rank elliptic curve in the solution of the Gauss class number problem for imaginary quadratic fields, that only finitely many elliptic curves in each characteristic p are supersingular yet every elliptic curve over Q has infinitely many primes of supersingular reduction (though the set of such primes is 0% in the non-CM case), and that the Gross-Zagier paper devotes a lot of space to primes of supersingular reduction.

Actually, the whole process of awarding Fields medals reminds of “top 10 world’s best place to travel” published by many magazines. Yes, Paris is great, but a mountain campsite can be pretty awesome too. The difference is that nobody really takes these top 10 lists seriously. On the other hand, I suspect there are many more travelers qualified to compare various travel destinations then there are mathematicians qualified to judge top 4 mathematicians under forty.

Speculation from outsiders is one thing, but leaks from insiders would be shocking IMHO.

I do hope the “female winner rumor” is true though, there are several exceptional ones this time around, so statistically out of four winners there’s no reason for it not to happen.

In his Oxford lecture “Problems in the theory of automorphic forms — 45 years later” Robert Langlands had some noteworthy points to make about the field. Specifically in the survey pdf that goes with the video lectures there are comments on the effects of Fields medals and on specialization.

p. 5: “Since Fields medals are taken quite seriously by a large number of mathematicians and by whatever part of the general public takes an interest in mathematics, the lack of perspective is regrettable. […] These specialists are all excellent mathematicians, but one wonders again whether the influence of these medals, and other prizes with wide recognition, on the course of mathematics and on our understanding of it is entirely beneficial.”

and more on specialists:

on p. 8 “I find, however, that they are excessively focussed. We can, I am sure, not do without them; they however appear to prefer problems with circumscribed, more modest goals! I fear that those with a larger view of the subject will have to learn from them; on the other hand, they themselves, so far as I have observed, have no desire to learn anything was ihnen nicht in den Kram paßt.”

On the other hand, regarding the broader perspective of geometric Langlands correspondence and in particular its claimed incarnation within string theory:

p. 6 “I am still puzzled by the notion of mirror symmetry and, in general, am curious what ‘theoretical physicists’ have in mind when they speak of the “Langlands program.”

More of this cautioning that there remains mathematical substance to be added to a handwaving story is in the first video lecture and was also pronounced earlier this year in a coment to a message to Sarnak:

“One popular introduction to the topic is Frenkel’s Bourbaki lecture, ‘Gauge theory and Langlands duality’. On the first page, he describes electro-magnetic duality as an aspect of the Maxwell equations and their quantum-theoretical form or, more generally, as an aspect of four-dimensional gauge theory. This duality is quite different than the functoriality and reciprocity introduced in the arithmetic theory. It entails a supplementary system of differential equations. Moreover, it has to be judged by different criteria. One is whether it is physically relevant. There is, I believe, a good deal of scepticism, which, if I am to believe my informants, is experimentally well-founded. Although the notions of functoriality and reciprocity have, on the whole, been well received by mathematicians, they have had to surmount some entrenched resistance, perhaps still latent. So I, at least, am uneasy about associating them with vulnerable physical notions. On the other hand, as strictly mathematical notions this duality and various attendant constructions, such as the Hitchin fibration, appear to have proven value, especially for topologists and geometers. Whether it is equal to that of functoriality and reciprocity is open to discussion.”

This seems to show that even the idea that the string theoretic argument is supposed to be a purely mathematical argument remains unclear.

In my own talk at the end of the Oxford meeting I took the liberty of picking of that theme, arguing that what is urgently missing is a general mathematical theory of higher codimension quantization that would allow to turn statements about ‘t Hooft operators, Wilson operators etc. into precise definitions, clear statements and formal proofs. Some indications in this direction are here.

To put the Bhargava, Skinner, Zhang result in perspective, we’ve known for 80 years that the Hodge conjecture is true for almost all abelian varieties (in terms of the moduli space), but this has helped not at all in our efforts to prove the Hodge conjecture for all abelian varieties.

Urs,

Is there a transcription error in this sentence from your quotation of Langlands?

Chris W., that seems a weird question of yours, unless I am missing something. Follow the link I gave to see that this is not transcribed but typed by RL himself.

Also, your suggestion for how to change the wording indicates that you may be missing what I think is the point of the statement. The point of the statement is that RL does not wish to have associated with the mathematical program named after him non-rigorous claims whose alleged validity he feels might well not stand the test of time, to the extent that they are based just on physics-inspired ideas and not on secure proofs.

The search for archived ICM papers can be improved by a factor > 5

http://arxiv.org/find/grp_math/1/OR+co:+AND+ICM+2014+co:+AND+ICM+proceedings/0/1/0/2014/0/1?per_page=100

As a member of a standards committee involved the removal of the mentioned algorithm from a standard, none of the members believe the “innuendo” theory, and all believe it was deliberately weakened. In fact, the possibility of potential weaknesses were noted during the NIST analysis. At a minimum, there is something very, very fishy about the algorithm in question.

This site:

http://blog.cryptographyengineering.com/2013/09/the-many-flaws-of-dualecdrbg.html

Has an excellent discussion of the flaws in Dual_EC_DRBG. Either (1) NSA was ignorant of these mathematical details, and therefore can’t be trusted based on cryptographic incompetence, or (2) they knew exactly what they were doing, and can’t be trusted because they intentionally put a backdoor in the algorithm. Your choice.

Someone really needs to publish a rebuttal to what is probably NSA backed propaganda in a published journal.

I have never heard of any proven weakness in a cryptographic algorithm that’s linked to NSA; just innuendo.This could be true, with less subtlety than the “100% versus all” (100% is less than all by something of measure 0). The author may have only read about proven weaknesses, not heard of them, and so on.

Peter, actually, giving Fields medals to two would make sense since it would take the pressure off one of them being the first female Fields medalist. There are female candidates, like Mirzakhani or Serfaty, who definitely deserve it as much as anybody else.

Any news of who will get the Fields next week?