A few mathematics items:

- David Ben-Zvi’s overview talk about
*Representation Theory as Gauge Theory*given last month at the Clay conference in Oxford that I attended is now available online, as slides and video. Other talks from the conference are here. - My fantasy that I might try and understand arithmetic algebraic geometry by reading Tate’s collected papers keeps getting delayed as the AMS puts off publication (now scheduled for January 18 of next year). While the books are not available, at least Milne’s review is.
- A couple weeks ago there was a Beyond Endoscopy conference at the IAS, at the same time I gather functioning as an 80th birthday celebration for Langlands. There’s a write-up by Langlands of his talk here. I think it can be described as the current Langlands take on “Geometric Langlands”.
- No recent news I’m aware of concerning Mochizuki and the the abc conjecture, but Inference magazine has just published a long article by Ivan Fesenko giving his take on “Inter-universal Teichmuller Theory”.
- The Breakthrough Prize symposium this year is scheduled for December 5 at UCSF, so I guess that means the prizes will likely be announced and awards ceremony held December 4, if things go like in recent years. I have no idea who will get the $3 million math prize since it’s a relatively new prize and there is a whole world of accomplished mathematicians who would make good candidates. One can be pretty sure though who won’t get it, arguably the most accomplished young mathematician around, Peter Scholze (since he turned down the junior version last year).
I have a modest proposal for whoever is awarded the prize: if you’re financially pretty well set already, how about doing the math community a huge favor? Donate the money to your university to endow a faculty position, then use the influence and moral high ground this will buy you to try and convince the Breakthrough Prize people to make this a policy. In the future, the winner gets a $3 million check made out to their institution to endow a position in their name. Then they could even try again with Scholze and perhaps get him to accept.

At the same time, there will also be a $3 million physics award. For a while these things were going pretty uniformly to string theorists, then they turned around and started giving them to experimentalists. I have no idea what they’ll do this year.

Breakthrough Prize in Physics to the LIGO guys? Kip S. Thorne, Rainer Weiss and Ronald Drever? I thought they were a lock for the Nobel. Then again, Higgs and Englert had to wait a year after the CERN announcement.

David,

They already gave those three (and the rest of LIGO) a Special Breakthrough Prize earlier this year.

https://breakthroughprize.org/News/32

“Like many budding number theorists,” writes Milne, “Tate’s favourite theorem when young was Gauss’s law of quadratic reciprocity.”

John Tate shared with our undergraduate number theory class in the spring of 1977 (then designated Math 101) that as a teen he came across Quadratic Reciprocity and immediately closed the book to attempt a proof himself. For several months, Prof. Tate recalled, he persevered but without success.

Prof. Tate wistfully noted to the class that by contrast the young Gauss had developed four different proofs. He then passed around Disquisitiones Arithmeticae in Latin. John never spoke to us of his own enormous contributions to generalized reciprocity laws.

Good idea, Peter, to endow new positions. Let’s keep this suggestion alive! We’ve got money coming in at my state university to support a chair in honor of a deceased colleague. However, we would need permission to hire into an existing tenure line and there’s a hiring freeze due to state budget problems. There seems to be money around for other things. (I discreetly avoid giving the name of my institution, but you could probably track me down.)

Peter, you may be interested in the recent arxiv posting by Kevin Costello, with what I think is the first mathematically rigorous construction of a particular type of (perturbative) M-theory. http://arxiv.org/abs/1610.04144

I would not be surprised if Maryna Viazovska — whose pioneering work* on sphere-packing in dimensions 8 and 24 (The latter being a collaborative effort.) has caused something of a sensation — is the recipient of the $3-million Breakthrough Prize.

Her status as a potential future Fields Medalist ought also not to go unremarked-upon….

* Of which I understand only the occasional punctuation mark.

I would guess Yitang Zhang for his work on bounded gaps between primes.

I’m not sure if some people, Scholze for example, would like to endow a professorship in ‘their name’.

anon,

Yes, a better idea would be for the winner to choose the name of the chair, perhaps with a default suggestion that the name would become that of the winner at their retirement. It would also be fine to just name these things “Breakthrough Prize” or “Milner-Zuckerberg” chairs. The suggestion of using the winner’s name is just that the main idea behind these prizes is to get more public recognition of the prize winners.

P.S. Thanks much for the link to the article by Ivan the Arithmetical. (I can hardly wait for the English translation.*)

* A remark that should * not * be taken as a dig at Fesenko: rather, it is my way of expressing my own bewilderment in the face of ideas whose depth, subtlety, and complexity will likely render them impregnable to any and all efforts on my part to understand them!

Peter,

I am not sure that doing anything more than making the endowment an option would work. Not all universities are run by honest people. The winner may choose to do something better with the money, like a school or charity in another country.

I don’t know if this is off-topic. I wonder what the intended audience of Inference magazine is. As someone who is moderately educated in math, I find the article by Ivan Fesenko very hard to follow when he starts talking about class field theory. Likewise, another article in the same issue about linguistics is almost incomprehensible to me as a lay person to the field. I suppose the magazine is not for a “broad audience”?

Michael Weiss: Your story of Tate and quadratic reciprocity reminds me of another. Tate learned about the K2 of fields from Hyman Bass when both were in Paris in Fall 1968. Bass asked Tate if he could compute K2(Q). According to legend, Tate was working on the question in the presence of Bass late one night when what he was doing reminded him of something in one of Gauss’s proofs of quadratic reciprocity. The two went out and found a copy of Gauss’s Disquisitiones in a second-hand book store, and sure enough Tate was able to find in Gauss’s work what he needed to complete the calculation.

I am an educated non-specialist in number theory (a physicist). Fesenko’s paper claims that IUT “settles the abc-conjecture”. Is Fesenko claiming that in fact Mochizuki has a proof of abc? Or is he just serving as a reporter and repeating Mochizuki’s claim? I believe that the consensus at present is that one should not claim that Mochizuki has a proof. Is that wrong?

I like the idea of the endowed chair, but I think arguments would erupt over where that endowed chair should be. Would the endowed chair be at the university where the recipient of the award currently resides, or at the university where the recipient did the bulk of his/her award-winning work?

I’m not sure how things tend to go in math, but in physics those are often not the same.

Jim Given,

The current consensus among experts is that Mochizuki has not yet produced an acceptable proof, since no one seems to be able to explain his proof to others in a convincing way (Fesenko’s article for example does not do this). Many experts tell me they think it is plausible that Mochizuki does have a proof, but the way he has written it up means experts cannot follow it or check it. Things seem to be at a standstill, with progress possible only if Mochizuki writes a less problematic version of the proof that others can follow, or someone else manages to see their way through what he has written and explain it to others.

G.S.,

I think questions about the details of the chair could be left up to the winner. The important point would be that the funds in the long term support mathematics research by endowing support for a permanent position for a mathematics researcher, rather than ending up as personal assets of the winner.

This would most affect winners with financial problems who need the money. But winning the prize would automatically increase their value in the job market and likely the salary they could command. If the terms of the endowed position are up to them, this would also put them in a very strong negotiating position with whatever institution they work for, or any interested in hiring them (they could bring the endowment with them).

So would it be ok for them to occupy their own endowed chair?

ronab,

Sure, that might even be the default arrangement.

Don’t quit yet on arithmetic algebraic geometric. My minimal suggestion is: start (and keep on with) with Silverman+Tate’s “Rational Points on Elliptic Curves”, and whenever you feel bored switch to Hindry+Silverman’s “Diophantine Geometry: An Introduction”. (Checkout Silverman’s W³ pages on both books.) There are many other options but my suggestion was minimal.

Peter, if the Breakthrough Prize Foundation informed you that in recognition of some outstanding results you had achieved, they wanted to award a large prize

to someone elsein your honor, wouldn’t you feel that a nasty joke was being played on you?Chris Austin,

Exactly my point is that such a prize should not go to a person, but to an institution for support of mathematical research. At this point, at least in the US, I think the best way to do that would be to endow permanent positions at research universities. If the Breakthrough Prize people want to endow such a position somewhere in my name, that would be great. I’d even consider dressing up in a monkey suit and attending their ceremony (by the way, I’m free Dec. 4 and was even thinking about a trip to the Bay Area around then, so they just have to let me know where and when to show up).

Better though would be if they endow the position here at Columbia, because then the administration and my colleagues would treat me even better than they do now. I might finally get a new carpet and paint job for my office…

PedroJVM,

Thanks! That looks like excellent advice.