Every year in Tucson the Arizona Winter School takes place, with a five day program on some topic in arithmetic geometry aimed mainly at advanced graduate students, designed to get them involved in current research-level topics. This year’s topic (Perfectoid Spaces) is drawing a huge number of people there next month, with about 450 participants expected (in the past numbers were more like 100). This should be a veritable Woodstock of arithmetic geometry, with no one I’ve talked to quite able to figure this out, thinking that there probably weren’t 450 people worldwide interested at all in arithmetic geometry. It seems everyone in the field will be there and then some.

Peter Scholze is the opening and closing act. The other lecturers who will take the stage have started to put lecture notes for their lectures on the school website.

Some are dubious that there really are 400 or so students in the world with the background necessary to understand this material. See for example MathOverflow where nfdc23 isn’t very encouraging to a student who doesn’t know any rigid analytic geometry, but plans to attend the AWS. In any case, I hear Tucson is quite nice in March.

At some kind of other end of the spectrum of such things, a couple months later experts will gather in Germany to discuss this field (see here). Also for about five days, at the Schloss Elmau Luxury Spa and Cultural Hideaway, the sort of place heads of state go for G7 meetings. Rooms run $600 a night or so, but in this case the tab is being picked up by the Simons Foundation. Sorry, by invitation only.

The end result of this, of course, is that grad students pick research topics based almost completely on perceived status rather than technical merit. By the time they have the background, they’ve devoted 5 years of their life specializing in a subfield, and they are too wedded to the subfield to assess it objectively. This generational inertia lets fads persist longer than they would otherwise.

The put-down comment by nfdc23 at MathOverflow could perhaps be contrasted with the call, by Francis Su, for quite the opposite behaviour in the math community, compare To Live Your Best Life, Do Mathematics.

As we say in French “Le ridicule ne tue pas”, fortunately because otherwise that would be a real hecatomb these days in the US.

“Business casual clothing should be worn during the symposia.”

The first time I remember hearing about dress code for a scientific meeting.

“Business casual clothing should be worn during the symposia.”

That’s probably to prevent a real Woodstock to happen.

From what I can tell from the titles, there will be no discussion of Mochizuki’s inter-universal Teichmüller theory…

Business casual? Wow. That reminds me of the first real conference I went to, at UCLA, in ’91, when I was working on my thesis. My wife insisted I go out and buy nice clothes, I kept telling her “it’s a math conference, no one will dress up.” Suffice it to say I was better dressed than any of the professors.

Maybe the posh conference hotel requires some level of formality so that the other guests there are not traumatised 😉

Argh, I had not noticed that the dress code was for the Simons conference and not the winter school…thus people can dress like in Woodstock at the winter school.

More seriously, the theme of the Simons gathering is “p-adic Hodge theory” which encompasses a larger area than the one covered by perfectoid spaces (on the geometric side), typically arithmetic applications of p-adic Hodge theory like you will find in the proof of the STW conjecture, Serre’s conjecture and so on.

I think I’m missing some context for this story. Can someone explain (or link to an explanation) to a non-mathematician what the sudden appeal of Perfectoid Spaces?

Re: nfdc23, etc.

“Young man, in mathematics you don’t understand things. You just get used to them.”

– John von Neumann

Sometimes, not understanding the foundations allows one to skip to new ideas. There is a also a level of fear in every social group concerning outsider penetration.

“Sometimes, not understanding the foundations allows one to skip to new ideas. There is a also a level of fear in every social group concerning outsider penetration.”

Although there is some truth in this assertion, this is the typical justification all crackpots give when their proof of the Riemann hypothesis is refused because it’s pretty clear they don’t know the basics of analytic number theory after reading a few lines (same with Fermat and Arithmetic Geometry and so on).

simplicio,

I’m in no position to explain what a “perfectoid space” is, but the context is that it’s a new idea about arithmetic geometry due to Peter Scholze that in recent years has been used to solve a range of problems in that subject. For an outline of such applications, one place is Scholze’s ICM talk

http://www.math.uni-bonn.de/people/scholze/ICM.pdf

As lf points out, the connection to the topic of the Simons workshop (p-adic Hodge theory) is just that there have been some applications of perfectoid spaces there. I’m guilty of putting the two topics close together because I couldn’t resist the amusement value of the contrast between them.

And I should make it clear that while I’m making a bit of fun of the AWS and Simons workshop, the “ridicule” is affectionate, with no intention of trying to kill off either, they’re valuable efforts (unlike when I make fun of some things going on in physics, which I really do wish could be killed off…)

@John Fredsted:

The comment I made on Math Overflow was not at all intended as a put-down, but rather as a reality check to save a person from wasting a lot of time struggling to understand papers and references on a subject when they have not yet learned much more basic things that every author of the more advanced works take as known by their readers. I have seen too many instances of students trying to leap ahead to “study” very advanced topics in mathematics without first learning something serious about the more basic contexts out of which the advanced ideas arose. There’s certainly nothing wrong with skipping some steps here and there when seeking to get a sense of what a subject is about, but this is not the situation that the person was describing when mentioning that they didn’t know about rigid-analytic geometry.

There are nowadays quite a number of references from which one can learn about rigid-analytic geometry and Berkovich spaces, either of which provide good preliminary experience to then try to get a feeling for the more sophisticated structures involved with creating and especially using perfectoid spaces. My point was that one absolutely have to learn something serious about at least one of those areas before making an attempt at learning about perfectoid spaces (let alone to attend a 5-day workshop on the topic). Anyone who doesn’t have some reasonable familiarity with either of those frameworks for doing non-archimedean geometry has no chance to understand anything serious with perfectoid spaces that will be lectured about at the AWS.

It is akin to trying to read advanced textbooks and papers on General Relativity without prior study of undergraduate physics or multivariable calculus. When I was an undergraduate, after taking a graduate course in functional analysis (which I knew also provided the mathematical framework for quantum mechanics) I decided to try to sit in the back of the room in an undergraduate course in QM in the hope that I could learn the subject despite that I had never studied serious classical mechanics (just the simple freshman-level E&M + SR). Needless to say, my experiment was a flop.

I am not claiming that one has to go through all prior historical developments before entering a new subject, or that reading backwards can’t be productive (within reason: one ought to have some knowledge of the basics). But if one is really expecting to get something out of a 5-day instructional workshop (not a semester-long course, just 5 days!) then one absolutely must learn the basics first and not rush to the seminal papers on the very advanced stuff first. That is why I said “Life is not a race”.

@Tom Andersen:

I don’t know if your background is in math or physics (or both), but in pure math having no understanding of the foundations of a subject (by which I mean no understanding of the more basic relevant mathematical ideas and how they are used in proofs and/or important examples, not about set-theoretic/logical foundations) makes it impossible to extract any real understanding about new ideas in more advanced areas of the subject. It may be that in physics (which I don’t know much about at a technical level) one really can “skip to new ideas” in a meaningful way in an advanced topic without understanding the foundations, but in theoretical math this is basically impossible (due to the nature of the subject). The issue is not about fear of outsider penetration, but about the way the human mind digests abstract concepts.

Perhaps the example par excellence of someone who made substantial contributions to cutting-edge pure math without a systematic prior study of earlier ideas was Ramanujan. But this is the exception that proves the rule: he rediscovered on his own everything he needed from the earlier ideas of others, and his mastery of examples was singularly spectacular. So even his case fits the paradigm of how one reaches the stage of making creative new contributions to a highly-developed subject in pure math (though the road he took was one that nobody else could have traveled).

In many (though not all) topics at the cutting edge of arithmetic geometry nowadays, it is more often structural rather numerical examples that guide the development of new ideas (that is certainly the case with perfectoid spaces and how they are used), so one really needs some framework to develop intuition for organizing these sophisticated concepts in one’s mind. Trying to proceed in such directions without an understanding of (relevant) foundations is not going to put one in a position to make creative use of those ideas. In fact, if the “insiders” in this area really did want to prevent outsider penetration (in terms of creative new ideas from “outsiders”), encouraging the study of the advanced work without knowing the more basic ideas out of which it emerged would be an excellent strategy. 🙂

@lf: in Tucson in March, dressing like Woodstock is encouraged!

@ds: indeed, I should have guessed it, but I “prefer” the spa and the swimming pool of the Schloss Emmau…

@nfdc23: Ok, point taken. But rereading your comment on MathOverflow, I still think that your message could have been conveyed in a more empathetic way. At least, if I had been the poster, I would have been put down by the very phrasing of it. But perhaps I am just being (too) sensitive.

Clarification: By ‘the poster’, I mean of course ‘the original poster’, i.e., the recipient.

I might not be the right person to give career advise but entering the nesting ground

of giants (especially if these monsters are so young) can mean living a very competitive life. There certainly will be tears. You are the tortois running after Archilles. Once you are where Archilles was, the giants have already gone much further. Having been close to giants I know the experience: it is exciting but also humbling. It is maybe the competitive career market but both in physics as in mathematics, the herd instinct is currently particularly strong. Fortunately, both in mathematics and physics, there are many nesting grounds and some are less crowded. But the one covered in this conference seems particularly grand. Maybe its just that most of the crowd will be tourists who want to have a glimpse at the superstars in that field which rapidly appears to become classical mathematics. And like Woodstock, also math has its legendary moments. The ICM’s of 1900 or ICM 1950 (http://abel.harvard.edu/history/icm1950) must have been such, or the Woods hole conference from 1964, at a time when the index theorem was exploding. I attended once a conference /summer school at Hillerod, where lots of giants were present http://abel.math.harvard.edu/~knill/history/hillerod Fantastic? Yes. But also a bit intimidating. I myself enjoyed once a summer school in les Houches, http://www.math.harvard.edu/~knill/various/chamonix , where my wife participated and I actually enjoyed the tourist status very much. Now a bit older, I only can give the advise, go and enjoy as much as you can but also know about other nesting grounds where you can find one or the other leafs not yet trampled upon. Its also good to be aware of one important fact which applies to almost everybody: there are always mathematicians/physicists who are (much) better than you!

Peter, OT. A colloquium by John Ellis at PI on where is particle physics going

http://pirsa.org/displayFlash.php?id=17020013

Maybe the MAA or AMS or the meeting organizers, seeing what is occurring, could also organize a few informal seminars, study groups, etc. at a more introductory level in conjunction with the main meeting for those who are not really prepared to understand much of what is being presented by the experts. This might turn a potentially negative event into something more enjoyable and valuable for all.

@sigoldberg1: Indeed the AWS does precisely that sort of thing, every year. AWS has been around for two decades and is a rather mature entity by now, with many people putting a lot of care into making it work well for as broad a range of students as possible.