From a comment by Tamas Gabal:

“I also agree that many 'applied' areas of mathematics do not have famous open problems, unlike 'pure' areas. In 'applied' areas it is more difficult to make bold conjectures, because the questions are often imprecise. They are trying to explain certain phenomena and most efforts are devoted to incremental improvements of algorithms, estimates, etc.”

The obsession of modern pure mathematicians with famous problems is not quite healthy. The proper role of such problems is to serve as a testing ground for new ideas, concepts, and theories. The reasons for this obsession appear to be purely social and geopolitical. The mathematical Olympiads turned in a sort of professional sport, where the winner increases the prestige of their country. Fields medals, Clay’s millions, zillions of other prizes increase the social role of problem solving. The reason is obvious: a solution of a long standing problem is clearly an achievement. In contrast, a new theory may prove its significance in ten year (and this will disqualify its author for the Fields medal), but may prove this only after 50 years or even more, like Grassmann’s theory. By the way, this is the main difficulity in evaluating J. Lurie's work.

Poincaré wrote that problems with a “yes/no” answer are not really interesting. The vague problems of the type of explaining certain phenomena are the most interesting ones and most likely to lead to some genuinely new mathematics. In contrast with applied mathematics, an incremental progress is rare in the pure mathematics, and is not valued much. I am aware that many analysts will object (say, T. Tao in his initial incarnation as an expert in harmonic analysis), and may say that replacing 15/16 by 16/17 in some estimate (the fractions are invented by me on the spot) is a huge progress comparable with solving one of the Clay problems. Still, I hold a different opinion. With these fractions the goal is certainly to get the constant 1, and no matter how close to 1 you will get, you will still need a radically new idea to get 1.

It is interesting to note that mathematicians who selected the Clay problems were aware of the fact that “yes/no” answer is not always the desired one. They included into description of prize a clause to the effect that a counterexample (a “no” answer) for a conjecture included in the list does not automatically qualifies for the prize. The conjectures are such that a “yes” answer always qualifies, but a “no” answer is interesting only if it really clarifies the situation.

Next post: Graduate level textbooks I.

I'm a complete outsider, but it seems that Lurie has been involved in solutions of known problems too: (1) with Gaitsgory he has solved a conjecture of Weil-Tamagawa, there's a talk online http://www.cornell.edu/video/jacob-lurie-the-siegel-mass-formula and material from a course here http://www.math.harvard.edu/~lurie/283.html (2) Gaitsgory has posted several preprints in 2013, including one where he writes "All of these ideas became available as a result of bringing the machinery of derived algebraic

ReplyDeletegeometry and higher category theory to the paradigm of Geometric Langlands. We learned about these subjects from J.Lurie."

So all this would maybe tend to show that Lurie's ideas already have an impact beyond its original setting.

Dear curiousnonymous,

ReplyDeleteThanks for the info.

Unfortunately, I am unable to wath video recordings of mathematical talks; I am bored to the death after 10 minutes. At the same time I am able to follow the same talk if I am physically present, and I am able to work with the written down expositions too. This is a little mysterious.

But you point out some texts. I will look at them. "Weil-Tamagava" sounds impressive, but which conjecture? I am not sure that only one has this name, and, probably, one or two were proved long ago. Actually, I noticed this talk about Siegel mass formula. The Siegel mass formula is something wich I am very curious about. Several years ago I was searching for an accessible and sufficiently advanced exposition, but failed to find anything suitable. It seems that the best expositions are papers and lecture notes of Siegel himself, which are not easy to read for a non-expert 60 years after they are written, and a book of M. Kneser. But that book is in German. I can read at least the mathematical German, but this is much harder than to read in English. Since the topic is hardly related to my own research, I don't have enough motivations to read a book in German about it (may be enough for not very long paper). May be you know a good reference; I would appreciate any help a lot.

Perhaps, J. Lurie already has enough applications. On the other hand, I would be even more happy if the Fields medal committee would be able to award a medal without any famous problem solved.

By the way, Geometric Langlands is an example of an area more or less closed to outsiders. I attempted to read an exposition by E. Frenkel in the Bull. AMS. It begins very friendly. A few pages later the reader expect to know the class field theory at the level of A. Weil book. Apparently, the paper is readable only if the reader knows almost everything except the Geometric Langlands. I suspect that the set of such readers is empty. At the same time Geometric Langlands seems to be an accessible substitute for the real problem: the original Langlands program.

Lurie and Gaitsgory today at the Gelfand Conference at MIT gave back-to-back talks in which they presented their proof of the function field analogue of Weil's conjecture on the Tamagawa number of a simply connected, semisimple group. Probably these notes explain it: http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf

ReplyDeleteI see those notes were already linked to. Well perhaps this is the paper you are looking for then: http://arxiv.org/pdf/1108.1741.pdf

ReplyDeleteDear Sowa,

ReplyDeleteIf you decide to make a list of your favorite textbooks, it would be great if you could also suggest sequences of books that take a student from the introduction to a subject all the way to current research.

Dear shopkins,

ReplyDeleteThanks for the references. I will look at them, but right now I do not have time. The first weeks of the term are always quite hectic at our university.

The participation of Gaitsgory, in fact, scares me. As far as I know, he belongs to or at least was educated within the Gelfand school. People from the Gelfand school are almost never wrote for outsiders, especially if the representation theory is involved.

Dear Tamas Gabal,

ReplyDeleteActually, I did compile a list of books. But it turned out to be fairly long, and right now I am at loss: I have no idea how made it useful. As is, it is just a part of my biography. I followed the following criteria. To be included, a book should in one of few other personal lists, I should be meaningfully familiar with it (if I read it not from cover to cover, then I at least read some non-introductory chapters), and I should like it a lot.

This list cannot serve as list of recommended books to study mathematics or any branch of it. (Surprisingly even for myself, this list contains very few books related to my own work.) And these books are usually too advanced to serve as a pleasant bedtime reading.

If you can be more specific, then, may be, I will be able to turn it into something more useful. Alternatively, if you believe that even this list is interesting, I can post it in some form (a pdf file, may be).

Dear Sowa,

ReplyDeleteYes, I would be interested to see your list even without specific recommendations.

Me too!

ReplyDeleteDear Tamas Gabal and Ravi and all,

ReplyDeleteThanks for the interest! I will look into what can be done without writing reviews of all these books (and may others too: some are missing not without a reason). Right now the list is practically random, being a juxtapostion of few lists ordered alphabetically.

Dear Sowa,

ReplyDeleteWhy do you stop update your bolg? Although I'm not a mathematics student , I'm quite interesting in your blog and benefit from it a lot.

Dear Owl,

ReplyDeleteI would also be interested in your list of books. (And in your ideas of books to write!)

Best wishes,

Marius Kempe

To wxnfifth: it looks like my teaching responsibilities this term were consuming all the energy I have. I did not expected this, I expected that the course assigned to me will be rather easy and pleasant to teach. I was wrong. There are other reasons; explaining them will lead me partially into politics, partially into non-mathematical aspects of my life. Neither is suitable for this blog.

ReplyDeleteTo all:

ReplyDeleteMy teaching responsibilities for this term will end soon. I am already back here. Right now I hope that I will be able to post something about books soon. At the very least, the first post about books is ready; but I would like to go over my lists over again before publishing it.

At the same time, I am not quite sure what direction this blog should take. I have no desire to devote a significant portion of my life to writing a commentary to the Gowers's output. A couple of months ago he posted an amusing paper about computers proving theorems. There are a lot of things to comment on in this paper. On the other hand, I feel like I already said most of what I can say. The idea of writing a manifesto outlining my view of mathematics naturally suggests itself (it will replace line by line comments to Gowers), but I am not really a fan of this literary form of expression.

Anhow, the discussion here already had overgrown the framework of discussing Gowers's ideas.