Cédric Villani is in town today, giving a talk at the French consulate. He’ll discuss his book, recently translated into English (I wrote a bit about it here). Yesterday, despite the lack of suitable bread and cheese, he was in Princeton, where he gave a public lecture at the IAS. The New Yorker has a story about him by Thomas Lin, entitled The Lady Gaga of French Mathematicians Comes Stateside.
If you’re not listening to Villani tonight, you could be watching a PBS Nova program on mathematics, The Great Math Mystery. Among the mathematicians interviewed will be my colleague Dusa McDuff. As for the question on the PBS site:
Is math a human invention or the discovery of the language of the universe?
the answer is the latter.
What some mathematicians might consider the “Great Math Mystery” is whether Mochizuki really has a proof of the abc conjecture. There finally will be the topic of a workshop involving experts in the field, to be held this December in Oxford. Still no paper from Go Yamashita about this, but here you can find some photographs of the boards from his talks in Kyoto last month. Mochizuki himself has a new paper, inspired by conversations with Fesenko.
Also in New York this week, Bjorn Poonen will be speaking on Thursday. His topic is a heuristic argument that there is a finite bound on the rank of elliptic curves. For notes from a talk of his about this last year, see here.
Update: The Villani IAS talk is available here.
Update: At David Mumford’s blog he has a long and very interesting posting about the state of mathematical research publishing.
Update: One more piece of math news. Dan Rockmore has set up a public version of his Concinnitas Project, which lets people post, with explanation, a picture of their choice of a “most beautiful mathematical expression”. See here for details.
Will the Villani IAS lecture be posted on their website afterwards?
Fesenko’s impressive survey of IUT,
https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf
cited in the last line of the “here” link above, ends with this pointed and invaluable remark:
An opinion of R. Langlands on current trends about supporting long-term fundamental research work can be heard during the 52nd minute of his video lecture [20].
Some roots of the decline of support to long-term fundamental work, such as the shortsighted race to higher number of publications and higher citation index, which often results in pressure to produce short-term work that consists essentially of minor improvements to known results, originate from causes external to the mathematical community. To do well in their academic career, young researchers are very often pushed to go along this path which typically implies a very narrow specialisation. The latter leads to the emphasise on technical perfection as opposite to innovation and on presentation rather than the substance of work. Following this path eventually makes it more difficult to think broader, to learn new areas, to develop in more directions. Lack of inventiveness, more widely spread imitation, fear to stand alone in scientific endeavour, fear to look too far away are associated issues. Some roots, such as the unnecessarily strong emphasis on concrete applications, originate from within the mathematical community. Changes are needed.
There is an issue about attitudes of number theorists towards the study of IUT and their unusually sluggish response. Reasons for this are related to the topics discussed in the third paragraph of 3.3 and in the previous paragraph. It seems that the number theory community is suffering from the problems listed there even more than other mathematical communities.
Never mind, Villani’s talk has already been posted:
https://video.ias.edu/villani-publiclecture-2015
PointedRemark,
I think Langland’s point is excellent and quite important, but I’m not convinced it’s relevant to the IUT/Mochizuki story. Here the question is not why people aren’t being inventive and developing their own long-term research projects, but why they’re not signing on to Mochizuki’s, which is different. This has a very high profile in the math research community, everyone in the field knows about it and has looked at it. My interpretation of what has happened is that a lot of experts have just decided that this claimed proof is not yet in a state where it can be evaluated by the usual methods and the methods used are not obviously convincing enough to be worth the time needed to try and figure out how to use them. Given this, it’s quite reasonable to decide to not spend one’s time on IUT, but to instead work on ones own ideas and wait until either Mochizuki/Yamashita/someone else puts out a comprehensible version of the arguments. Mochizuki’s own reports on what is being done to check the proof aren’t inspiring confidence.
I think sooner or later though, enough people will put enough time into this for it to become clear what the value of the ideas is. Some day we’ll know whether or not there’s really a proof of abc and powerful new methods there, but in the meantime I don’t think people can be criticized for not finding this convincing and not wanting to abandon work on their own ideas in favor of working on Mochizuki’s.
Hi Peter,
“Is math a human invention or the discovery of the language of the universe? [T]he answer is the latter.”
Risking that this might be off-topic, I nevertheless need to ask something about this. It seems that you have a very strong opinion, and I am curious why? For example, how do you rate, say, the axiom of choice — is it a part of the language of the universe, or not? Note that if you say “yes” I’ll ask you further about the Banach-Tarski paradox (which seems anything but to hold in our universe), while if you say “no” I’ll ask you about the lack of power of the ZF without C, and the fact that most mathematicians today prefer ZFC over ZF, despite it not being the language of the universe (by assumption).
Or as another example, take the distributive law (of propositional logic) and note that it fails in the context of quantum mechanics (as discussed, say, in the Wikipedia article on quantum logic). If math is the discovered language of the universe, does distributive law hold or not? If not, how many theorems in standard math would go down the drain because their proofs rely on it?
There is also the question regarding Brouwer’s intuitionistic logic versus Hilbert’s formalist logic (discussed here) — which one is the correct language of the universe? I as suspicious that it can’t be both, so…
It seems to me that these examples suggest that math (being mostly built on top of some particular logic and some paricular set theory) is a language that sometimes nicely describes the real world, and sometimes less so, depending on the adopted axioms. This makes it more a human invention than a discovery of some property of nature, since otherwise issues like Brouwer-Hilbert controversy could be resolved experimentally. I am really curious to hear how do you deal with these issues, from your POV.
Best, 🙂
Marko
Marko,
Sorry, this really is off-topic. In my essay I tried to explain my point of view. The question is a completely ill-defined one, and people can use it for inspiration to discuss all sorts of topics. Questions about the fundamentals of mathematics (or physics) in terms of logic happen to be ones that have always left me cold, so I’ve neither the knowledge nor the interest to carry on my half of an intelligible debate about the issues you raise.
Last month, The Observer (which is the Sunday version of The Guardian) had a 2000-word article “Cédric Villani: ‘Mathematics is about progress and adventure and emotion’ “. The article focuses on Villani’s book Birth of a Theorem.
Speaking of the research climate for fundamental mathematics, what is the research climate like for pure mathematics in general? Obviously they’ve never needed huge grants ( in general anyway) but I’ve always wondered how hard it is to become a research professor in a subject* which is so controversial.
*I do realize this probably varies from subtopic to subtopic within pure mathematics.
gadfly,
I don’t think pure math is controversial, quite the opposite. Few people are interested or care one way or the other about it.
Academic math departments have a mix of pure and applied, and the pure component is in some sense mostly funded by the teaching mission: there are lots of students out there who need to learn calculus, etc. This is what most pure mathematicians are being mostly funded to do. The number of purely research positions, e.g. at research institutes, is a small part of the overall number of positions.
Like most of academia and the rest of society, there’s been a trend to a star system where some people do very well, most people not so much. So, people with permanent positions at major research universities are doing quite well and have good environments for their research, but such positions are hard to get (and, like Langlands one might worry this makes people risk-averse). Not so hard to get some kind of position, but often these are poorly paid adjunct positions, in which carrying on research can be quite difficult.
Peter and gadfly,
As an academic mathematician I’ll put my two cents in. Peter has it basically right. Math (pure or applied) is significantly easier than physics, in that lots of students need calculus, and not many need basic physics. At my university, we have 32 full time mathematicians (this includes math ed and statistics, if you take just math folks it’s 16). There are 5 physics professors. That said, getting an academic position in math is hard. When we have an opening, we’ll get 3 or 4 hundred applicants, most over qualified. I’m not at a research one school, you’re expected to do research but you teach 3 classes a semester. Our last hire has a Ph.D. from Cornell and a research postdoc at Rochester, and he was happy to get the job. Most of our adjuncts are honestly not Ph.D. mathematicians, that end of the job market actually gets covered in large part by jobs at community colleges and such, brutal, but at least with job security and vaguely reasonable pay. Not that there aren’t some Ph.D. adjuncts in math, just nothing as bad as physics, or art. And math, like physics, has an out, you can go work on wall street and make 10 times what I do.
I must be missing something, but I would very much like to know why Cédric Villani is called “the lady gaga of french mathematicians”.
Thanks.
EFT,
Probably like most people, the little I know about Lady Gaga is that she is a celebrity who favors flamboyant dress. I’d guess that’s the analogy in someone’s mind that the article’s author is referring to.
“a heuristic argument that there is a finite bound on the rank of elliptic curves”
Oh my. Is he gunning for the Birch and Swinnerton-Dyer Conjecture?
Zimriel,
I don’t think Poonen’s heuristic argument really addresses Birch Swinnerton-Dyer. Even if you prove no elliptic curves of arbitrarily high rank, you still have to prove BSD for curves up to some finite rank, and that’s completely open.
@Zimriel,
there are still infinitely many curves of rank greater than, say 18, (and conjecturally, only finitely many above 22 or so) so there’s no hope of just enumerating the rank greater than 1 curves, and using Bhargava-style results on density (100% curves blah blah) to get BS-D.
Columbia was certainly well-represented on the Nova program (I counted 3 reps), even if Dr. Tegmark got the bulk of the airtime. Does someone there know somebody at PBS?
Flashy graphics and hip engineers didn’t rescue this program from boredom, imho. One bet they missed: instead of rhapsodizing about the universality of gravity, explain that most folks are currently doubling down on the theory, invoking unobserved dark matter to explain discrepant galactic motions, in direct analogy to the Uranus orbit discrepancy leading to the discovery of Neptune. (Rather fewer are betting that the correct analogy is to the Mercury orbit discrepancy.)
EFT, the New Yorker article just says “Villani has been called the Lady Gaga of French mathematicians” which it probably got from one of the many earlier articles that says the same thing. However, a bit of Googling indicates that it comes from Villani himself in an interview with Télérama in 2011, where he comments on the attention in French media after he won the Fields metal; a lot of focus on his clothing style, little on what he actually did or on fellow French winner Ngô Bảo Châu: “Je suis un peu la Lady Gaga des mathématiques.”
Peter & Obs,
Thank you very much for the clarifications.
I finally watched the nova program. The Tegmark section caused me to laugh out loud. The point where he implied that nature was contained in its entirety in the standard model and that this somehow furnished his argument that math and nature are one and the same reconfirmed my view of him as a nigh unparalleled munchkin.
It is obvious that the standard model doesn’t actually describe the vast majority of reality… right?
gadfly,
I haven’t watched the Nova program yet (it’s on the DVR…) but while I disagree with Tegmark about a lot of things, it sounds like this isn’t one of them. The Standard Model + GR is an amazingly successful and comprehensive fundamental theory, and the connections between these theories and mathematics are very deep.
There is a big difference though between being a successful fundamental theory and “describing reality”. A fundamental theory is just not relevant for understanding most of science, not to mention things besides science.
Peter,
Intriguing. The SM+GR does indeed have a far more mathematical character than say, the statistical mechanics of classical liquids. But I have always heard that GR and QFT contain a relatively small amount of pure mathematics; a dash of continuous group theory in the latter and a bit more than a dash of differential geometry in the former. Is this true?
What do you make of the alternative viewpoint that more exotic mathematics is a result of poor intuition? We have a poor intuition of subatomic, high energy systems, and a poor intuition of black holes and cosmological scales, so we need a more subtle and complex language to describe them; in some regards, it is a form of packaging ignorance. You can find sophisticated mathematics elsewhere in physics where there is a breakdown of intuition; people have tried to use topological techniques to study the configuration spaces of biomolecular systems, for instance.
gadfly,
Speaking of reliance on intuition versus mathematical formalization, see this fortuitous post from Steve Hsu.
One can argue that the breakdown of intuition is inevitable; science can’t help but advance into realms where intuition is not helpful, and worse yet, can never really be developed. Extreme combinatorial complexity might be one such realm. In such realms the human mind needs powerful aids.
gadfly,
No, it’s simply not true at all. SM + GR contain a huge amount of deep, non-trivial mathematics.
Mathematics is a language that can be of great use in situations where our intuition fails us, but typically it is not sophisticated mathematics that is of use, but more basic mathematical ideas. What’s amazing about SM + GR is that deep mathematical ideas are precisely what is needed to state the theory.
Peter, I learned of your site after finding John Baez’s site (or series of postings), around 2003-4. I was pretty down on “string theory” at the time, having been unconvinced by the semi-popular version that all particles are some kind of violin strings. So your site, and the books by you and Smolin were refreshing. (I am less down on ST now, despite the untestable part, but at least things have moved beyond the guitar and violin metaphors. Also, ST no longer seems to be the 800-lb gorilla. Perhaps I am wrong in terms of who is being hired, but certainly the shine has come off it a bit in the past 15 years. The AdS-QFT thing is really intriguing, and two of the most interesting talks I’ve been to at Stanford have been by Lenny Susskind!)
It seems to me that the category/topos and representation theory aspects of physics are central, though not always necessary for “shut up and calculate” work, and that these involve some of the deepest aspects of work done by Grothendieck, Serre, Mac Lane, and others. (Maybe my connection to computers and programming theory have influenced my interest in type theory, category theory, and “topology via logic” points of view, which are along the lines of “algebraic geometry” a la Grothendieck, Lawvere, Abramsky, and others.)
I’m just a retired physicist who did some fun work for Intel in the 70s and 80s and who took Jim Hartle’s GR class in 1973. (Wish I’d learned more…seems black hole horizons are a lot more confusing than I seemed to think back then! As the current slogan goes, it’s not the singularity at the center that is interesting, but the event horizon.)
BTW, some of the most invigorating stuff back then was in analysis and point set topology (mostly class notes, but Kelly and Dugundji were often used). Utterly unconnected with physics, to me at that time, but I keep coming back to it again and again. And it of course relates to quantum theory in interesting ways–lattices, von Neumann algebras, even intuitionistic logic.
Thank you for your site.
–Tim May, California
LadyGaga is famous for her outrageous costumes, esp her shoes, but she seems to have tamed it down some now that she is engaged/married. She does have a very good singing voice and is trained.
Her greatest accomplishment is getting everyone to call her ‘Lady Gaga’ and be known across the globe.