Robert Langlands: The Greatest Mathematician You’ve Never Heard Of? | Hacker News Reader

Robert Langlands: The Greatest Mathematician You’ve Never Heard Of?

thewalrus.ca

124 pointsColinWright73 comments

If you haven't heard of Langlands but have heard of Andrew Wiles and his proof of Fermat's Last Theorem, you may be interested to learn that the two are somewhat closely related. Wiles actually proved a part of the modularity theorem, which had been shown by Frey, Serre, and Ribet to imply FLT in the 80s.

The modularity theorem is very much a Langlands-style theorem and could be seen as a more concrete version of many of the ideas and conjectures that form the Langlands program. The conjecture now known as the modularity theorem was formulated as early as the 50s and 60s by Taniyama and Shimura, thus predating the Langlands program, and it was taken seriously once Weil gave conceptual evidence for it (but did not come close to a proof).

In fact, the modularity theorem is just a very oddly phrased reciprocity law. General reciprocity laws often look astonishingly nothing like the simple law of quadratic reciprocity, or they require some clever squinting to see the relationship. Modularity gives you for any rational elliptic curve E a modular form which is a simultaneous eigenvector for the Hecke operators (one for each prime number p) and whose eigenvalues give the solution counts of the elliptic curve equation modulo p for various primes p. These eigenvalues are also the coefficients in the Fourier expansion of the modular form.

Oh, Robert Langlands!

I worked on a short film about him, for the Abel Prize ceremony, last March.

http://www.abelprize.no/artikkel/vis.html?tid=73176

This is a charming film! How did you decide to include things like that little bit about him paying for the lectures he had posters of? Or the scene where you were peaking in from outside (which I suppose is a metaphor for the craft of creating character films in the first place).

Nice short film. Langlands was a theory builder as opposed to someone like Erdos who was more interested in solving problems. Theory builders are often admired, but because the endeavor is so broad, very few of them emerge and even fewer are actually successful.

I like the part where he said he began to write before he understood everything, and in order to write he had to discover many things, and even had to discover them after he started to write.

It underscores the crucial role of writing in discovery. Most writers will tell you they are exploring the space during the writing process. Writing isn't a process of committing what you already know to paper; it's a process of learning what you don't know and or haven't considered. It often leads you down paths you would never expect. (this happens to me with my HN comments too -- I often myself writing a very different comment from the one I set out to write)

This is why I think a Ph.D. dissertation should be a continuously evolving collection of notes, and not something you "write-up" in the end after all the work is ostensibly done.

> as opposed

From The Rising Sea: Grothendieck on simplicity and generality by C. McLarty:

Grothendieck describes two styles in mathematics. If you think of a theorem to be proved as a nut to be opened, so as to reach “the nourishing flesh protected by the shell”, then the hammer and chisel principle is: “put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks—and you are satisfied”. He goes on to say: "I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!"

Incidentally, I have heard Serre's work described as the exemplar of the hammer-and-chisel approach. (McLarty goes on to say that Bourbaki's work fits in the rising-sea approach, which is surprising to me.)

That's also why I detested the formal, algorithmic writing method you learn in English classes. They always wanted you to turn in an outline before you had even begun your paper. How are you supposed to organize a paper before you even have anything to organize?

> Langlands was a theory builder as opposed to someone like Erdos who was more interested in solving problems.

No 'was' about it, by the way; he's still alive, and (last I heard from him) still working.

What is the background music you used for this video? Thanks

More like greatest mathematician whose work I’ve never understood. Sadly. At least I can reduce Grothendieck to algebraic topology or Wiles to number theory plus that one old problem. Terry Tao to analysis. But Langlands? Interconnectivity of all sub-fields? I’ve got nothing.

I highly recommend reading Edward Frenkel's memoir, Love and Math, which is more or less an up close and personal popular account of the author's involvement with the Langlands Program:

Perhaps the most remarkable part of the book though is the way it makes a serious attempt to tackle the problem of explaining one of the deepest sets of ideas in mathematics, those which go under the name of the “Langlands program”. These ideas have fascinated me for years, and much of what I have learned about them has come from reading some of Frenkel’s great expository articles on the subject. To anyone who wants to learn more about this subject, the best advice for how to proceed is to read the overview in “Love and Math” (which you likely won’t fully understand, but which will give you a general picture and glimpses of what is really going on), and then try reading some of his more technical surveys [...]

http://www.math.columbia.edu/~woit/wordpress/?p=6266

Thanks! I'll take a look.

He has a lecture series I've posted about a few times - https://video.ias.edu/The-Practice-of-Mathematics

Working mathematician here --

1. Langlands is indeed a great mathematician, whose work has been enormously influential.

2. Most of us aren't all that eccentric. He wrote his paper in Russian, a language which he (presumably) does not natively speak, apparently just for the heck of it? That's just weird.

Most of the mathematicians I know, including the most influential ones, are relatively normal people. And they want as many people to read their work as possible, and don't throw up artificial roadblocks. (Most working mathematicians cannot read or write Russian.)

3. There is not that big of a conflict between pure and applied mathematicians. (Except when they're competing for money, or trying to decide whom to hire in their departments.)

The most common attitude among pure mathematicians is: we work on mathematical questions for their own sake, and don't think too much about applications to the "real world" -- but we are happy if it is brought to our attention that such applications exist.

In the library here they have a book called "Russian for Mathematicians".

Maybe if you need to look up some soviet era literature?

The weird thing is that while modern-style mathematics with lots of display maths is supposedly to be less dependent on subtleties of text (as opposed to the arguments of old-style topology books like Munkres, which use display equations a lot less and pack a lot of oomph into dense paragraphs with inlined symbols). But mathematical terminology actually changes a lot -- it's enough to open a page in Wikipedia and switch languages to see.

A "field" in the algebraic sense (the real numbers are a "complete ordered field") in portuguese is corpo, but vector fields are campos. And both in Portuguese and French, manifolds and algebraic varieties are the same word (variedades/varietés).

Maybe if one starts graphing these separations and collisions across many languages some structure emerges -- obviously no one mixes up algebraic fields with physics fields, but the manifold/variety collision hints at some historical commonality that matters for expressive power at the level of a Grothendieck trying to say something sweeping about the entire landscape of mathematics.

To be fair, a ‘campo’ is also a ‘corpo’ that is commutative.

An observation that illustrates my point.

Can you elaborate? Vector spaces and fields differ in how they define multiplication. In the former, we multiply vectors with scalars while in the latter we multiply two scalars.

Also, vector fields are functions from points to vectors whereas field is an algebraic structure.

I'm not entirely sure what he meant, but the flow maps of vector fields are semigroups. If irreversible, they're groups. And a field is like a group and a ring over the same set, right?

A group is a set with a single operation defined on it that abides by certain axioms. A field is a set with two operations defined on it. But the field operations must abide by more axioms than a group operation. While vector spaces and fields are very similar (two operations, similar number of axioms), vector spaces are defined over fields (for example, an element cv is defined where v is a vector and c is a scalar) while fields are not defined over anything -- they are just a structure with two operations and a number of axioms (there's no element cv in a field, but it does have an element c_1 * c_2 where c_i is a scalar).

That said, vector field is a different object. It's a function. Likely named so by physics people while the structures like group/ring/field/etc were named by math folk.

I have no idea what flow maps of vector fields are, but if you give me their definition, it'd be trivial to check if they form a semigroup under a certain operation: we'll just check it for associativity.

To get a hang of this stuff I recommend the following books:

Book of Proof by Richard Hammack (tools of the trade)

Linear Algebra by Kuldeep Singh (rigorous tutorial: combines the rigor of a textbook and the ease of use of tutorial)

Abstract Algebra by the Dos Reis (rigorous tutorial)

Real Analysis by Lara Alcock (this books makes the rigorous definition of sequences trivial)

Real Analysis by Jay Cummings (contains much more info than the one above and is very similar in spirit)

Real Analysis by Rafi Grinberg (takes you from reals to Euclidean Spaces and Metric Spaces)

After that you ccan start reading intro level mathematical physics books to get an easy intro to differential geometry, manifolds and analysis in abstract spaces. Once you get an intuitive hang of this stuff, you can come back to the more brutal pure math setting.

Here, I like Modern Math Physics by Peter Szekeres. It's gentle and more about geometry and less about analysis.

Flow maps: take for example an autonomous ODE

    x'(t) = f(x(t))

f(.) describes a vector field, right? A flow map is a function w_t(u) = x(t) that solves the ODE with x(0) = u, for fixed t. If f(.) is invertible, then each flow map is a group in the very same way rotations of a Rubik cube are a group. If not, it's a semigroup, which is a group without an invertibility. The former describes systems that you can track back in time and calculate initial conditions only from looking at the present state.

My dissertation was actually about numerically integrating symplectic vector fields; I spent a lot of time hunching over Arnold's "Mathematical methods of classical mechanics".

What I meant was, in some languages 'corpo' is a more general notion of the (algebraic) field, the non-commutative version of which is known in English literature as the "skew field."

If I understand you correctly, you are saying vector fields and skew fields are the same object. But that's not true, though. The former is a function, the latter is a structure.

In other words: 'campo' = (commutative) field, 'corpo' = field (possibly skew); 'campo vectorial' is something else, of course.

According to Wikipedia: " Langlands likes to learn foreign languages, both for better understanding of foreign publications on his topic and just as a hobby. He speaks French, Turkish, German and Russian".

I'd say it's not _that_ weird, if he likes foreign languages, to combine both his passions and write one paper in Russian :) Also considering there are actually a lot of mathematicians who can read and write Russian.

A half-remembered, and possibly apocryphal, story which another mathematician once told me:

Langlands was once invited to lecture in France, and he chose to give his talk in French. Evidently his accent was not all that good, and the audience found it a bit painful.

Jean-Pierre Serre, one of the leading mathematicians of the 20th century, and a Frenchman, was attending the lecture that day. He interrupted to ask a question, in English.

Declining to take the hint, Langlands answered in French and carried on with his lecture.

Frenchmen are legendarily intolerant of badly-spoken French. This was a recurring theme in my couple of years as a student at Alliance Française - when in France (most people hoped to go to college or grad school there) if you're not solid don't even try.

In Brazil people will be glad you're trying and try to speak slowly in return.

Speaking bad French to French people helps them to suddenly remember that they also speak English.

German has a similar effect, too.

Yeah, this is my experience too. You should always attempt to use the language of where you are except in France. They would rather you spoke English than butchered their language. I usually keep it to please and thank you in french as a result. Although I did once use my high school french to ask where the swimming pool was once and had a bit of trouble deciphering the response I got (in french). So swimming pools and libraries and please and thank you :)

I think problem is, English accent is probably the worst. This is because pronunciation-wise there hardly are other two languages as different as English and French. (The French return the favor, obviously - their accent sounds just as terrible.)

Accent is such a non issue, in the UK the accent can be very different even 30 miles from one another, so let alone someone speaking it with a French, German or Spanish accent. It's still English and 99% of the time you can understand it. I assume the same is true of other languages too.

> Accent is such a non issue

But for the French it apparently is (and understandably so).

I meant that you can still understand the speaker, and if so it is courteous to answer. Refusing to understand because you don't like the accent is hardly courteous.

Which is ironic, given that English developed as basically French + German (or some old versions of those languages).

This is completely at odds with my experience of speaking French in France, both in major cities and small villages. My spoken French is appalling, and yet I've encountered nothing but good will from the French, happy to encourage me, and to work together to figure out what I'm trying to say.

Obviously YMMV, but your assertions are contrary to my experience.

Edit: Down-voted. Thanks for the reality check.

2nd edit: Up-voted, perhaps to compensate. Thank you to whoever did that.

There's sort of an uncanny valley sort of situation, in my experience (as an anglophone canadian with the french of a 3-4 year old): If you're clearly a foreign visitor, and you're speaking bad french, it's often appreciated and maybe found slightly charming. But once you get to the point where your french is reasonably good, but you have an obvious accent, then (some) people (might, sometimes) be more inclined to be a bit snooty about it.

Precisely. If you try a few tourist phrases, or very basic French, it will be mildly appreciated. But if you speak fluent French with a bit of an accent, particularly in the big cities, you'll frequently be met with arrogance and attempts to switch to English, even if your French is far better than your interlocutor's.

That said, in small towns and villages, this doesn't happen as much. Seems to mainly be a phenomenon in Paris and other large cities.

Also, in Quebec this is never an issue. As long as you speak fluently, Quebecois are happy to speak with you in French.

> Frenchmen are legendarily intolerant of badly-spoken French.

It may have changed now. I have been living in Paris for several years, speaking strongly accented french (but mostly understandable), and I have had no tolerance problems so far, except the occasional person who prefers to speak in english to me.

I'm fluent in French. Quebecois French. In France they're just rude about it, even when they speak terrible english. I would absolutely do the same thing.

Presumably that's the paper in Russian - https://publications.ias.edu/sites/default/files/iztvestiya_...

It's understandable, but with lots of grammatical, spelling and stylistic mistakes. Basically it reads like something Google Translate would produce.

He has a discussion at the very end (on the last two pages) about his decision to write in Russian, in which he acknowledges that he may not have done very well. Google Translate's version of the final paragraph:

> This article is a consequence of two pushes, first of all, an attempt to understand the nature of the geometric theory, to form a clear idea about the difference between it and the arithmetic theory and their similarity. Here I think I was successful, although I do not argue a little. Secondly, I wanted to significantly improve my knowledge of Russian. Here I had only limited success. For me, Russian is on a completely different level than the two foreign languages with which I am familiar, French and German. As I noted above, Russian is much more difficult than I appreciated, even more than Turkish, another language in which I have a limited but hard-to-work ability. Therefore, my efforts and the efforts of friends and acquaintances who encouraged me as I wrote this article had limited success. I am still pleased that, despite my age, I do not regret either the time or the effort that I have given her.

> Most mathematicians also agree that the Langlands Program could help find a proof for the Riemann Hypothesis, probably the most famous unsolved mathematical problem (about the distribution of prime numbers). These problems are just as abstract as Langlands’s own work, however, which means his research program as it was originally conceived has little relevance to everyday life.

I dunno, figuring out the Riemann Hypothesis could have some far reaching consequences for our understanding of prime numbers, and therefore cryptography, no?

No, for any practical applications, we might as well work assume it is true, the same way we assume quantum physics or relativity are true when building stuff.

And if we actually prove it's not true?

E. Frenkel who was also mentioned in the article (as being in a kind of opposition to Langlands), is an interesting figure in his own right. His own account on his career in mathematics is fascinating.

I took linear algebra from Professor Frenkel. He was indeed one of the most eccentric professors I had. He made a film called Rites of Math and Love whose trailer was very popular (as a laughing material) in our class.

HN has heard of him: https://hn.algolia.com/?query=Langlands%20points%3E3&sort=by...

I've heard of him and I'm just a regular everyday normal guy who codes for a living.

You'd be surprised how "regular everyday guys" haven't heard of him.

In fact I'd be surprised if most US citizens can even name the vice president of the US, even with all the hoopla around Trump's presidency...

Conversation topic transition diagram in the past few years:

Politics -> Trump

History -> Trump

Computing -> Trump

...

Mathematics -> Trump

Trump -> Trump

* -> Trump

That gets old pretty soon.

Is this how Godwin's law ends?

“Langlands’s”! Now that’s eccentric.

It's Robert Langlands[1], for anybody who doesn't want to give in to the clickbait title

1: https://en.wikipedia.org/wiki/Robert_Langlands

We should stop abusing the term clickbait for everything and anything.

Journalists (and editors) used juicy titles since centuries, as part of the art of writing a piece, even when the piece was buried well inside a newspaper or magazine that you would have already bought to see it anyway...

"This is an article about Robert_Langlands the mathematician" doesn't strike as nice as a title.

This is a typical journalistic title, and not at all the same as the modern notion of clickbait (not to mention the article is a legit article, and not some clickbait listicle or similar BS).

>This is a typical journalistic title, and not at all the same as the modern notion of clickbait

I can see where you're coming from (since the article does contain interesting details and isn't just restating the headline), but I think this title sits closer to modern clickbait, in the spectrum that runs from dry description to full-on clickbait.

I will admit, that this style of title has always infuriated me in particular: it feels like the editor is infantilising their audience by imagining the public only know the things that journalists decide to write.

Also I'd personally expect a journalistic title to at least mention the subject. Even something as simple as "Robert Langlands: the mathematician you've never heard of" would be a big improvement

There should be a submission guideline that disallows such useless titling.

You can downvote and flag the post, but the title of the submission is the same as the title of the article, so I see no problem. Unless of course the article is pure clickbait.

I don't think it worth flagging, maybe some HN readers are interested in Langland's work? Just retitle it, prefix [note for clickbait-aversed: it's Langland]

I wouldn't do that, I'd just prefix `Langland: `, so the title remains as in the article, but a bit like a subtitle.

(I Would Also Remove Clickbait Case But Maybe That's Just Me!)

Well, The work of Saints is done through others. It's been fixed by forces higher than us.

What's Clickbait Case?

Many, many times I've put in the time and taken care to create a better title than the one on the article, only subsequently to have it changed by the mods. I've given up doing that.

But did he win the Putman?

It seems to me that doing well in competitions is a separate question from going on to do amazing research. There are plenty of people who did phenomenally well in competitions and then went on to be successful being mathematicians in other fields, working at Big Co, or starting up their own company, but most of the really, really top mathematicians that I know didn't go in for competitions.

There are exceptions, but it's almost like being a top research mathematician is independent (in the technical sense) of doing well in competitions.

I've had a couple of experiences with math olympians. They were both very bright (I don't know how they're not followed by throngs of beautiful ${people who are attracted to men} -- solving compass-and-straightedge problems can be as performative and awe-inspiring as an impressive guitar solo), but they lacked in other hard-to-define areas. They had thin skins for criticism (people like me who have failed in competitions and gotten low grades etc. have a lot more emotional scar tissue) and tended to overestimate themselves. Interestingly, they were never arrogant, they just overpromised a little.

They were still valuable workers, but took a strange onboarding process compared to regular dummies (like me) who just want to be the best they possibly can under skull constraints.

> women

Or men, for that matter.

Fair enough. Fixed!

I'm pretty sure the comment you're replying to wasn't so much a serious question as a reference to this little incident on HN: https://news.ycombinator.com/item?id=35079

Indeed, but I like to play with a straight bat, and I think the point was worth making regardless.

Indeed. I'm glad at least one person got the joke.

Reply on news.ycombinator.com