Robert Langlands, who linked number theory, analysis, and geometry (2015) | Hacker News Reader

Robert Langlands, who linked number theory, analysis, and geometry (2015)

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231 pointsmathgenius54 comments

A lecture series by Langlands for a lay audience:

https://video.ias.edu/The-Practice-of-Mathematics

The video is not great, but the content. From the summary:

"In spite of forty years as a mathematician, I have difficulty describing these problems, even to myself, in a simple, cogent and concise manner that makes it clear what is wanted and why. As a possible, but only partial, remedy I thought I might undertake to explain them to a lay audience."

> If it works out, I would like to continue in following years on classical fluid mechanics and turbulence

Since it's dated 1999, I guesss he didn't continue with fluid mechanics?

It's about Robert Langlands of https://en.wikipedia.org/wiki/Langlands_program

>linking math’s main branches — number theory (once called arithmetic), harmonic analysis, which includes calculus, and geometry.

>To mathematicians, this is mind-blowing stuff. The branches deal with completely different things: number theory is about, yes, numbers, harmonic analysis studies motion and geometry deals with shapes. They may as well be different planets.

Different planets on which you find all of Shakespeare's works, depsite no apparent common origin.

I think that may be an exxageration. I study maths at a much lower level and the teacher regarily points out links between these domains.

I believe number theory is already strongly linked with analysis.

Those links only became apparent after centuries of development in total isolation. Now that we know they are there, we involve them in teaching because they change how we see things.

Somewhat randomly, I've been working with a mathematician that has convinced me to study the Langlands program with him. I feel ill-equipped for the journey, having boycotted number theory ever since they tried to get me to learn the multiplication table. But on the other hand, there is plenty of geometry involved! Who would have thought that the dodecahedron is involved with solving the quintic? It really does seem like this area (the Langlands program) is at a vast nexus of many different branches of mathematics. It's intense.

Shapes just come about when you are talking about objects with certain types of symmetries. One of the main "attacks" on polynomials is to realize that their roots are related to eachother, in terms of total, product, difference, etc.

The simplest of these methods is just Vieta's formulas:

https://en.wikipedia.org/wiki/Vieta%27s_formulas

I highly recommend Edward Frenkel's book, Love and Math. It was a great read about the mathematics itself, and also his troubles as a Jewish academic in Soviet Russia.

Love and Math: https://amzn.to/2qcqUb4

Right, but the dodecahedron is a modular curve, and I'm guessing this is how Hermite came up with the solution to the quintic using modular functions in 1858. If there are any real number theorists here they might be able to clarify this some more. Modular functions, modular forms, automorphic forms, these are all key ingredients of the Langlands program.

I believe you're talking about the free algebras on groups of modular forms of a certain 'weight.' That's not exactly the modular forms themselves.

Here's some more information: https://johncarlosbaez.wordpress.com/2017/12/31/quantum-mech...

That john baez article is about something else.

Try this: https://arxiv.org/pdf/1308.0955.pdf

What do you mean when you say the dodecahedron is a modular curve? Is there some N for which X_1(N) can be naturally thought of as a dodecahedron or something?

Edit: Okay glanced at the arxiv paper linked below. The dodecahedron is not a modular curve per se, but the link is as follows:

The quotient X(5) -> X(1), which geometrically is a quotient from P^1 (the sphere) to itself, can also be thought of as quotienting the sphere by the group of symmetries of its inscribed dodecahedron. I believe this group can naturally be thought of as PSL_2(Z/5Z)

It's quite complicated (https://en.wikipedia.org/wiki/Bring_radical) ..amazing he derived it so long ago , long before the internet and modern research tools were invented. he had to scour for all that information and derive this solution.. even with the help of the internet I find it hard to understand

The book of Frenkel (the guy of the photo), Love and Math, is one way to go from zeto to not being 100% clueless about the Langlands program, and enjoy the time at the same time. Very likable.

A somewhat accessible account of the Langlands Program can be found in Edward Frenkel's book Love & Math.

Here is Frenkel giving a talk on the geometric Langlands program [1]. I wouldn't say it is "accessible", but if you have seen some physics maybe give it a go.

[1] "What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?" http://online.itp.ucsb.edu/online/bblunch/frenkel/

For the Langlands program he was awarded the Abel prize just a few weeks ago, following the link "Read more about Robert P. Langlands" at the end of the official page [1] for the Abel prize you can read a short and informal explanation of his work written by Alex Bellos

[1] http://www.abelprize.no/nyheter/vis.html?tid=73025

Discussed a bit at https://news.ycombinator.com/item?id=16632364.

What's the intended purpose of this article? Why was it written?

It's always frustrating to read pieces about mathematicians written by people whose attitude is "[I have no chance of ever grasping what ultimately makes this article possible]" and "'[Cartesian geometry?] I can recall sitting in that class,' I said, lying.".

I know writing such things is often supposed to make the article more palatable to a lay audience who probably feels the same way, but honestly all it accomplishes is reinforcing to that audience that mathematics is an esoteric topic for geniuses.

This kind of article, which pretends to be bringing mathematics to the masses, is just another one of the reasons students feel so comfortable incessantly returning to saying "I'm not a math person" instead of being empowered whenever they have some success with math.

This is so frustrating because of course I approve of the apparent motivation behind writing an article like this.

His seminal paper, Problems in the Theory of Automorphic Forms is here:

https://publications.ias.edu/sites/default/files/problems-in...

It is inspirational that he is still active at 80+. Frankel’s videos are great, though I am still building to a real grasp of Langlands work.

The way the title is formatted made me think this would be an obituary. I'm really glad that's not the case.

> “What normal person cares whether the square root of two is a rational number?”

egh, is it?

Wasn't Category theory supposed to unify everything? Has it failed in doing so?

Category theory is basically a language. If one wanted to formulate an explicit conjecture corresponding to the original Langlands program, it could possibly be phrased as some kind of equivalence of a category of automorphic representations and a category of motives. Precisely defining those two categories is the hard part.

So even with this language, one still has to do the work of actually proving these things. It's like having a nice programming language; you still have to write the code to do the thing!

I see.thanks for the info and light

Hahaha I love how the title of this post implies the idea that the idea of a Canadian being a good mathematician is a shocker. In my head I read it the same as "John, an octopus, is the world's greatest mathematician".

The Star is a Canadian publication. The clear interpretation is of national pride.

Still somewhat over the top. They mention he’s a Canadian three times in the first half of the article. In case you missed it, he’s a Canadian. Although looking him up it seems he’s also an American.

It's kinda like Alexander Graham Bell, claimed by the Scots (in England they call him British) the Canadians and the Americans.

>It's kinda like Alexander Graham Bell, claimed by the Scots (in England they call him British) the Canadians and the Americans.

... and "universally" (i.e. by the British, the Canadians and the US people) attributed the invention of the telephone, which was invented by an Italian, instead ...

https://www.theguardian.com/world/2002/jun/17/humanities.int...

Though the Canadians are still not convinced:

https://en.wikipedia.org/wiki/Antonio_Meucci#2002_U.S._Congr...

Yes, it's a little over the top. But as a Canadian, I can say that Canadian publications (especially left-of-center ones like the Star) tend to be a little nationalistic whenever a local does well. It's a natural response to the brain drain of many talented Canadians to our neighbor to the south.

Welcome to the other side. It's common place here to see the same sort of nationalism for the US, but no one ever comments on it. For those of us outside of the US, it's extremely obnoxious though.

Right in line with that, the minute any comment or article even slightly nationalistic about another country, you get tons of comments about how untrue it is and how it's still the US that is amazing, not the other country.

I’m not from the US and don’t really notice this.

I have no idea why you were downvoted. As a Canadian whenever I see a news article that writes like this I cringe so hard. Being a Canadian had nothing to do with this guy's success. That's just how it is here sometimes.

The headline of the article, which was the original title of the submission, is less ridiculous. But for some reason it has been changed.

And it's not particularly rare. To take one example, Curtis McMullen is both a Fields Medal winner and the thesis supervisor of another Fields Medal winner. He also happens to be Canadian (at least by birth).

Canadian media treat the word "Canadian" as an honorific and optimize for the number of times they can use it. It has been that way for as long as I can remember.

We changed the title to use a representative phrase from the article, and added the year.

With respect - you are both being overly general about Canadian media, and showing an American bias. The Star is not representative of all Canadian media, and there are many American publications that trumpet American exceptionalism at every opportunity.

I should hope it shows Canadian bias since that's where most of my experience has been! I mean it affectionately and with no implied comparison.

I know it's dodgy to post occasional personal comments with a moderator account, but it's also dodgy if the moderators aren't community members too. And we were community members long before we became moderators. So I make a point of doing it sometimes.

Tangentially related, Jakow Trachtenberg invented a new and much faster system for mental arithmetic [1] while imprisoned in a Nazi concentration camp. It's a far cry from Common Core math, not easy to learn but once you know it, wow, you can multiply arbitrary 3-digit numbers in your head nearly instantly. 4-digits and higher gets a little trickier but the mental difficulty increases slowly.

[1] https://en.m.wikipedia.org/wiki/Trachtenberg_system

How is this related?

Tangentially related as well, I had 1 blueberry muffin for breakfast and 2 slices of bacon

Side bacon or belly bacon?

Thinking in a brand new way about Math, a way no one else has, to come up with novel new connections?

Arithmetic is to mathematics as typing is to being a writer. If the subject of an article is a brilliant author, then a cool new keyboard that makes it easier to type really isn't related, not even tangentially.

However, tangents themselves are part of geometry, which is tangentially related to Langlands' work.

Damnit, you win this round. :-)

Arithmetics is part of math. (And typing is not part of being a writer.)

Perhaps a more apt analogy is that mental arithmetic tricks are to math what stenography is to writing.

The Trachtenberg system is number theory.

Basically, every good scientific math publication does just that (on the level of the Trachtenberg system). What makes the Langlands program stand out is that it connects entire branches of mathematics previously thought to be unconnected; that means decades long legacies of journal publications of people working within their branch (each time through a novel discovery) and not noticing the connection. The two are not really comparable.

Perhaps there exists a framework isomorphic to Langlands' theories that unite the three disparate topics of this subthread:

1. The perfunctory machinery of arithmetic 2. The theoretical connective tissue between entire branches of mathematics 3. The "egotistical creatures", as he put it, who downvote and give someone a hard time for posting a comment.

I don't know how related it is, but I'm glad you shared that link.

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