
Robert Langlands, who linked number theory, analysis, and geometry (2015) - mathgenius
http://projects.thestar.com/math-the-canadian-who-reinvented-mathematics/
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jessup
A lecture series by Langlands for a lay audience:

[https://video.ias.edu/The-Practice-of-Mathematics](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."

~~~
hyperpallium
> 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?

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

~~~
7dare
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.

~~~
whatshisface
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.

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mathgenius
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.

~~~
goldenkey
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](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](https://amzn.to/2qcqUb4)

~~~
mathgenius
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.

~~~
goldenkey
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...](https://johncarlosbaez.wordpress.com/2017/12/31/quantum-mechanics-
and-the-dodecahedron/)

~~~
mathgenius
That john baez article is about something else.

Try this:
[https://arxiv.org/pdf/1308.0955.pdf](https://arxiv.org/pdf/1308.0955.pdf)

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jesuslop
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.

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Koshkin
A somewhat accessible account of the Langlands Program can be found in Edward
Frenkel's book _Love & Math_.

~~~
mathgenius
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/](http://online.itp.ucsb.edu/online/bblunch/frenkel/)

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ginnungagap
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](http://www.abelprize.no/nyheter/vis.html?tid=73025)

~~~
dang
Discussed a bit at
[https://news.ycombinator.com/item?id=16632364](https://news.ycombinator.com/item?id=16632364).

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kirkules
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.

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MaysonL
His seminal paper, _Problems in the Theory of Automorphic Forms_ is here:

[https://publications.ias.edu/sites/default/files/problems-
in...](https://publications.ias.edu/sites/default/files/problems-in-the-
theory-of-automorphic-forms.pdf)

------
charlescearl
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.

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

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

egh, is it?

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

~~~
qmalzp
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!

~~~
oggyhead
I see.thanks for the info and light

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Brozilean
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".

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

~~~
ajeet_dhaliwal
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.

~~~
tensor
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.

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

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cobbzilla
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](https://en.m.wikipedia.org/wiki/Trachtenberg_system)

~~~
y7
How is this related?

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

~~~
nkoren
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.

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

~~~
nkoren
Damnit, you win this round. :-)

