
An Unheralded Breakthrough: The Rosetta Stone of Mathematics - ColinWright
http://blogs.scientificamerican.com/guest-blog/2013/05/21/an-unheralded-breakthrough-the-rosetta-stone-of-mathematics/
======
yaakov34
I think that in their desire to explain this popularly, which is legitimate,
they made a rather confused version of both the history and the mathematics.
First, neither Weil nor Deligne are unheralded - they are about as famous and
acknowledged as mathematicians can be, aside from a few universal geniuses
such as Gauss and a few people who have solved a long-outstanding problem and
were mentioned in the press a lot (e.g. Andrew Wiles). (Incidentally, Weil
once joked that many generations from now, people will probably think that
André Weil and Andrew Wiles, both professors from Princeton, were the same
person).

Second, it's an exaggeration (to put it mildly) to say that Weil is the first
one to think of a connection between numbers and geometry. I mean, this goes
to Descartes at the latest. And such sophisticated tools as the fundamental
group of a topological surface were introduced in the 19th century.

For anyone confused by the statement that the solutions to x^2+y^2=1 in
complex numbers form a sphere: they obviously don't form a sphere in the usual
sense, since the solutions are not bounded (since there is a root of every
polynomial in complex numbers, you can find a corresponding y for every x,
indeed usually two such y's). What's meant is that the Riemann surface defined
by the graph of this function (i.e. y=sqrt(1-x^2), which is two-dimensional
surface embedded into four-dimensional space, has genus 0, i.e. it is
topologically equivalent to a sphere, which is to say it doesn't have any
holes in it. I think you are supposed to prove that this is so by considering
this surface on the complex plane minus the set [1,+inf], seeing that the
function has two distinct continuous branches on the remaining set, each of
which is equivalent to a sphere since it's a simple one-valued function over
the complex plane, and then gluing the two spheres along the line [1,+inf]
like you would glue two regular spheres along edges of cuts made in both. You
end up with another sphere (from the point of view of topology).

~~~
stiff
I don't see any place in the article where they say that Weil was the first to
form a connection between numbers and geometry. The only thing close they say:

 _Weil suggested that sentences written in the language of number theory could
be translated into the language of geometry, and vice versa._

And since this is elaborated upon later on in the article I don't see anything
wrong with the sentence.

~~~
yaakov34
My problem is not with this sentence, but with this passage from the second
paragraph:

> ... number theory is the study of numbers [...]

> Geometry, on the other hand, studies shapes [...]

> But French mathematician André Weil had a penetrating

> insight that the two subjects are in fact closely related.

This pretty clearly says Weil came up with the idea of connecting number
theory and geometry - or at least connecting them closely - which is very,
very wrong.

------
pfortuny
Deligne's work is certainly impressive and he deserves the prize.

However, it is a real pity that they have not even tried to bestow the Abel on
Grothendieck [1].

Yes, he will certainly decline or even not answer but, honestly, they are
missing the chance of saying 'hey, we really appreciate his work although he
is not going to accept the price'.

Probably neither Deligne nor Atiyah nor so many people would have done what
they have without Grothendieck's astounding contributions. But I digress.

[1] <http://en.wikipedia.org/wiki/Alexander_Grothendieck>

~~~
stiff
From the Wikipedia page:

 _He gave lectures on category theory in the forests surrounding Hanoi while
the city was being bombed, to protest against the Vietnam War (The Life and
Work of Alexander Grothendieck, American Mathematical Monthly, vol. 113, no.
9, footnote 6)._

Wow, just wow...

~~~
pfortuny
Well, yes he was really special.

EDIT: That behaviour show what having a conscience means when you take it to
the limit, which -to me- is really inspiring...

It is not only that. He left the IHES as a protest because it was _partially_
founded by the NATO.

For non-specialists, the IHES is equivalent in 'scientific value' to any
Insitute of your choice: Princeton, the Newton Institute, Max Planck, CERN...
whatever. It is one of the few 'best of the best'.

He just left.

He became a recluse long ago, but that does not diminish his scientific
merits.

------
Osmium
If ever there was an article in need of some good illustrations... Does anyone
have an example of one of these "solutions in natural numbers modulo N give us
other, more elusive, avatars"?

~~~
vlasev
Here is an illustration I made with Mathematica just for you

<http://i.imgur.com/8JIocR6.png>

------
carlob
Deligne once built an igloo on the IAS grounds and slept in it with wife. A
friend, post-doc there, had to beg him for a week to let him sleep in it,
before he relinquished it and let him try it for a night.

I think this is a wonderful example of how someone at the top of his career
(this was only a couple of years ago), can still maintain a childlike wonder
towards the world around him.

~~~
StavrosK
Was it an igloo, or a quinzhee? <http://en.wikipedia.org/wiki/Quinzhee>

The latter is easier to build, and the whole process looks to be a lot of fun.
I'll have to try it out.

~~~
carlob
Definitely an igloo. He used a squarish paper bin to compress snow and get
bricks.

------
zafka
This stuff is very cool, and I feel extremely outclassed. Is there a math camp
for adults who just made it out of engineering school, but never absorbed the
real beauty of math? People post this very interesting stuff, and I see
something sparkly and say wow! I must say the writing of Feynman gives me a
little hope, but so far most of the language of math is beyond me.

~~~
habitue
I would recommend the Princeton Companion to Mathematics[1]. Its got a broad
range of topics in short digestible articles. (though the first section has
preliminary material for understanding the later articles)

[1][http://www.amazon.com/gp/aw/d/0691118809/ref=redir_mdp_mobil...](http://www.amazon.com/gp/aw/d/0691118809/ref=redir_mdp_mobile)

~~~
flatline
This is a good reference. It is basically an encyclopedia, so some sections
are better than others. I'm not a big fan of Gowers' intro, it is very heavy
and could have done much more in the way of explaining how to navigate the
volume itself. It is generally assumed that you know integral and differential
calculus.

If you are daunted by mathematical formulas and don't have a solid basis in
math generally, I would recommend Pickover's "The Math Book"[1]. Very
engaging; good short, non-technical descriptions of many of the same topics in
Princeton; lots of pretty pictures.

[1] [http://www.amazon.com/Math-Book-Pythagoras-Milestones-
Mathem...](http://www.amazon.com/Math-Book-Pythagoras-Milestones-
Mathematics/dp/1402788290)

~~~
alok-g
How deep does Pickover's book go? I read in the reviews that each topic is
limited to one page only, so I presume the real mathematics would not be
covered. In other words, after reading this, will I understand the topics or
would just understand things about them?

I also see a Physics book by the same author, but have the same question there
too: [http://www.amazon.com/Archimedes-Hawking-Science-Great-
Behin...](http://www.amazon.com/Archimedes-Hawking-Science-Great-
Behind/dp/0195336119/ref=cm_cr_dp_asin_lnk)

~~~
flatline
IIRC there are a few things that are simple enough to actually explain in a
page, but yes, one page per...artifact? Very little real mathematics, you will
just understand things about them. It is, however, the best written of these
pop-math type books that I've come across. If you are comfortable with
undergraduate-level mathematics, the Princeton Guide is a much better book for
getting an idea about a specific topic. Pickover's book is more for getting a
feel for what topics are out there. They were written for two very different
audiences and I like them both for different reasons.

------
nraynaud
I hope the scientific details are a bit more true than the french work day
length. It's 7h50 of work/week but we usually have a long lunch break so we
end up the day quite late anyways, and we generally work 8h and take some
complete or half days off (Friday afternoon is a common choice) to make up for
the time

~~~
wtvanhest
_It's 7h50 of work/week but we usually have a long lunch break so we end up
the day quite late anyways, and we generally work 8h and take some complete or
half days off (Friday afternoon is a common choice) to make up for the time_

As an American, that is such a foreign statement to read. I literally laughed
out loud when I read it. The craziest part to wrap my head around is the
concept of working during the week, then saying, "I have worked too many
hours, I have to go home Friday afternoon". I can only imagine the work that
isn't being done and being left behind.

Do CEOs and investment bankers only work 50 hours a week? What about
investment managers?

[ADDED question]

So the hours rules only apply to hourly workers? If that is the case, it is
similar to America because companies have to pay overtime for hourly workers
so they try not to let them work past 40 hours anyway.

~~~
nraynaud
managers and higher ranking people are paid by the day, so their hours are not
counted. I have been paid by the hour only the first 4 months in my ~10years
of working carrer.

edit: but on a side note France (and Germany) ranked very high in output per
worked hour the last time I checked, way higher than the US. Kind of not
messing around in the office people.

~~~
polshaw
>managers and higher ranking people are paid by the day, so their hours are
not counted

Do you have any more info on this? Are you basically saying that the law is
ignored? Because AFAICT the French and EU (working time directive) are both
quite strict and there don't seem to be any opt-outs.. their hours would just
be logged as '7h' regardless of reality? (and if the employee were to kick up
a fuss, they would no doubt win any case?).

(i've been interested in hiring in French jurisdiction, but put off by working
time law)

~~~
nraynaud
ouch, you're going in very specific here. It's going to hurt.

About the EU law : you are not subject to it, it's countries who are subject
to it. It work by fining the infringing country. EU laws generally mandate
countries to have national laws in this or that way. If a citizen gets a
condamnation by an infringing national law, and after having tried every
possible national appeals, he can go to the EU court and get his country
condemned. I'll stop here on that subject.

So, on a French level, we have work laws (Code du travail), regulations and
"collective agreement" (I hope my google translation is correct). In computer
science the collective agreement is called "syntec" it has force of law (it's
not the case for all agreements). Collective agreements can offer better
benefits than the law, but not worse. In the case of computer science, the
employees are classified in 2 layers, basic employees (secretaries etc.) and
"Managers and Engineers" (Cadres et Ingénieurs) in startups, almost everybody
is an engineer. Then engineers are classified with a number, depending on
experience, autonomy, degree of skills the job needs, and people managed. This
number (coefficient) correspond to the minimum salary you have to pay them,
and the other way around, if in everyday work, their job goes into a higher
level, their minimum salary have to be increased (if you were already paying
them high than the minimum, you don't care). For computer science we are
generally way over the minimum, but it's good to have someone check the
employment contract before signing it. Employers tend to give lower number at
employment, I tend to raise it to their true level, because I value my team on
paper too. People whose number is high enough meaning their seniority is high
enough are to be paid by the day and not the hour. Those people are paid to
work 218 days a year (waiting for some backlash here:) ).

Then you have the limits on the number of hours worked, like no more than 13h
in a row, sleep time etc. It's not really about salary, it's about security
and well-being. As far I as know it's not a issue in computer science.

------
contingencies
Practical challenge: automatically generate a natural language plus visual
diagram series explanation of arbitrary equations marked up in MathML or
similar.

~~~
gwern
Here's a start: [http://gowers.wordpress.com/2013/04/14/answers-results-of-
po...](http://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-a-
brief-description-of-the-program/)

~~~
contingencies
Interesting but I wouldn't call that 'natural language' - I'd call that
mathematical jargon! I was thinking more like something that anyone could
grasp, optimally visually. So, for example, given some idea about a
particular, practical problem (engineering, surveying, agriculture,
theoretical complexity of software, etc.) an explanation could access
relatively advanced mathematical knowledge and present it clearly to the
uninitiated.

------
gangst
"Deligne’s most spectacular results are on the interface of two areas of
mathematics: number theory and geometry"

This is very compelling. Can anyone suggest topics or materials of study to
explore this interface?

~~~
yaakov34
Study complex analysis. It is a very pretty field which certainly has
connections both to algebra (in fact, complex analysis grew out of the study
of roots of polynomials) and to geometry - there are a lot of fascinating
geometrical results in it, such as the famous Euler's formula for angles, e^i
_Theta_ =cos _Theta_ \+ i sin _Theta_ as one of the simplest examples. There
is also the theory of conformal mappings and all sorts of beautiful results
for analytic functions, e.g. the fact that a complex function which is smooth
(i.e. differentiable) and non-constant must take every possible complex value
(except possibly one point) - certainly not true for the real numbers! Once
you have a good understanding of complex analysis, you can continue to study
Riemann surfaces and topology. A lot of modern geometry and number theory
grows out of these studies, e.g. the Riemann hypothesis which is very
important in number theory, and Riemann surfaces which are strongly connected
to models of spacetime, both started as part of complex analysis. Also the
field has a huge number of applied results, e.g. in the area of differential
equations and Fourier analysis. Once you have a good grounding in complex
analysis, you can decide if you want to move into the later results (such as
Weil's), which tend to be very sophisticated.

------
vlasev
> The avatars of algebraic equations in complex numbers give us geometric
> shapes like the sphere or the surface of a donut; solutions in natural
> numbers modulo N give us other, more elusive, avatars.

If anyone is wondering what those "avatars" for solutions in natural numbers
modulo N give, here is my plot for N from 2 to 22 that I made with
Mathematica.

<http://i.imgur.com/8JIocR6.png>

------
danbmil99
"no one can patent a formula."

No one _should_ be able to patent a formula. FTFY

