An Unheralded Breakthrough: The Rosetta Stone of Mathematics

http://blogs.scientificamerican.com/guest-blog/2013/05/21/an-unheralded-breakthrough-the-rosetta-stone-of-mathematics/

by ColinWright • 13 years ago

146 42 13 years ago

blogs.scientificamerican.com

zafka• 13 years ago

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• 13 years ago

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...

aethertap• 13 years ago

Thank you for that recommendation, I've been looking for a book like that for ages.

gsk• 13 years ago

Thank you!

flatline2• 13 years ago

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...

drhodes• 13 years ago

these nine videos are fantastic http://www.dimensions-math.org/Dim_tour_E.htm

here's a whirlwind tour http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

nickpinkston• 13 years ago

Can't recommend Ian Stewart enough!!

kamaal• 13 years ago

Learning math is pretty much like learning programming.

Sit down to read books, it gets boring. Get a real problem to solve, and you learn tons of things along the way.

Don't learn mathematics. It's not supposed to 'learned'. Math is basically what you indicate in notation without you would other wise denote in words while you explore something.

So don't learn, pick up a problem and explore.

glhaynes• 13 years ago

I think that your point about learning being easier when you've got a problem is probably true. But as someone without such a problem immediately at hand, I could really use a list of interesting problems to give me a starting point. And, even if I had a problem I wanted to solve, I expect I might not even know where to begin on it (so then I'd have two problems, as they say). So in practice I'm not sure what to do with your advice.

Osmium• 13 years ago

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"?

vlasev2• 13 years ago

Here is an illustration I made with Mathematica just for you

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

pfortuny3• 13 years ago

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

stiff2• 13 years ago

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...

pfortuny3• 13 years ago

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.

pfortuny3• 13 years ago

[Cannot edit the parent, sorry] There is a somewhat embarrassing (to my taste) discussion here

http://gowers.wordpress.com/2013/03/20/pierre-deligne-wins-t...

in which they claim that it is policy of the Abel Foundation not to bestow the prize on people who would not attend the ceremony.

This strikes me as surprsing at the very least but you can never tell with foundations.

I am thinking in three centuries' time people will say: "Of course, they gave Grothendieck one of the first Abel prizes" and receiving a sorry answer "well, I say, no... it was a question of policy".

Kings are not kings if they do not survive someone rejecting their bounties.

carlob2• 13 years ago

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• 13 years ago

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.

carlob2• 13 years ago

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

nraynaud4• 13 years ago

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• 13 years ago

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.

nraynaud4• 13 years ago

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.

Gmo3• 13 years ago

That's not at all how it works. This is completely absurd.

First, you need to plan your days off. So in reality, you will work 40hrs 1 week and 32 the next one for instance.

Secondly, when you have a certain status (called "cadre"), you are actually giving away the exact count number of hours for a number of days of work. Those work days can be (but not necessarily all the time) more than 8 hours. Actually, any work days is usually more than 8 hours because, like the previous poster says, we take a long break for lunch and that's not part of the hours.

Finally, Americans always have this idea that the French are super strict with their hours, but that's true only in jobs where they are actually clocked by the hour. There a many job where it's not the case and someone committed won't mind staying a few hours extra a week if it means having the job well-done.

In contrast, I've been working in the Netherlands for 6 years now, and I see this behavior much more often : after having stayed the mandatory 8.5 hours (30 minutes lunch break), the person will leave, no matter what.

nraynaud4• 13 years ago

to your added question: yes.

sometimes we compute "fake hours" for people working by the day but it's for administrative purpose (like tax stuff, or unemployment benefits).

I think in the context of the mathematician during the war, we can assume he was thinking of a worker in an factory in peace time or something very cliché like that. At the time, 40 hours weeks and paid vacations were a quite "new" (like less than 10 years) thing.

Gmo3• 13 years ago

Yeah I don't know anybody that leaves after 7h being at work ...

Dewie• 13 years ago

When I read that, I wondered if it was to poke fun at the French (although more hours doesn't necessarily mean more output... neither does more output imply a better quality of life). It's such a subtle prod that I almost feel like I'm making a mountain out of a molehill by just mentioning it.

gangst• 13 years ago

"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?

yaakov343• 13 years ago

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.

contingencies2• 13 years ago

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

gwern• 13 years ago

Here's a start: http://gowers.wordpress.com/2013/04/14/answers-results-of-po...

contingencies2• 13 years ago

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.

yaakov343• 13 years ago

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).

stiff2• 13 years ago

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.

yaakov343• 13 years ago

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.

vlasev2• 13 years ago

> 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• 13 years ago

"no one can patent a formula."

No one should be able to patent a formula. FTFY

An Unheralded Breakthrough: The Rosetta Stone of Mathematics

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