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).
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.
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.
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.
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).
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.
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.
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"?
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.
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.
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)
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.
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?
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.
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.
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.
Write simulators that use the mathematics you want to learn. One way or another you will learn the mathematics if you do this. A game engine with bone animation is great for really getting familiar with 3D vector mathematics. Electrical circuits simulator and a Physics engine are great follow ups. Fluid dynamics is another one. A symbolic math engine will get you even more familiar with lots of other pure mathematics.
Honestly speaking I've only built the circuits simulator, partially built the symbolic math engine and currently working on the physics engine. I have learnt a great deal though.
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
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.
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.
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.
>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.
I am not trying to start an argument with you (and France seems like a great place to live and work) but in Googling around, I found the following, which reports that German GDP per hour worked is .92 of the American and French is .958 of the American:
If you click on the "Show information" tab on the right, you see that "At the beginning of the year 2012, GDP per hour worked was significantly revised for a large number of OECD countries". Could it be that you relied on the "pre-revision" figures?
More generally, do you remember where you got your data?
>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)
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.
The EU directive put a legal up limit on the number of hours you can work yes. But this limit is quite high. Besides, this limit does not apply when the person goes home and work in the evenings from there on her own will (and for certain positions, that's frequent).
Secondly, it's especially written in the law that with the "cadre" status, you are to work a number of days annually, not hours (it's called "au forfait" if I'm not mistaken).
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.
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.
Practical challenge: automatically generate a natural language plus visual diagram series explanation of arbitrary equations marked up in MathML or similar.
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.
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.
> 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.
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).