An Unheralded Breakthrough: The Rosetta Stone of Mathematics
blogs.scientificamerican.com
blogs.scientificamerican.com
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).
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.
> ... 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.
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).
Wow, just wow...
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.
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.
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.
The latter is easier to build, and the whole process looks to be a lot of fun. I'll have to try it out.
[1]http://www.amazon.com/gp/aw/d/0691118809/ref=redir_mdp_mobil...
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...
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...
here's a whirlwind tour http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...
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.
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.
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.
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.
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)
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).
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.
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:
http://stats.oecd.org/Index.aspx?DatasetCode=LEVEL
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?
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.
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.
This is very compelling. Can anyone suggest topics or materials of study to explore this interface?
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.
No one should be able to patent a formula. FTFY