Alexander Grothendieck is Dead

Bourbaki
Bourbaki: On November 13th, 2014, Alexander Grothendieck, easily one of the greatest mathematicians of all time and perhaps its greatest visionary, died at age 86.

I'd like to dedicate this short post to some of his infinitely many achievements. For the sake of brevity I will include no personal biographical information.

At 18 it was apparent Grothendieck had a knack for generality. He produced independently a very general integration manuscript which was turned down as generalization for the sake of generalization by one of the French greats (Dieudonné or Schwarz If I remember correctly). He was given the problem of finding the right topology on the tensor product of topological vector spaces, a problem which began his functional analysis era. He brilliantly solved the problem and proceeded to revolutionize the field by solving many more difficult problems aswell as introducing central concepts such as reflexive Banach spaces and nuclear spaces. He did not return to the field later in his life.
In 1955, Grothendieck turned his attention to fields more fitting his proclivity for abstractness. He began working on homological algebra and sheaf theory. Grothendieck introduced categories into many then-nascent realms of mathematical research, thereby both unifying and clarifying their similarities and formalism. He introduced a vast number of invaluable concepts, to cite a few: abelian categories, derived functors, etale cohomology, grothendieck topology, topoi, schemes, stacks, motives, and many more. He is the father of modern algebraic geometry and indirectly of many related fields. Among his countless contributions, it was essentially his work that made possible the solution of the Weil conjectures by his student Pierre Deligne.
Grothendieck was perhaps the greatest mathematician of all time in the sense of abstraction and generality. Much can be learned from his history about structure and pattern in mathematics, aswell as the inherent value of generalization and in particular categorification.
(Edited by Bourbaki)
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The13th
The13th: Interesting. Thanks for posting. If I have nine lives, I want to spend a couple of them as mathematician.
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Bourbaki
Bourbaki: It's never too late to start!
(Edited by Bourbaki)
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The13th
The13th: Not this life though, abstract mathematics is a bit beyond my graps. In this life I can perhaps try to understand div, curl and all that - and thats probably it haha.
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winglady
winglady: before this i didn't know who that guy was
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Bourbaki
Bourbaki: There was really no way for you to have known about Grothendieck if you're not in mathematical circles
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CoIin
CoIin: I'd never heard of Mr Grothendieck either, Bourbaki. Thanks for enlightening me.

Um, I hope this isn't a silly question, but would it be possible to give us a sample of some of his work in language that we might understand?

For example, you mentioned "generalization" more than once. I've no idea what that is with respect to mathematics. And I think it's about time somebody explained to me what a "functor" is.
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The13th
The13th: What impress me more is this guy actually fought in 1905 Russian revolution and escape firing squad because he is a boy. But still get jail until 1917.
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The13th
The13th: Or did I get it right? You meant he is at least 110 years old or older? But didnt you said that he is 86?

Nah, now I got it. Its his old man that fought in that 1905 revolution.
(Edited by The13th)
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Bourbaki
Bourbaki: It is not possible to give any sample deserving mention without the details. I suggest you read about 'topoi', which are essentially universes with their internal logic (these are, as expected, categorical constructs). Generalization will be the topic of a separate post. As for functors, you can find their definition on wikipedia or the nlab; it would be cumbersome to try and explain what they are here without TeX.
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Page2000
Page2000: I was wondering if vector space is some kind of matrice.
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Bourbaki
Bourbaki: Your question is extremely vague, but conceptually speaking the answer is no. Vector spaces are the setting for linear operators. In the case of finite dimension, linear operators can be represented by matrices. Upon choice of basis, this representation becomes unique. There does not, however, exist a representation which is natural in the sense of category theory. Finding bases for which the representing matrix has a convenient form is a central problem in linear algebra. There is much more to be said about this of course, but not here.
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The13th
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Bourbaki
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CoIin
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Page2000
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Bourbaki
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Captain Canada
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tallboy61
tallboy61: it all adds up now.
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Bourbaki
Bourbaki: Huh.. strange.. I do not recall deleting the posts above. I do not recall ever deleting a post of Colin's..
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The13th
The13th: But come to think of it we don't really have any earth shattering mathematicians in the last 50 years is it? People of Gauss, Poincare, von Neumann sort of calibre?
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tallboy61
tallboy61: is he ?
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