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rgold
Trad climber
Poughkeepsie, NY
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Topic Author's Original Post - Nov 14, 2014 - 08:31am PT
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One of the greatest 20th century mathematicians and a real iconoclast.
His influence, from Wikipedia:
Within algebraic geometry itself, his theory of schemes is used in technical work. His generalization of the classical Riemann-Roch theorem started the study of algebraic and topological K-theory. His construction of new cohomology theories has left consequences for algebraic number theory, algebraic topology, and representation theory. His creation of topos theory has appeared in set theory and logic.
One of his results is the discovery of the first arithmetic Weil cohomology theory: the ℓ-adic étale cohomology. This result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne. To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with applications to the Langlands program.
Grothendieck influenced generations of mathematicians after his departure from mathematics. His emphasis on the role of universal properties brought category theory into the mainstream as an organizing principle. His notion of abelian category is now the basic object of study in homological algebra. His conjectural theory of motives has been behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.
For those who know about such things, not mentioned here is that Jean Dieudonne (his thesis advisor) compared Grothendieck’s early work in topological vector spaces as equivalent in influence to Banach
In the light of all this, the following quote (translated from French of course) is interesting, perhaps especially in view of some of the other discussions about mathematics that have occurred on this site.
...I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
It has been said archly, that at either end of the economic spectrum, there is a leisure class. Perhaps to this we should add that at either end of the intellectual spectrum, there is a dumb ox.
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MH2
climber
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Nov 14, 2014 - 08:47am PT
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It is impossible for me to guess how much trust to place Grotthendieck’s statement about himself, but it sounds similar to what Patrick Billingsley said one evening at that drinking establishment near the University of Chicago: "Doing mathematics takes a lot of time and hard work."
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yanqui
climber
Balcarce, Argentina
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Nov 14, 2014 - 11:44am PT
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Grothendieck probably had as much to do with the mathematics I used in my thesis as anyone: topological tensor products, nuclear spaces, topological algebras, derived categories, D-modules; I used all the big guns. He was definitely an interesting (excentric? insane?) guy who, for a period, worked extremely hard in mathematics and was incredibly productive. Then he just stopped. Malgrange once told me that he felt Grothendiek had worked so hard that something broke in his brain. Was Grothendieck a mad man? a genius? He was the most talked about mathematician among my peers when I was a graduate student. To a large extent, he was rejected by the mainstream when he began to publicly protest the compliance of professional mathematics with the military-industrial complex. He went to Brazil to teach and publish his new developments on topological vector spaces and then, latter on, went to the jungles of North Vietnam to teach category theory to the Viet-Cong. I wonder if those guys complained that mathematics (as Grothendieck saw it) wasn't useful? In the end he just disappeared from public life and never returned. RIP Grothendieck.
PS: plus his name is pretty much impossible to spell correctly.
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jgill
Boulder climber
Colorado
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Nov 16, 2014 - 03:33pm PT
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Alexander can't even hold his own against Zsa Zsa Gabor in the Supertopo page levels. (rgold)
“Indeed, the trend toward increased generality and abstraction, which can be seen across the whole field since the middle of the 20th century, is due in no small part to Grothendieck’s influence.” (NYT)
AG's contributions and influence were phenomenal, but his legacy is generally incomprehensible to most normal, college-educated people*. A hundred and fifty years ago or so it may have been possible for mathematical discoveries to be appreciated by many in the educated population, but for the last hundred years levels of abstraction have risen to the point that, apart from a few number theory conjectures and geometric conundrums, a lot of what has transpired is as unintelligible as a peculiar foreign language. Of course there are always a few entertaining analogies that convey quite a bit: e.g., the discussion that a topologist is someone who cannot tell the difference between a doughnut and a coffee cup!
In graduate school almost fifty years ago I had the option of going into "soft analysis" (abstract), but chose to turn to the past for (modest) new discoveries in "classical analysis"(less abstract). That served me well, for I don't think I had the ability to make contributions in a very abstract mathematical world.
* Tim or Rich or others might disagree. If so I'll learn something!
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rgold
Trad climber
Poughkeepsie, NY
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Topic Author's Reply - Nov 16, 2014 - 05:38pm PT
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AG's contributions and influence were phenomenal, but his legacy is generally incomprehensible to most normal, college-educated people.
I think it is fair to say that a large fraction, perhaps a majority, of working mathematicians don't understand Grothendieck's work and contributions; it is in some ways very specialized even though in other ways extremely general. You probably have to be working in algebraic geometry or some allied field to really know what Grothendieck was up to.
That said, I think an analogy, no doubt flawed in more ways than I realize, might be made with, say, Isaac Newton. Newton was in possession of Kepler's calculations of planetary orbits, based on Tyco Brahe's astonishingly accurate celestial observations (done without a telescope, indeed with wooden instruments, with a resolution about the width of a human hair held at arm's length). Kepler had made some utterly remarkable observations, in some sense the least of which is that the planets travel in elliptic orbits with the sun at one of the two foci. And the question was is this always true, and if so why?
Newton looked past the exceptional complexity of the observations and the almost mystical nature of Kepler's insights and instead sought basically simple fundamental structures whose existence would entail Kepler's observations as consequences. These he found, in the form of his three laws and the supposition that the gravitational force between two bodies is inversely proportional to the distance between the bodies. Newton was obliged to invent calculus along the way in order to work out the consequences of his basic laws. What we got was an exceptionally fruitful theory that is the basis of almost all world-scale enterprises involving force and motion.
Finding simplicity where almost everyone else sees overwhelming complexity is the work of geniuses. I think one could argue that Grothendieck did something similar in the fields he contributed to, primarily algebraic geometry. I don't think one could make a case for simplicity here, but we could say that Grothendieck found abstract structures that had consequences that were conjectured but beyond the realm of proof. And abstraction, being essentially the ignoring of details, is really a form of simplicity, even when it is hard to learn.
Playing the role of Kepler's observations were the Weil conjectures, which seemed as if they might be true because of analogies with other branches of mathematics, in particular algebraic topology, but which lacked the structural underpinnings (eg continuity) to be verified in their own context. Something akin to the concept of cohomology with rational coefficients in the realm of algebraic topology was needed in the realm of algebraic geometry. Grothendieck built a whole abstract edifice that made the analogies possible and conferred new and unanticipated powers on the workers who mastered it.
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zBrown
Ice climber
Brujò de la Playa
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Nov 16, 2014 - 05:57pm PT
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Alexander can't even hold his own against Zsa Zsa Gabor in the Supertopo page levels.
AG's contributions and influence were phenomenal, but his legacy is generally incomprehensible to most normal, college-educated people.
Well, it's probably (mathematical construct, no?) worse than you have stated. The Zsa Zsa story is most likely a fake.
Likely too, there are more than a few non-college-educated readers on the ST.
Anyway, while I can't understand it, I can appreciate it.
RIP Mr. Grothendieck.
algebraic geometry
theory of schemes
classical Riemann-Roch theorem
algebraic and topological K-theory
new cohomology theories
algebraic number theory
algebraic topology
representation theory
topos theory
arithmetic Weil cohomology theory
the ℓ-adic étale cohomology
Weil conjectures
To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with applications to the Langlands program.
Any (simple) beta on these routes?
I thought so.
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Andrew Barnes
Ice climber
Albany, NY
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Nov 16, 2014 - 07:31pm PT
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Grothendieck's brilliance is universally acknowledged in mathematical circles. As people in this thread have pointed out, he has revolutionized
algebraic geometry and number theory - the notion of a scheme and the
simplifications it has engendered are hard to describe.
His personality and life trajectory are not too surprising, even though
unconventional (for mathematicians). There was nothing he could have gained
by feigning respect for the mathematical establishment, and he didn't. If he
"retired" from mathematics early, that was his prerogative. A day or a
month of Grothendieck is much more than a lifetime for most regular
mathematicians.
When one has seen the light and scaled the divine celestial heights, there is nothing to be gained from the pedestrian bickerings of academia, tenure, grants, publishing etc. No prizes or marks of human recognition can add to
the lustre of Grothendieck. He was a mathematician who discovered eternal truths, of the kind that time does not diminish.
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MH2
climber
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Nov 16, 2014 - 07:41pm PT
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This thread may not occupy much space on SuperTopo, but I like the way it recalls the math (and physics) backgrounds of my earliest climbing mentors.
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zBrown
Ice climber
Brujò de la Playa
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Nov 17, 2014 - 07:47am PT
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Nice tribute Mr. Barnes.
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Ed Hartouni
Trad climber
Livermore, CA
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Nov 17, 2014 - 09:45am PT
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I'll try later, but maybe one of our house mathematicians could come up with some examples of the "practical utility" of some of these ideas, and perhaps the more general beauty of the ideas themselves...
it's hard, but our crew is capable!
I just got back from a long weekend trip to S. Korea so I'm not back up to speed yet...
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yanqui
climber
Balcarce, Argentina
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Nov 17, 2014 - 10:02am PT
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I think it would be unrealistic to imagine that Grothendieck's break from his life as a professional mathematician was the happy separation of a man who was too authentic to fit into the the phoniness of academic life. Certainly the break had elements of authenticity, but it also seems to have had elements of madness and it's not at all clear that Grothendieck ever found what he might have been looking for. For anyone who might be interested, here are some in depth considerations from a biographer who made the effort to wade through the thousands of pages of Grothendieck's meditations:
http://www.ams.org/notices/200808/tx080800930p.pdf
I also think it's inaccurate to hold Grothendieck, in some sense, as responsible for a higher level of abstraction in mathematics. Early on, Grothendieck integrated smoothly (ha ha: math metaphor!) into the elite group of French academics of his day and participated in (more like: became a leading figure in) the kind of mathematics that was already going on in that time and place. In this sense, a new level of abstraction was not a revolutionary feature of Grothendieck's work.
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Reilly
Mountain climber
The Other Monrovia- CA
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Nov 17, 2014 - 10:31am PT
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"Alexander can't even hold his own against Zsa Zsa Gabor in the Supertopo page levels."
He likely would not have lost much sleep over that. But then ZsaZsa couldn't hold a candle to
threads about pooping on ropes or wolves running free, either, which I doubt would have
troubled her.
Both of them don't seem to be as relevant as rapping with a garden hose, either.
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JEleazarian
Trad climber
Fresno CA
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Nov 17, 2014 - 11:08am PT
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Oh come on, Reilly. Don't dismiss the importance of mathematics to climbing and climbers. At Berkeley, several of us believed that math, music and mountains went together. My undergrad math major advisor, Alan Durfee, was a climber and a harpsichordist. I would see him in Evans Hall, in the practice rooms in the basement of Morrison Hall, and in Yosemite. I met Mike Spivak at Indian Rock before I realized he was a mathematician. (Incidentally, I thrilled my nephew, a recent Ph.D. in algebraic topology, when he discovered I had a copy of Mike's monogrph Calculus on Manifolds).
There were limitations, though. David Altman and I discovered empirically that we couldn't learn Abstract Nonesense very well in one rainy day in the Lodge lounge.
John
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Marlow
Sport climber
OSLO
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Nov 17, 2014 - 11:25am PT
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There's some surprising branches on the Supertopo forum-tree. This is one. Much appreciated...
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jgill
Boulder climber
Colorado
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Nov 17, 2014 - 07:38pm PT
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The article in the AMS Notices cited by Yanqui paints a sad picture of a genius in psychotic decline. It reminded me of articles I read years ago about Georg Cantor.
"Abstraction" in math is frequently equated to "generalization." I'm not sure I entirely agree with this, but my argument would be weak. Like a giant snowball rolling down a slope math keeps accumulating as its outer "shell" grows, new definitions appearing to describe categorization processes. A simple example: some complex functions may be differentiated, thus leading to a class of complex functions called "holomorphic." Some holomorphic functions are "injective" or one-to-one, and are then labeled "univalent." On and on it goes . . . and where it ends, nobody knows. (Used to enjoy that guy! Recognize him?)
Much of "soft" analysis revolves around functions that are "linear" (very well behaved) and abstractions abound, some in functional analysis for instance providing the kinds of overviews mentioned above as AG's wonderful insights. But once one leaves linearity and moves into non-linear territory - analytic theory of continued fractions, e.g. - not a lot of cosmic perspective is available. When I was active in a small international group years ago we would talk about a hypothetical generalized theory that might explain analytic continued fractions like the wonderfully simple structures of power series - or a hypothetical metric that might open new vistas - but these things didn't happen. Still awaiting someone like AG . . .
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rgold
Trad climber
Poughkeepsie, NY
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Topic Author's Reply - Nov 18, 2014 - 09:22pm PT
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I also think it's inaccurate to hold Grothendieck, in some sense, as responsible for a higher level of abstraction in mathematics. Early on, Grothendieck integrated smoothly (ha ha: math metaphor!) into the elite group of French academics of his day and participated in (more like: became a leading figure in) the kind of mathematics that was already going on in that time and place. In this sense, a new level of abstraction was not a revolutionary feature of Grothendieck's work.
Allyn Jackson, writing in the Notices of the American Mathematical Society (AMS), said "He had an extremely powerful, almost other-worldly ability of abstraction that allowed him to see problems in a highly general context, and he used this ability with exquisite precision."
I think it fair to say that Jackson wrote the definitive commentary on Grothendieck's unusual life and his remarkable contributions to mathematics. It is a two-part series in the Notices of the AMS, October-November 2004:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
Whether Grothendieck's contributions involved a "higher" level of abstraction, or simply "leveraged" the current level in exceptionally original and productive ways could be a subject for debate I guess...I think it is fair to say that the level of abstraction Grothendieck brought to algebraic geometry in the 1960's was arguably "higher" than the existing state, and that this "higher" level of abstraction is what everyone at the time knew would be necessary in order to find a context for the Weil Conjectures that would make them proveable. "Everyone" "knew" but it was Grothendieck had the vision required to see the way.
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neebee
Social climber
calif/texas
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Nov 18, 2014 - 09:37pm PT
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hey there say, ... wow, i wish my daddy would have talked to me more, i always wanted to understand how and when, all this math and all the etcs, became part of him...
and wished i could have had him share about it all... :(
thanks for posting, guys...
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