Why is Alexandre Grothendieck so revered by mathematicians

most pure mathematicians don't care about applied stuff...that is what applied mathematics is for

Sure, so you shouldn't praise them. Euler is a genius cause his mathematics has found so freaking many real life applications.

Grothendieck's mathematics has not, at least yet, so he and algebraic geometry are nearly on Ramanujan's level by now.

What are you on about? Mathematicians think he's a great Mathematician. Mathematicians don't care about real life applications.

>Be one of the founders of an entire field
>"LOLZ u rnt a genius until it's applied"
>A century later someone else applies your work to a "real life" problem
>Magically become a genius
>Literally nothing about your work has changed
>In fact you've been dead for the past 3 decades

You're a retard.

Never mind him, user. He's made his attitude perfectly clear.

For him, the idea of appreciating math on an aesthetic level is absurd.

Also this guy gets it.

>It doesn't help at defense industry, it doesn't help at medicine.
Almost nothing in math and science helps those things.

Why OP? Why? Why you keep making these threads?

I believe you meant to say almost nothing doesn't help them.

Half Jewish from his father's side, so not really Jewish.

Grothendieck developed incredibly powerful tools in algebraic geometry, which today we are still trying to harness the full power of.

He was also extraordinarily gifted with handling abstraction, and his rather unique style of solving problems (making a series of apparently weak reductions and to the problem until it and many problems like it literally dissolve into triviality) is deep and inspiring.

There is an article by McLarty called "The Rising Sea" about some of Grothendieck's ideas and contributions to mathematics, well worth a read.

This, that's the issue with geniuses, it takes the brainlet engineers a few hundred years to figure out what the fuck the mathematicians are doing.

I understand your point about matrilinearity, but if you're jewish enough to get railroaded into a concentration camp, then you're jewish.

True, since I thought to check the wiki, it is never expressly stated in that sketch that Grothendieck himself is kept in camps /because he is "jewish"/, but the idea that there is no possible situation in which he might have been Concentrated for that exact reason, stretches credulity. Let's look at this. The father was a /jewish anarchist ffs/, which is everything that the nazis hate. Of course they would naturally want to destroy his entire family, whatever their circumstances, if they had had it within their power to do so.

As it happened, the mother and the son were kept at various other camps for being "undesirable foreigners", as opposed to jews. But it is very easy to imagine a situation where the entire Grothendieck family missed the right train only to board the wrong one together, rounded up as both "jews" (father and son), and "jew associate" (mother).

What? I heard it was his mother who was teaching him those jewish we-are-above-others things.

What's so special about algebraic geometry? Also Grothendieck has no theorems, theories or whatsoever outside that algebraic-topological-so-called-geometrical abstract nonsense, which is pretty disturbing. Every other real genius has some descent results there. Remember Poincare, Hilbert, Kolmogorov, ... But Grothendieck? Why the fuck algebraic geometry is something we should praise?

You're probably mis-remembering, but let's suppose for discussion that that anecdote is true. However much she might profess to be a radical, biological imperatives have different plans in mind for humans. And most people need some outlet for their own private chauvinism, even if it's only in their own heads.

It wouldn't be the first time that a woman tried to get in good with the husband's culture, in a bid to mate "up". They did produce a child with a superior intellect, after all.

An aberrent version of the above: it often happens, especially in America, that low class, undesirable, or simply fat white women (who black guys happily go for when they can pull it off) will "act black" to a certain extent, to try and ingratiate themselves to their apes. You might say that they ape their apes.

What about the Grothendieck–Katz p-curvature conjecture?

>Grothendieck–Katz p-curvature conjecture
> It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form.
>The general case remains unsolved
Well, ok

Those people had the luxury of living at a time where you didn't have to specialize (not much at least).

Dude, Grothendieck's unpublished ideas are a million times more valuable than most mathematicians' published work. AFAIK he didn't publish anything about n-categories either (and certainly didn't work out the general theory) but he had the idea, which is huge.

he was a genius

>Those people had the luxury of living at a time where you didn't have to specialize (not much at least).
Kolmogorov lived at his time, Hilbert worked at 20th century too. What a poor excuse.

>Dude, Grothendieck's unpublished ideas are a million times more valuable than most mathematicians' published work
Wow, isn't that kind of religious statement?

1. He did prove theorems in functional analysis during his thesis
2. He developed entire new fields of study (ie. tools) in algebraic geometry and asked a great deal of interesting questions, which, from a researcher's point of view, is just as valuable as proving theorems: a researcher doesn't care so much about what you do as they do about how they can use what you do for whatever it is they are working on

> a literary theorist doesn't care so much about what you do as they do about how they can use it for what they're working on

So you've convinced me that grot was extremely important to the incestious circlejerk that is ""academic """ research"""""

why do you give a fuck what academia cares about? you're clearly too idiotic to do anything worthwhile

Don't think he did a significant work in functional analysis though. It seems the only one who praised it was his own advisor.

As mathematicians better understand Mochizuki's work, some are saying that IUT is how number theory will be done in the future.

By this metric, Mochizuki will be the Grothendieck of number theory. And like Grothendieck, Mochizuki completely reinvented his field.

>Kolmogorov lived at his time, Hilbert worked at 20th century too. What a poor excuse.

Grothendieck was born 25 years after Kolmogorov, which for math in that period was not a trivial difference. You could argue, ok, Grothendieck could have chosen an area of more fundamental importance than algebraic geometry, or he could have branched out a little more. But I'm not aware of any mathematicians since him who have had the same combination of wide-ranging vision and technical ability.

>Wow, isn't that kind of religious statement?

No...I literally just gave an example of one. But of course, if you think it's just "abstract nonsense" then you won't be able to appreciate it.

You could say Grothendieck's greatest contribution was essentially to show people what "applied category theory" looks like.

Who?

this lmao

>By this metric, Mochizuki will be the Grothendieck of number theory.

Grothendieck's work changed number theory too. Where do you think the Arithmetic Geometry IUT is based off of came from?

if IUT is successful will Mochizuki surpass Grothendieck?

Jeebus Im getting more and more impressed by this guy.

Highly doubtful...even if it's a great achievement it's still just one isolated achievement. Voevodsky has a greater claim than Mochizuki to that status IMO.

what is Voevodsky doing?

Drinking Vodka

Univalence/Homotopy type theory (which has already revolutionized math) and some stuff with motives that I'm not familiar with.

no seriously, is he doing revolutionary work or something?

Vodka drinking is serious business

isn't Univalent Foundations considered constructive mathematics?

Yes? Not sure what your point is. It accommodates everything you can prove in normal set theory while also giving a natural framework for infinity-groupoids, which is huge.

Nothing, I just thought constructivist/intuitionist mathematics was looked down upon among the pure mathematics community for some reason

If you're talking about finitism, V=L, that kind of stuff then yes. Intuitionistic type theory is constructive but it's a hell of a lot more elegant. It's not super well known compared to set theory but it will be.

>8245325

It's going to be interesting to see the future of Homotopy Type Theory and Univalent Foundations as an alternate to set theory

because their are still alot of mathematicians that are loyal to set theory

Hugh Woodin is creating his V=Ultimate L theory

>because their are still alot of mathematicians that are loyal to set theory

idk about "loyal." Some fields (analysis, diffgeo, combinatorics, universal algebra, etc.) haven't really succumbed to the category theory meme yet, but it's more out of laziness than anything. No one gives a shit about Woodin or set theorists' work in general; it ceased to be foundational a long time ago.

Univalent Foundations isn't Homotopy Type Theory.
Vovo sperged out that the comp sci kinda stole his project, wrote that book and distanced himself from the latter.

He also wrote a Wikipedia article about his project, and you can read it by him
en.wikipedia.org/wiki/Univalent_foundations

>Univalent Foundations isn't Homotopy Type Theory.

What is the difference though, honestly? Seems more like a political split.

He united classical problems in geometry, finally giving a setting where both affine and projective varieties could be easily classified and compared, and in fact extended the field so that it can encompass algebraic number theory as well, after many failed attempts to do so.

The generalization even in the relatively "classical" cases also allowed for geometric realizations of algebraic infinitesimals using the same techniques. I.e. it is able to talk about objects which look the same as spaces, but one may have information about infinitesimals stored in it. In this way, we can distinguish "functions" even if they take the same values. This allows for much more accurate algebraic description of singularities.

In doing so, he helped lay the groundwork for applying geometric and topological techniques to numerical problems, such as using cohomology, a way of attaching "numerical" data to spaces, to detect information about solving problems about solutions to equations in commutative rings in a totally general setting. The ability to transport numerical questions into spatial questions and hence made spatial tools relevant made things like Wiles' proof of FLT possible.

He also extended galois theory to be able to talk about a much more general class of algebras.

In doing so, he also helped take category theory beyond just a nice descriptive tool, but actually showed how essential it is in algebraic and geometric mathematics. For example, a "side-effect" of his generalization of spaces to sites not only met the needs of developing a way to apply spatial techniques to things which are not obvious spatial, but in fact leads directly to topos theory.

Topos theory in many ways vastly generalizes constructions like the Lindenbaum-Tarski algebra to FOL, and provides an infinite class of categories other than Set where higher-order logic can be interpreted, and is closely related to type theory and certain forms of the lambda calculus.

so what is the current progress of Univalent Foundations right now?

If you include HoTT - I get the feeling that people are working out the details, but that progress has slowed somewhat. At least, they haven't been writing a whole lot lately. You can check this out too:

ncatlab.org/homotopytypetheory/show/open problems

Homotopy Type Theory is a particular type theory, a variant of Intuitionistic type theory (and as such capable of expressing fundamental mathematics, and thus general mathematics), and with semantics in higher category theory, and with primitive concepts that make encoding of homotopy theory possible.

Univalent foundations is the perspective of taking homotopy types as primitive concepts (as realized in Homotopy Type Theory), e.g. as opposed to sets, and as such it's a system that can be tried and implemented (via Homotopy Type Theory, say). The creator is a (weird edgy Russian) fields medal winning homotopy theorist.

The people in the former subject (comp sci people) do basic formal how to realize this stuff math. One of their main challanges is to realize the Univalence Axiom as a constructive principle, give meaning to it.
The latter one guy does and thinks homotopy type theoretically, abstract math. He came up with that axiom, and in higher category, stating it just means "I'll only consider systems where it holds". Just like in set theory, if you require well-foundedness for your sets, then you just disregard all set concepts that doesn't have it (like non-well-founded set theory).
He's more on the categorical semantics side and not tied to the above type theory.

I don't think there is or can be quick enough progress to make it really interesting not directly involved with the people who work on it. To be quite honest.

>He's more on the categorical semantics side and not tied to the above type theory.

okay, but the HoTT community is still actively exploring variations on the system. I don't think anybody thinks it's reached its "final form" yet.

No, and I'd say the problem is that we're miles away from anything that you'd use for normal mathematics. Or programming.

I think it would be cool if you could actually start thinking in homotopy types - but then again I don't even know if there are common problems where that makes more things possible.

My hope was (is) that people use strongly typed and expressible languages. I hope the HoTT reasearch makes such things more popular.

Also, it's sort of ironic that you mention Hilbert and Poincare. A lot of Grothendieck's work directly extends theirs.

His construction of the category of schemes can be seen as a generalization of the nullstellensatz to what it "should" be (i.e. CRing and AScheme are dually equivalent, with polynomials over ACFs being a "nice" case); and as I mentioned his construction of toposes is an extension of set-theoretic model theory and its connection with geometry a la L-T and stone spaces of types, also in a more "natural" way (the adjoint functors in a geometric morphism correspond to a logical and spatial part). In fact, Deligne's theorem that a coherent topos has points is a geometric proof of the completeness theorem for FOL. Grothendieck's work on motives and cohomology of toposes is a direct descendent of Poincare's algebraic topology.

Well, regular dependently-typed languages are slowly finding applications in practical things (i.e. miTLS written in F*) for some value of "practical", at least. So maybe it's just a question of time when someone finds and application for HoTT.

Quick survey, what did you study in undergrad, and if you are in grad school (or past that) now, what are you studying/working on/building? I'm curious as to how many came out of *just* mathematics backgrounds and how many comp-sci, how many mixed, neither, etc. Also, how many do "applied" work alongside "pure" work?

I know that he did a lot of generalizations. I can take it only as an attempt to gain the fame for free or as some sort of hobby cause I'm sure that Riemann-Roch theorem did well enough without any of his generalizations, as well as galois theory, as well as anything outside alg geometry that he attempted to generalize.

>We [Dieudonn´e and Schwartz] received Grothendieck in oc-
tober, 1951. He showed to Dieudonn´e a 50 pages paper on ”In-
tegration with values in a topological group”. It was exact, but
with no interest at all. Dieudonn´e, with all the aggressiveness
he could have, ([and he could a lot] gave him a severe ticking-
off, arguing that he should not work in such a manner, just
generalizing for the pleasure of doing so...Dieudonn´e was right,
but Grothendieck never admitted it..

I (, ) am just a math hobbyist, and grad student in something rare enough on this board to give away my user status so no thanks. I've taken a few math courses in undergrad and grad tho, but not these subjects. My work is theoretical but in an "applied" field you could say.

What Grothendieck did for Algebraic Geometry alone makes him arguably the greatest mathematician of the 20th century

you really seem to hate Grothendieck alot man

a) algebraic geometry definitely could not do what it does now before Grothendeick. There was no category which included affine and projective varieties as well as integral and rational "spaces" and the appropriate morphisms between them.

b) mathematicians are not computer scientists. Having a statement that is natural and easily generalizable is as important as having something that just "works". Grothendieck's Galois theory, AG, and toposes are all more flexible and lead to generalizations more naturally than Galois', Hilbert's, and Frege's in many respects and arguably reveal "why" all these things are true in a more straightforward manner.

Yes, he proved quite a few theorems. However, we need to take a moment here to think about the goal of a mathematician, or of mathematics in general. Is it to acquire some encyclopedia of known, verified facts that we can write down in textbooks? No, it is to achieve a deeper understanding of why certain, often very simple, things are true. This is the real reason a mathematician loves proofs, and the reason why some proofs are considered better than others. A good proof completely illuminates the "why." Grothendieck did this, and did it as well as anyone of that century. He took a field on shaky foundations and unified it to a grand viewpoint, one in which one can phrase and answer a lot of questions very naturally; one so far reaching that it extends into mathematics beyond just algebraic geometry. His idea is that if a proof is not trivial, then you are not looking at things from the correct perspective. These ideas were his contribution.

That's a nice idea about how mathematics should be but it becomes so much less when you take the field of study into account.

He solved the formula that allow him to be able to grow ten dicks.

here, I'm just a pleb who stumbled upon HoTT through Robert Harper's blog, sorry.

algebraic geomertry is becoming more and more commonly used in computer vision which certainly has applications in defense

don't make baseless statements you can't back up

This is me: Math background, undergraduate but I took a lot of grad courses. Wouldn't say I do "applied" work exactly, at least not in compsci but I do programming too so I can see HoTT making potential impact on both sides.

>beta instead of esszett

>algebraic geomertry is becoming more and more commonly used in computer vision
It is not commonly used in computer vision. And what's being used is usually a groebner basis. Hilbert scheme and things like that are not practical. In practice it's always better to use some kind of neural network and get a nice result, usually faster.
>don't make baseless statements you can't back up
Oh, I think you just did one.

Why does Mathematics have to have applications?

>defense industry
the most-expensive fraud in history

This gets me every single time

>people solve problems to overcome hardship
>solutions to problems make life more convenient
>people are born in convenience, never know hardship and can afford to ask such questions as, "what is the point of solving problems to overcome hardship", legitimately unaware of why they think they can ask such questions
>next generation does not know how to solve hardship because last generation never taught them because they questioned it's use
>hardship returns
>repeat

Or people start solving problems that don't help anyone overcome hardship, and ignore the ones that do.

And, it always amazes me that the folks who tend to bitch about social welfare programs the loudest don't see that the defense industry is a giant corporate welfare system.

who is the next most influential figure in algebraic geometry after Grothendieck?

Can we not go down the path of who has it harder between mathematicians, physicists and engineers?

Uhh...Serre?

I'd say whoever did GAGA. Don't remember the guys name.

Yes, you're both referring to the same guy. This is certainly correct.

Take your /a/lgebraic geometry back to pedophile.

Fun fact: my advisor wouldn't let me refer to some result as "long known" in a paper because it was most likely due to Serre, who is still alive, and we don't want to insinuate that living mathematicians are old a dirt.

google literally already implements computer vision techniques in google maps/earth that come from algebraic geometry and representation theory to reconstruct 3d images from 2d fragments, where grassmannians show up everywhere

and there's more to algebraic geometry than grobner bases and hilbert schemes...

even aside from computer vision, AG has applications in robotics, cryptography, coding theory, the list goes on

can you back up this 'useless' claim?

and even aside from this, developments take time

number theory was 'useless' until public key cryptography.

relativity was 'useless' until nukes destroyed japan and ended their war stunt.

>it's a "what use does this have in the real world?" thread

>google literally already implements computer vision techniques in google maps/earth that come from algebraic geometry and representation theory to reconstruct 3d images from 2d fragments, where grassmannians show up everywhere
Oh, can you please show me exactly what kind of algebraic geometry solves this problem in practice?
To reconstruct 3d images from 2d fragments is an old known problem which is efficiently being solved without _any_ of algebraic geometry. Keep in mind that the data is always kind of fuzzy.

>robotics, cryptography, coding theory, the list goes on
I would like to discuss in details that so-called list too.

Who comes after Grothendieck in terms of importance in algebraic geometry?

Smallthendieck

didn't realize Foucault was also a mathematician

>not using B
GauB would be disappointed

...

what is GAGA?

archive.numdam.org/ARCHIVE/AIF/AIF_1956__6_/AIF_1956__6__1_0/AIF_1956__6__1_0.pdf

Serre wrote an influential paper connecting analytic study of complex manifolds to the algebraic study of schemes.

Kleinendick