
Who Is Alexander Grothendieck? (2008) [pdf] - benbreen
https://www.ams.org/notices/200808/tx080800930p.pdf
======
bollu
I am currently learning modern algebraic geometry (scheme theory) from Ravi
Vakil's algebraic geometry in the time of COVID:
[https://math216.wordpress.com/agittoc-2020/](https://math216.wordpress.com/agittoc-2020/).

It's a great ongoing course that offers amazing intuition into scheme theory.
We're divided into "working group(oid)s" where we discuss the mathematics and
solve the weekly homework assigned by Ravi. Best of all, anyone [all over the
world] can join in on the fun! You get added to a discord and a zulip group,
where all the discussion happens.

My favourite article about Grothendieck is the one titled "The Grothendieck-
Serre Correspondence": [https://webusers.imj-
prg.fr/~leila.schneps/corr.pdf](https://webusers.imj-
prg.fr/~leila.schneps/corr.pdf)

This article describes both the mathematics, and the lives of Grothendieck and
Serre through their exchange of letters. It's powerfully written, and provides
great insight into both the mathematics and the two as people. The last line
of the article is absolutely beautiful.

~~~
markhollis
I'm also learning some algebraic geometry at the moment:

Some resources I have used are:

* video lectures by Richard Borcherds

* Lecture notes by Andreas Gathmann ([https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-200...](https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf))

* This blog: [https://rigtriv.wordpress.com/ag-from-the-beginning/](https://rigtriv.wordpress.com/ag-from-the-beginning/)

* An infinite large napkin, by Evan Chen ([https://venhance.github.io/napkin/Napkin.pdf](https://venhance.github.io/napkin/Napkin.pdf))

It also found it helpful to learn some (algebraic) number theory, to get a
sense of where some of the motivation comes from (e.g. elliptic curves,
modular forms). Grothendieck's work is abstract, but he was always motivated
by concrete problems (e.g. Weil conjectures).

------
bhopballooo
Stories like these suggest to me that genius level creativity equals intrinsic
motivation plus time.

They also demonstrate that such motivation cannot be adjusted at will. It
amounts to one's deepest understanding of where personal progress or best
direction lies. Thus when Grothendieck or Swedenborg had religious experiences
later in life their technical output ceased because their motivations had
changed irrevocably.

~~~
ChrisLomont
> their technical output ceased

I think Grothendieck was still writing multi-1000-page mathematical
manuscripts up until he died.

I know there are such unpublished manuscripts floating around the mathematical
community.

As Wikipedia states " he lived secluded, still working tirelessly on
mathematics until his death in 2014".

[https://en.wikipedia.org/wiki/Alexander_Grothendieck](https://en.wikipedia.org/wiki/Alexander_Grothendieck)

~~~
bhopballooo
Good point and it's not cut and dried but the character of his output had
changed to the extent that it was not published in technical journals as it
had been:

 _> Relatively little of his work after 1960 was published by the conventional
route of the learned journal, circulating initially in duplicated volumes of
seminar notes; his influence was to a considerable extent personal. His
influence spilled over into many other branches of mathematics, for example
the contemporary theory of D-modules. (It also provoked adverse reactions,
with many mathematicians seeking out more concrete areas and problems.)_

My guess would be that he was working on himself rather than on the
mathematics. Mathematics was merely the means and the medium. But I haven't
read the material...!

~~~
ChrisLomont
>but the character of his output had changed to the extent that it was not
published in technical journals as it had been

Grothendieck didn't publish much in journals at all, ever. He was famous for
writing books, not technical articles, such as the EGA and SGA series. His
later work was the same, and the resulting manuscripts were widely sought
because they were really good stuff.

Before he died he also claimed to have many tens of thousands of pages more
that would never see the light of day, and people have sought those documents
ever since.

I'm quite familiar with his output because my PhD is in algebraic geometry,
and I've worked through his (known) writings extensively.

Here [1] is his complete list of publications. You'll notice by far his output
is books and lecture notes, with very few papers at any point in his career.

[1] [https://webusers.imj-
prg.fr/~leila.schneps/grothendieckcircl...](https://webusers.imj-
prg.fr/~leila.schneps/grothendieckcircle/AG-bibliotex.pdf)

~~~
bhopballooo
Thanks!

------
ur-whale
Grothendieck's work is, for me, a rather perfect example of what's happened in
the 20th century to mathematics, where the boundaries of knowledge have been
pushed so hard and so far that there is almost no hope for anyone to even
begin to understand the substance of the work outside a very tiny circle of
people who pretty much dedicated life to that arcane corner of the field.

I really wish vulgarization was an actual scientific disciple rather than
being done in an ad-hoc fashion by people who happen to be good at it.

I also really wish I could read a vulgarized summary of Grothendieck's body of
work, and specifically:

    
    
        . why everyone is so excited about the work
        . what potential practical application do (or will) exist.

~~~
impendia
As another commenter mentioned, currently Ravi Vakil is teaching a course in
modern algbraic geometry.

The course is largely based on Vakil's book in progress, which he titled _The
Rising Sea_ , after a quote from Grothendieck:

> I can illustrate the ... approach with the ... image of a nut to be opened.
> The first analogy that came to my mind is of immersing the nut in some
> softening liquid, and why not simply water? From time to time you rub so the
> liquid penetrates better, and otherwise you let time pass. The shell becomes
> more flexible through weeks and months — when the time is ripe, hand
> pressure is enough, the shell opens like a perfectly ripened avocado! . . .

> A different image came to me a few weeks ago. The unknown thing to be known
> appeared to me as some stretch of earth or hard marl, resisting penetration
> ... the sea advances insensibly in silence, nothing seems to happen, nothing
> moves, the water is so far off you hardly hear it ... yet finally it
> surrounds the resistant substance.

Here is, roughly speaking, a theorem that illustrates what modern algebraic
geometry is good for.

 _Theorem_. If you have two conic sections in the plane, which don't share a
common component, then they intersect in exactly four points.

Here a "conic section" is just any quadratic equation in two variables, e.g.
x^2 + y^2 + 3xy + 4x = 1. "Common component" means that the conic sections are
identical or overlap in an entire line.

Now, here are some "examples".

Concentric circles, e.g. x^2 + y^2 = 1 and x^2 + y^2 = 4. They don't intersect
at all, right? Ah, except you forgot to count points over the _complex_
numbers, where they do.

Tangency, e.g. y = x^2 and x^2 + (y - 1)^2 = 1. These curves are tangent at
(0, 0), so you have to count this point with multiplicity.

One major goal of modern algebraic geometry is to wrap these sorts of
considerations into the foundations. So you have to work _a lot_ harder to
even say what a conic section _is_ , or what it means for two of them to
intersect. But, once you've laid the foundations in this manner, there are no
special cases.

~~~
eafer
> Concentric circles, e.g. x^2 + y^2 = 1 and x^2 + y^2 = 4. They don't
> intersect at all, right? Ah, except you forgot to count points over the
> complex numbers, where they do.

How is this possible? Are you saying that there are pairs of complex numbers
(x,y) such that x^2 + y^2 = 1 = 4?

~~~
alcolade
You need to also count "points at infinity" in the "projective" plane. So the
"projectivized" equations are actually X^2 + Y^2 = 1 _Z, and X^2 + Y^2 = 4_ Z.
The intuition is similar to how 2 parallel lines will meet at the horizion
(infinity).

~~~
eafer
Thanks, I think I got it.

------
m4lvin
If you read German or just want more pictures, here are slides and handouts
from a talk about Grothendieck
[https://jonathanweinberger.wordpress.com/2014/08/27/alexande...](https://jonathanweinberger.wordpress.com/2014/08/27/alexander-
grothendieck-necessary-solitude/)

------
MaxBarraclough
This PDF weighs 11MB, for no good reason (there are few images in the document
but nothing to justify 11MB). The web server is pitifully slow. Curiously,
curl tells me the HTTPS cert has expired, but Firefox doesn't.

Unfortunately I wasn't able to find a decent mirror.

~~~
yesenadam
This 500KB version is the first result on google scholar searching by title.

[https://www.jeanpierrevarlenge.com/app/download/826854410/Wh...](https://www.jeanpierrevarlenge.com/app/download/826854410/WhoIsAGrothendieck.pdf)

Also I think it should say (2006) not 2008:

> This article is a translation of the article “Wer ist Alexander
> Grothendieck?”, which originally appeared in German in the Annual Report
> 2006 of the Mathematics Research Institute in Oberwolfach, Germany

~~~
MaxBarraclough
That's much better, thanks. (I overthought it and tried googling for a short
extract from the article, should have done the obvious thing.)

------
seesawtron
>> Alexander Grothendieck was a mathematician who became the leading figure in
the creation of modern algebraic geometry. His research extended the scope of
the field and added elements of commutative algebra, homological algebra,
sheaf theory and category theory to its foundations, while his so-called
"relative" perspective led to revolutionary advances in many areas of pure
mathematics. He is considered by many to be the greatest mathematician of the
20th century.

Wiki:
[https://en.wikipedia.org/wiki/Alexander_Grothendieck](https://en.wikipedia.org/wiki/Alexander_Grothendieck)

~~~
throwgeorge
what is the point of this? usually this type of comment is when there's
something inscrutable in the title of a post but the title of this post is
literally "Who Is Alexander Grothendieck?"; if you didn't know who he was
before clicking you certainly know afterwards.

~~~
seesawtron
When the link doesn't load and yet you are stuck with this boiling question in
the title, you start to look for answers in the comments to decide how long
you are willing to wait for the said link.

