
Peter Scholze and the Future of Arithmetic Geometry (2016) - kercker
https://www.quantamagazine.org/peter-scholze-and-the-future-of-arithmetic-geometry-20160628
======
m00n
I remember the buzz in Germany's math graduate community, when word of
Scholze's perfectoid spaces began to go around. Reading groups spawned
everywhere trying to get a grasp on the technicalities:

Work in number theory very often deals only with one of two fundamentally
different settings.

* Either objects where multiplication with any natural number can be inverted ('characteristic 0', examples for such objects are the rational or complex numbers or "something inbetween the two"),

* or objects where a certain prime number p has a special role; namely, the multiplication by p is the 0-map. This sounds horrible, but it actually has a great implication: (a+b)^p = a^p + b^p, because the middle binomial coefficients are multiples of p. This makes x -> x^p a multiplicative and additive (!) map, the FROBENIUS.

Scholze introduced a way to pull the Frobenius map over to characteristic 0.
He could do this 'tilting' in towers and in this way compared the theory of
towers in characteristic 0 and p. For details, see his famed answer here [1].

Very soon it became clear, that this tool had remarkable applications and his
thesis explored only one of them: a proof of the monodromy-weight conjecture
in characteristic 0 by tilting results in characteristic p.

This result alone made the characteristic 0 neck hair stand up :-)

[1] [https://mathoverflow.net/questions/65729/what-are-
perfectoid...](https://mathoverflow.net/questions/65729/what-are-perfectoid-
spaces)

~~~
auggierose
Just skimming the answer by Peter Scholze reminds me why I switched from
Mathematics to Computer Science: Too much to learn for too little benefit.

~~~
Chris2048
Are Scholzes lectures online? They are described as intuitive and accessible
to undergraduates?

While I'm a little skeptical[0] of the tone of this article, I hold out hope
this just is the Feynman (is able to communicate complex ideas well) of Math.

Sometimes, I'd like time with someone with advanced knowledge of such things,
and be able to shoot my naive questions at them. I'm sometimes shocked by how
subjects like this are not just non-intuitive, but outright _obs_ by curious
omissions[1] by experts in their explanations.

For example, look on HN/mathOverflow, and ask for an _intuitive_ explanation
of something, and you'll often get "there is no such thing, you just have to
work through it - math is as simple as it can be". Then, someone eventually
provides an explanation that _is_ more intuitive than the norm, proving that
it was failure to communicate in simpler terms in the first place.

this article:

"Unlike many mathematicians, he often starts not with a particular problem he
wants to solve, but with some elusive concept that he wants to understand for
its own sake"

"[Scholze] would never lose himself in the jungle, because he’s never trying
to fight the jungle. He’s always looking for the overview, for some kind of
clear concept."

I can't find them now, but I'm sure I've read sentiments from mathematicians
such as "don't try to get an 'intuition', there is no such thing in advanced
math where there are few mappings to real life things, and such metaphors will
only create misunderstandings" and "we'd all like a 'map' of mathematics, but
math is not neatly ordered like a landscape, it's not possible to even
visualise all the links between math fields".

So this guys method is exactly what other mathematicians warn against? Trying
to understand?

[0] - He is depicted so far ahead that mathematicians look forward to him
entering their field? Even if true, I'm skeptical most would admit something
like that - also anything that appears in wired will have a certain minimum of
hot-air..

[1] - e.g. the norm distribution id the convergence of the binomial
distribution (which can in turn, be constructed), which is why it turns up so
often whenever a large number or random variables are involved. I've yet to be
taught stats where this is explained, before I realised it was as if norm was
just something that appeared by magic.

~~~
qmalzp
If you're talking about Scholze's lectures about his cutting-edge work, there
is not a chance they are accessible to undergraduates; when I attended one as
a fourth-year PhD student specializing in algebraic number theory, I would say
I only kind of got the gist of it.

Re: your skepticism... The guy is a once-in-a-generation talent; his
constructions were able to vastly simplify multiple very long, very
complicated proofs that _groups_ of the top people in this field were working
on. This is in a field (algebraic number theory) which is considered one of
the more saturated and technically difficult within all of mathematics
(admittedly, I am likely biased on this point). That being said, all of his
work so far has been in the ballpark of Langlands/p-adic/arithmetic geometry,
so I would be surprised if he achieved significant results that strayed _too_
far from this stuff.

I'm not sure what you mean about Feynman; Peter's genius is not so much his
ability to _communicate_ complex ideas in a simple way, but rather he was able
to come up with constructions (or if you like, abstractions) which
compartmentalize the complex ideas in the right way so that they are easier to
deal with. To make an analogy with computing, think of the concept of a
"thread". Without the concept of a thread, you'd have to do so much manual
maintenance that you could never dream of building say Google. Scholze's
perfectoid spaces are analogous; their definition would have been understood
by mathematicians 50 years ago, but no one really got that this was the right
thing to consider.

~~~
soVeryTired
Sounds more like a Grothendieck than a Feynman.

~~~
qmalzp
I concur.

------
contingo
> “To this day, that’s to a large extent how I learn,” he said. “I never
> really learned the basic things like linear algebra, actually — I only
> assimilated it through learning some other stuff.”

This was interesting to me. My math profs always admonished us to ensure
foundations are completely watertight before advancing to the next thing in
tiny increments. I've absorbed this to the point where I perhaps get stuck
filling in inconsequential gaps at roughly the same level, kitting out base
camp as fully as possible but postponing the ascent.

I have no dreams of becoming a professional mathematician, but maybe I'd have
quicker insights and more creative ideas if I tried this approach too,
tackling something impossible and working backwards.

~~~
kefka
> My math profs always admonished us to ensure foundations are completely
> watertight before advancing to the next thing in tiny increments.

Of course someone who gets paid to teach would highly stress learning tidbits
slowly and excruciatingly. That's their economic incentive. Schools also
stress learning detritus for "learning's sake", even if the very people who
teach it can't properly explain what it's actually for.

I also have learned compsci with his similar methods of finding interesting
areas and digging in. I know my programming knowledge has holes, and I fill
them in as I come to them. I like to know how things fit together, even if
they are cross-domain and seemingly disparate. I come on in, and go "see these
two areas are pretty similar, let me show you what can be done". And then I
look like a miracle worker, because I see the generalities.

Frankly, professors would be more useful to me, if I could purchase their time
by the hour over issues I don't understand. I can teach myself most things.
Sometimes, a professional helps with the jump-start to get a good grasp.

~~~
contingo
> Of course someone who gets paid to teach would highly stress learning
> tidbits slowly and excruciatingly. That's their economic incentive...

Well, I can see that's a factor, but not necessarily an overriding one. As
someone who's taught at uni myself (not pure maths) we are not usually that
cynical or fond of serving up "detritus". It's not as if we lack for valuable
and interesting stuff to teach if we go through foundations too quickly, and
we aren't paid just for teaching. Anyway, I think most do benefit from a quite
painstakingly incrementalist approach to maths; me taking that too far and
sometimes getting stuck is a personal failing.

(I recall a quote by a colleague of the group theorist Simon Norton, who
famously suffered a career collapse/hiatus after a series of brilliant
results, something along the lines of him having opened a doorway into a
wondrous realm of new mathematics, but ending up stuck there, at the doorway,
obsessed by the details of the doorframe.)

If I was teaching a linear algebra course, I'm not going to say "by the way
you can skip this subject entirely because it will just fall out of your
working backwards through Wiles' proof of FLT". For those of us without a
once-in-a-generation mind I think the traditional approach is the right one. I
was only trying to say, I personally sometimes get stuck and it will be
interesting to try the opposite approach.

> Frankly, professors would be more useful to me, if I could purchase their
> time by the hour

If you go to a good uni, at least by postgrad level you do have that kind of
access, and, if you get along, you retain it for free after you leave.

~~~
kefka
> Well, I can see that's a factor, but not necessarily an overriding one. As
> someone who's taught at uni myself (not pure maths) we are not usually that
> cynical or fond of serving up "detritus". It's not as if we lack for
> valuable and interesting stuff to teach if we go through foundations too
> quickly, and we aren't paid just for teaching. Anyway, I think most do
> benefit from a quite painstakingly incrementalist approach to maths; me
> taking that too far and sometimes getting stuck is a personal failing.

Possibly so, but I never went in any of the grad programs. Most of the lower
classes are taught by AI's and contract-based "instructors" paid by the uni on
a per credit-hour basis. And much of the time, the department shovels the
syllabus and required areas to them for the students.

And unfortunately, this avenue of teaching very much shows. You have
instructors who have some semblance of caring, but not terribly much. They
teach weeder classes with the intent of failing much of the class. Whomever is
teaching isn't always able to explain what's going on in an area - they can do
the process, but can't explain why their actions work.

Perhaps it is a jaded viewpoint. But after spending way too much money in
"Higher Ed", along with working at an institution, I know the game. And I'm
sure it's better if you're a post-doc with prestige or on that track. But the
rest of us are spoon-fed bland crumbs these days, and pumped-and-dumped for
excessive scholastic loans to get a job to pay the loans back with.

> If you go to a good uni, at least by postgrad level you do have that kind of
> access, and, if you get along, you retain it for free after you leave.

Yep, and if you're not on that track, the access isn't there. I'd be willing
to pay for it directly. Google had a program quite a while back, of paying
experts for direct guidance in specific fields. Too bad they cancelled it.

~~~
contingo
I did have bad experiences with bad lecturers as an undergrad (hello, here is
a handout, now I will project the handout on a screen, now I will read what's
on the screen without any elaboration, goodbye), but they were the exception.
Obviously this depends hugely on the exact institution in question. And yes,
many are now increasingly functioning as blandly corporate battery student
farms...

> Yep, and if you're not on that track, the access isn't there.

Actually I'd also be interested in such a scheme, now that I'm exploring ideas
far away from my original research area. Although if you have a bona fide
interest to discuss something technical and specific with an academic who has
the relevant expertise, I've found they can be pretty approachable, even if
you email them out of the blue to ask for a chat... but I do have the right
sort of background to do that I suppose.

------
adamnemecek
"I never really learned the basic things like linear algebra, actually — I
only assimilated it through learning some other stuff."

The hacker attitude seems to be the way to go even in fundamentally anti-
hacker environments.

------
Chris2048
New life goal: understand _at least one_ of the math articles on quanta..

------
Chris2048
Yeesh. I wish there were "for Dummies" books on this stuff..

~~~
auntienomen
I think most mathematicians feel the same. (Although as the article points
out, Scholze did write a 'for Dummies' version of Harris & Taylor's local
Langlands paper.)

For perfectoid spaces, the most accessible thing I know of is Jared
Weinstein's article in the Bulletin of the AMS.

[http://www.ams.org/journals/bull/2016-53-01/S0273-0979-2015-...](http://www.ams.org/journals/bull/2016-53-01/S0273-0979-2015-01515-6/)

~~~
Chris2048
> I think most mathematicians feel the same

About this in particular, or advanced mathematics?

If the latter, why don't those mathematicians do it one _they_ understand!?

~~~
dkarl
_“You have only some kind of limited capacity in your head, so you can’t do
too complicated things.”_

A Peter Scholze quote from the article that captures for me one of the two
common aspects of programming and mathematics, that we make progress by
building better and better conceptual machines (the intellectual equivalents
of levers, pulleys, inclined planes, etc.) to multiply our limited limited
capacity to solve more and more difficult problems.

Edit/PS: You can see how highly mathematicians respect simplification by the
story in the article about how Peter Scholze first became internationally
famous in the mathematical community, by discovering a new solution for an
already solved problem. His solution was much shorter (and apparently easier
to follow,) and that was taken as a sign of exceptional ability, just as it is
in programming.

