6 ms·

Peter Scholze and the Future of Arithmetic Geometry (2016)

67 points by kercker 9y ago

Chris2048 9y ago
Yeesh. I wish there were "for Dummies" books on this stuff..
- auntienomen 9y ago
  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/S0273-0979-2015-01515-6/ http://www.ams.org/journals/bull/2016-53-01/S0273-0979-2015-...
  - Chris2048 9y ago
    > 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!?
    - lmm 9y ago
      They do, gradually. Good review articles are respected and people put a lot of effort into them; eventually stuff percolates down to textbooks and so on. The kind of thing that an advanced grad student would struggle with a few decades ago becomes an undergrad exercise today. But meanwhile the frontier keeps advancing.
    - dkarl 9y ago
      “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.
quickthrower 9y ago
That was a real joy to read. Although I can't say I understand the maths I did get a sense of the man and he seems like the colleague/friend we'd all love to have.
m00n 9y ago
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-spaces https://mathoverflow.net/questions/65729/what-are-perfectoid...
- auggierose 9y ago
  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 9y ago
    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.
contingo 9y ago
> “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.
- Chris2048 9y ago
  I think the trick is figuring when underlying details are just details, and when you miss an important understanding in the underlying foundations that might spawn misunderstandings. I guess a good lecturer is one who can tell you - insisting on learning everything is inefficient, although less work for your lecturer..
- kefka 9y ago
  > 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.
  - 9y ago
Chris2048 9y ago
New life goal: understand at least one of the math articles on quanta..
adamnemecek 9y ago
"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.

67 points by kercker 9y ago

Chris2048 9y ago

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

auntienomen 9y ago
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/S0273-0979-2015-01515-6/ http://www.ams.org/journals/bull/2016-53-01/S0273-0979-2015-...
- Chris2048 9y ago
  > 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!?
  - lmm 9y ago
    They do, gradually. Good review articles are respected and people put a lot of effort into them; eventually stuff percolates down to textbooks and so on. The kind of thing that an advanced grad student would struggle with a few decades ago becomes an undergrad exercise today. But meanwhile the frontier keeps advancing.
  - dkarl 9y ago
    “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.

quickthrower 9y ago

That was a real joy to read. Although I can't say I understand the maths I did get a sense of the man and he seems like the colleague/friend we'd all love to have.

m00n 9y ago

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-spaces https://mathoverflow.net/questions/65729/what-are-perfectoid...

auggierose 9y ago
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 9y ago
  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.

contingo 9y ago

> “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.

Chris2048 9y ago
I think the trick is figuring when underlying details are just details, and when you miss an important understanding in the underlying foundations that might spawn misunderstandings. I guess a good lecturer is one who can tell you - insisting on learning everything is inefficient, although less work for your lecturer..
kefka 9y ago
> 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.
- 9y ago

Chris2048 9y ago

New life goal: understand at least one of the math articles on quanta..

adamnemecek 9y ago

"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.