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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

by 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.
  - throwawayjava 9y ago
    > So this guys method is exactly what other mathematicians warn against? Trying to understand? No, I don't think so. The "there's no intuition" advice is really good advice for undergraduates or lay people first encountering abstract concepts in pure mathematics. In fact there is an intuition, but the intuition is usually sort of on its own terms -- not so much grounded in physical life experiences. In other words, you will get an intuitive grasp on the ideas but that intuition will probably not be an analogy to your previous experiences in everyday life. When mathematicians say "there's no intuition" they usually mean "there's no physical experience that corresponds to the underlying intuition, and it's not really like anything you've seen before. It's a new experience and you need to experience it on its own terms. Stop trying to ground yourself in classical mechanics or whatever." But that's quite the mouthful, so better to just say "there's no intuition; just focus on working with the objects and you'll eventually get comfortable". This quote from the article captures it well, I think: > "Now I find real numbers much, much more confusing than p-adic numbers. I’ve gotten so used to them that now real numbers feel very strange."
  - qmalzp 9y ago
    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.
- kafkaesq 9y ago
  Too much to learn for too little benefit. It all depends on what you want out of life.
  - knn 9y ago
    Exactly - if you want to make choices based off what's economically the best (studying CS), that's fine. But there's so much beauty in the world that is "impractical" to reach for (like studying music, art, seeing nature, etc)
    - qmalzp 9y ago
      I'm happy with my path: Youthful idealism -> number theory Mature pragmatism -> pivot to CS It was entirely luck that my passion was so closely related to the current gold rush.
  - auggierose 9y ago
    Yep, it does. If you are ready to give one month of your life to understand "perfectoid spaces", that might be a wise choice, you can only really tell once you've done it, because there doesn't seem to be an upfront way of knowing what you are going to get out of it. On the other hand, I am currently investing two weeks into learning "geometric algebra" because I found a book that tells me in its introduction why it's worth for me to learn it.

by m00n 9y ago

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.
- throwawayjava 9y ago
  > So this guys method is exactly what other mathematicians warn against? Trying to understand? No, I don't think so. The "there's no intuition" advice is really good advice for undergraduates or lay people first encountering abstract concepts in pure mathematics. In fact there is an intuition, but the intuition is usually sort of on its own terms -- not so much grounded in physical life experiences. In other words, you will get an intuitive grasp on the ideas but that intuition will probably not be an analogy to your previous experiences in everyday life. When mathematicians say "there's no intuition" they usually mean "there's no physical experience that corresponds to the underlying intuition, and it's not really like anything you've seen before. It's a new experience and you need to experience it on its own terms. Stop trying to ground yourself in classical mechanics or whatever." But that's quite the mouthful, so better to just say "there's no intuition; just focus on working with the objects and you'll eventually get comfortable". This quote from the article captures it well, I think: > "Now I find real numbers much, much more confusing than p-adic numbers. I’ve gotten so used to them that now real numbers feel very strange."
- qmalzp 9y ago
  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.
kafkaesq 9y ago
Too much to learn for too little benefit. It all depends on what you want out of life.
- knn 9y ago
  Exactly - if you want to make choices based off what's economically the best (studying CS), that's fine. But there's so much beauty in the world that is "impractical" to reach for (like studying music, art, seeing nature, etc)
  - qmalzp 9y ago
    I'm happy with my path: Youthful idealism -> number theory Mature pragmatism -> pivot to CS It was entirely luck that my passion was so closely related to the current gold rush.
- auggierose 9y ago
  Yep, it does. If you are ready to give one month of your life to understand "perfectoid spaces", that might be a wise choice, you can only really tell once you've done it, because there doesn't seem to be an upfront way of knowing what you are going to get out of it. On the other hand, I am currently investing two weeks into learning "geometric algebra" because I found a book that tells me in its introduction why it's worth for me to learn it.