5 ms·

What is the Birch and Swinnerton-Dyer conjecture?

33 points by subnaught 11y ago

mjcohen 11y ago
This is quite advanced. At least graduate math level. This is supposed to be a "elementary introduction to the Birch and Swinnerton-Dyer conjecture". Ha.
- cperciva 11y ago
  This is quite advanced. At least graduate math level. Yes, I learned most of this when I took graduate mathematics courses. Soon thereafter I decided that I'd be better off studying computer science... This is supposed to be a "elementary introduction to the Birch and Swinnerton-Dyer conjecture". Ha. I don't think the word "elementary" means what you think it means. In number theory, "elementary" means "not involving complex analysis"; it does not mean "straightforward".
  - jordigh 11y ago
    In this context, they are not shying away from using complex analysis. They really do mean "elementary" as in "not for number theory experts", but for a general mathematical audience.
    - cperciva 11y ago
      They mention complex analysis in passing, but I wouldn't say that they use it here.
chrispeel 11y ago
...such a talk can easily succeed in convincing the audience that there is a dry, ridiculous, but famous conjecture coming out of nothing; crazy number theorists have wasted their life to contribute to the list of partial results, which cannot even be claimed to be a long list. Hopefully I have failed in that way. Ummm, I guess that even among techies on HN, there are only a small fraction who could understand this. So the author failed with me. Any one want to do an ELI5?
- daniel-levin 11y ago
  I think understanding that sentence reduces to parsing jargon. I'll provide an explanation I hope isn't in any way condescending. The main activity of research mathematicians is theorem proving. That is, turning conjectures into theorems. In the jargon, proven theorems (and equations, bounds, counterexamples etc.) are often called 'results'. Essentially, the author is saying that with very few constructs and tools, number theorists have been able to pose a conjecture that is comparatively simple to state and understand, but is frustratingly as yet unproven, in spite of decades of effort by many, very good mathematicians. So far, many special cases ('partial results') have been proven, but the highly desired goal that is the proof of the BSD conjecture, in full generality, has not yet been achieved. The consensus is that mathematicians are quite far away from resolving the BSD conjecture, but very strongly believe it is true.
- ColinWright 11y ago
  >> ... such a talk can easily succeed in >> convincing the audience that there is >> a dry, ridiculous, but famous conjecture >> coming out of nothing; crazy number >> theorists have wasted their life to >> contribute to the list of partial >> results, which cannot even be claimed >> to be a long list. >> Hopefully I have failed in that way. > Ummm, I guess that even among techies on HN, > there are only a small fraction who could > understand this. ^^^^ > So the author failed with me. Any one want > to do an ELI5? It's not clear what you mean by "this". You might mean this small section that you quoted, or you might mean the entire article to which this is an opening preamble. If it's the first of these, then here's a free translation: A talk like this can convince an audience that the conjecture is just a wild, crazy thing, produced by wild crazy people, and it comes from nowhere, without motivation, without provenance. If so, you'd have to believe that number theorists are wasting their lives chasing bizarre and pointless things, and really, not many of them either. I hope that this article *fails* to make you think that. I hope this article shows you that the BSD conjecture is a part of a larger, meaningful context. On the other hand, if your "this" refers to the article itself, then it's a rather larger task to help access it. I attended a talk a while ago that gave an excellent, high-level overview of this conjecture and the work being done on it, and I can see the structure of the arguments here. But it's not intended to be read like a novel by people other than experts in the area. If you're not an expert - and very few people are - it's an article you need to wrestle with, work on, and dive into. It's math, it's not a graphic novel, and math is not a spectator sport. It's requires participation and effort. So, is it the article itself for which you need an ELI5? If so, seriously, good luck. This is material at the very limits of modern math.

33 points by subnaught 11y ago

This is quite advanced. At least graduate math level. This is supposed to be a "elementary introduction to the Birch and Swinnerton-Dyer conjecture". Ha.

cperciva 11y ago
This is quite advanced. At least graduate math level. Yes, I learned most of this when I took graduate mathematics courses. Soon thereafter I decided that I'd be better off studying computer science... This is supposed to be a "elementary introduction to the Birch and Swinnerton-Dyer conjecture". Ha. I don't think the word "elementary" means what you think it means. In number theory, "elementary" means "not involving complex analysis"; it does not mean "straightforward".
- jordigh 11y ago
  In this context, they are not shying away from using complex analysis. They really do mean "elementary" as in "not for number theory experts", but for a general mathematical audience.
  - cperciva 11y ago
    They mention complex analysis in passing, but I wouldn't say that they use it here.

chrispeel 11y ago

...such a talk can easily succeed in convincing the audience that there is a dry, ridiculous, but famous conjecture coming out of nothing; crazy number theorists have wasted their life to contribute to the list of partial results, which cannot even be claimed to be a long list. Hopefully I have failed in that way. Ummm, I guess that even among techies on HN, there are only a small fraction who could understand this. So the author failed with me. Any one want to do an ELI5?

daniel-levin 11y ago
I think understanding that sentence reduces to parsing jargon. I'll provide an explanation I hope isn't in any way condescending. The main activity of research mathematicians is theorem proving. That is, turning conjectures into theorems. In the jargon, proven theorems (and equations, bounds, counterexamples etc.) are often called 'results'. Essentially, the author is saying that with very few constructs and tools, number theorists have been able to pose a conjecture that is comparatively simple to state and understand, but is frustratingly as yet unproven, in spite of decades of effort by many, very good mathematicians. So far, many special cases ('partial results') have been proven, but the highly desired goal that is the proof of the BSD conjecture, in full generality, has not yet been achieved. The consensus is that mathematicians are quite far away from resolving the BSD conjecture, but very strongly believe it is true.
ColinWright 11y ago
>> ... such a talk can easily succeed in >> convincing the audience that there is >> a dry, ridiculous, but famous conjecture >> coming out of nothing; crazy number >> theorists have wasted their life to >> contribute to the list of partial >> results, which cannot even be claimed >> to be a long list. >> Hopefully I have failed in that way. > Ummm, I guess that even among techies on HN, > there are only a small fraction who could > understand this. ^^^^ > So the author failed with me. Any one want > to do an ELI5? It's not clear what you mean by "this". You might mean this small section that you quoted, or you might mean the entire article to which this is an opening preamble. If it's the first of these, then here's a free translation: A talk like this can convince an audience that the conjecture is just a wild, crazy thing, produced by wild crazy people, and it comes from nowhere, without motivation, without provenance. If so, you'd have to believe that number theorists are wasting their lives chasing bizarre and pointless things, and really, not many of them either. I hope that this article *fails* to make you think that. I hope this article shows you that the BSD conjecture is a part of a larger, meaningful context. On the other hand, if your "this" refers to the article itself, then it's a rather larger task to help access it. I attended a talk a while ago that gave an excellent, high-level overview of this conjecture and the work being done on it, and I can see the structure of the arguments here. But it's not intended to be read like a novel by people other than experts in the area. If you're not an expert - and very few people are - it's an article you need to wrestle with, work on, and dive into. It's math, it's not a graphic novel, and math is not a spectator sport. It's requires participation and effort. So, is it the article itself for which you need an ELI5? If so, seriously, good luck. This is material at the very limits of modern math.