Timeline for Algebraic number theory and applications to properties of the natural numbers.
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2022 at 7:57 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Feb 24, 2010 at 20:59 | comment | added | Regenbogen | @Qiaochu. I already agreed that it is worthwhile to mention. The rest, I was just responding to your comment on "what is important in a concrete subject". | |
| Feb 24, 2010 at 20:51 | comment | added | Qiaochu Yuan | @Regenbogen: while I see your point, I don't understand why this is relevant to whether you should bring up practical applications in a class. | |
| Feb 24, 2010 at 19:37 | comment | added | KConrad | Here is a nice reason to mention the number field sieve at least in passing (if its details are too involved for the class). When you first learn about the ring of integers O_K in a number field K, they all look like Z[a]. Not every O_K has this special form, so from an abstract point of view you might start to look down on the rings Z[a]. However, computations in a ring Z[a] are far more efficient than in O_K (to start, try finding a Z-basis in each). For computational purposes, the number field sieve makes essential use of rings Z[a] in preference to O_K itself. | |
| Feb 24, 2010 at 18:49 | comment | added | Regenbogen | @Qiaochu. It is not just applications. For example, the Ramanujan tau function conjecture was proved by Deligne using Grothendieck-style algebraic geometry. On the face of it, when the conjecture was made, it would be hard to imagine that the solution would require such sophisticated methods. That is what I was saying; this kind of work is meant for really smart people like Ramanujan. If I get my hands onto it, I would likely produce only essentially worthless stuff. | |
| Feb 24, 2010 at 18:43 | comment | added | Qiaochu Yuan | What is important in a concrete subject is what would advance applications the most. This is much less ambiguous than what is important in an abstract subject. But as Terence Tao argues (arxiv.org/abs/math.HO/0702396) there is certainly a need for both points of view. | |
| Feb 24, 2010 at 18:05 | comment | added | Regenbogen | Yes, it is a good point to give various flavors to students. So it might be good to include this CS-style example also. That said, what I meant about getting lost is the following. One has to have a certain notion of what is important, and what is not. In abstract subjects, this distinction is easier to make. The more abstract, the "better" it is, in a certain sense! Also philosophically minded people can get away to some extent in algebraic geometry, etc.. | |
| Feb 24, 2010 at 18:01 | comment | added | Qiaochu Yuan | That's an interesting statement. Most people I know would find it much easier to get lost in abstract stuff. When designing an algorithm it is easy not to get lost: you are on the right track if the algorithm is fast and on the wrong track if the algorithm is slow. In any case, I don't see the point of making all these unnecessary divisions between subjects. Wouldn't you prefer that your students get a taste of many kinds of mathematics and of their unity? | |
| Feb 24, 2010 at 17:54 | comment | added | Regenbogen | Well, the design of algorithms is for really smart chaps. If I were to take up that subject for example, I might get lost into inconsequential stuff. Whereas in pure mathematics, certain subjects are so abstract that it is harder to get lost. | |
| Feb 24, 2010 at 17:49 | comment | added | Qiaochu Yuan | So suddenly prime factorization is not a mathematical activity? Whether the computation is a baser mental activity or not, the design of algorithms is a nontrivial mathematical endeavor. | |
| Feb 24, 2010 at 17:45 | comment | added | Regenbogen | Yes it does fit. However this is more like a CS-style application? I was having more pure math style in mind. Somehow I got brainwashed into thinking that blind (almost mechanized) computation is a sort of baser mental activity, compared to stuff involving ideas in the domain of thought. | |
| Feb 24, 2010 at 17:32 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |