I just added to the geometry front page, and (as I wrote in a comment there) I took the view-point that topology is a part of geometry. (I hadn't seen this comment thread at that time, but I seem to have acted in accord with its sentiment in any case.)
I added a brief description of geometry. I was motivated here by the somewhat standard tripartite division of mathematics into algebra/analysis/geometry, so my conception of geometry includes topology, for example (i.e. not just metric geometry). This seemed justified, since 3-manifolds was one of the existing links.
I added a short list of modern methods in geometry, together with some discussion of how geometry can be applied in other areas of mathematics. It would be good to flesh this out by adding more examples, and creating pages that illustrate them, with appropriate links.
A quick comment: I see that you put the group theory front page as the parent for this article, but that you didn't put a link in that article. I point this out in case you were expecting that link to be added automatically, since that does not happen. I've added the link myself now.
I had a vague plan to convert an old blog post of mine on solving cubics into a Tricki article. If so, then this Method 5 would fit in naturally there.
For now, the title of the method suggests a rather general idea, but I wonder whether it really is. Perhaps it should be given a more specific title such as "Think of a good substitution." In this case, the basic idea is to think of the two roots as and and solve for and which turns out to be easy. And one can motivate the idea by the observation that in specific instances we observe that the solutions are of the form
I feel as though if I thought about this for long enough and checked all the calculations then I would eventually see why you chose this particular substitution. But it would be nice if you could add a little here, to say something like, "We are trying to do such-and-such, so we want the to be in such-and-such a form. This gives us the following equations ... and solving those equations we arrive at the following substitution." Is it possible to present a fully motivated account along these lines? By the way, this is a great trick to have on the site: starting with something that looks terrifying and showing that in fact it isn't.
Actually the problem was that I'd put a full stop inside the link. I've been discovering quite a few dead-looking links to articles that do in fact exist, and often clicking on the link actually works. It's very helpful to have these pointed out, or just quietly corrected.
If you have to show something for all morphisms (of type P), factorise then (e.g. into mono/epi: abelian category, fibration/cofibration: model category, closed immersion/projection: projective morphisms -> this strategy is used in the proof of the Grothendieck-Riemann-Roch theorem) and show it for all the factorisations and that it is preserved by composition.
Edit: Misread the given condition . I had proved that there exists an element with order the smallest prime (which I had misread as ). I am working on it..
Just a note: there seems to be problems with apostrophes. In particular, the `How to use Zorn's lemma' article exists but is listed here as nonexistent.
Reduction to a univeral example. If there exists a classifying object (a moduli space) M with a universal object (universal family) X for objects (P) on it, i.e. every object with the property (P) is the pullback of X by a morphism to M, and if one wants to show something about all objects with (P) that is preserved by pullbacks, it suffices to show it for X/M.
Example: For the construction of Chern classes, it suffices to calculate the cohomology of infinite Grassmannians.
I like the following proof of the existence part of Sylow best:
1. Show that if there exists a p-Sylow subgroup for G, there exists one for every subgroup H of G (by using double cosets).
2. Show that there exists a p-Sylow subgroup of GL_n(F_p) (upper triangular matrices with 1 on diagonal).
3. GL_n(F_p) are cogenerators in the category of finite groups. (Every finite group can be embedded into a GL_n(F_p) for n = |G|.)
This shows also a general algebraic trick: If there is a class of generators of an (at the best abelian) category, show the result for the (simpler) generators, and then for all quotients of them. (Here, use the dual strategy.)
One of the best examples of a proof using group actions in a non-obvious way is Wielandt's proof of Sylow's theorem. (Other proofs use group actions, but it seems Wielandt's proof is the most popular.) It's a fairly well known example and can be found on the Wikipedia page for Sylow's theorem, but seeing as it's probably the easiest known proof of the most important theorem of finite group theory, one which was seen as an extremely difficult result when originally obtained, it warrants an extensive exposition here. I'll write something about it on this page at some point unless someone else wants to.
I've rewritten the article in a way that hopefully assumes less background of the reader, and also included Terence's example. This is indeed a very good example of what I am talking about. Is it a bad thing to have it written up here and in Tim's article, or should there just be a link to the most detailed instance of an example (Tim's in this case)?
I see that you too have written up the example that Terence Tao suggested. I think this is absolutely fine: the example can now be found in two genuinely distinct ways: one by somebody who wants to find a normal subgroup of small index (who will naturally arrive at this article) and the other by somebody who wants to understand why people are interested in group actions and how they manage to use them to solve problems that are not ostensibly about actions (who will be led to the article on proving results by letting a group act on a finite set. In general, I am all in favour of the same example popping up in more than one place, and wouldn't even be unduly disturbed if it was simply cut and pasted from one article to another, though that has not happened in this case.
Hmm, that's not quite what I meant. If you have a look at most subject-matter front pages (the combinatorics one is a bit of an exception) then you will see that they tend to consist of a lot of links and not much text. So what you now have as "Techniques for counting" is in that kind of Front Page style, and what you have as "Enumerative combinatorics front page" is not really in that style.
What I want to encourage is two routes to a typical article: one by means of gradual zeroing in on its subject matter, and the other by means of classifying the type of problem that the article can help you solve. So I would imagine an article entitled "Techniques for counting" as doing something like starting with a general description of what counting is, giving some kind of classification of counting problems into different types, trying to explain briefly which techniques are good for which types of problems, and giving links to fuller articles about those techniques. (For example, in the Princeton Companion to Mathematics, Doron Zeilberger gives a nice explanation of when you use straight generating functions and when you use exponential generating functions.) It seems to me that we'd get a better approximation to all this if we exchanged the titles of your two articles!
I've gone ahead and written up this example as a way of trying to get started an article on proving results by letting a group act on a finite set. I think it's a nice one. (I came across it last year when giving supervisions on a first Cambridge course in group theory: the lecturer had set it as a question.)
For me, it seems unnatural to say this without mentioning the symmetric group. I would give the proof as follows. The action of on is the same thing as a homomorphism . If is the kernel of this homomorphism then is isomorphic to a subgroup of and so of order dividing . It's clear that is contained in (indeed, it's precisely the intersection of with its conjugates).
I think you're right. I took what was the introduction and made it the Enumerative Combinatorics front page (still needs some tuning up) and re-wrote 'techniques for counting'. Presumably the distinction is that 'techniques for counting' should link not only to the combinatorial methods, but also to articles or proofs in non-combinatorial fields that make use of combinatorial techniques. That is, it should link to the answer to the question: what does [technique A] look like when applied in [field W]?
I can add a link for the probabilistic method, and will do so in a moment. We should perhaps think about whether this is the most appropriate title for the article – it would also be naturally described as "Enumerative Combinatorics Front Page," except perhaps for the brief mentions of probabilistic and extremal combinatorics. It's genuinely not obvious to me what the right way to organize things is here. One possibility might be to have a rather short navigation page that describes what the word "Counting" means to mathematicians (called "Techniques for counting") which could then link to the combinatorics front page, which is waiting for an enumerative combinatorics front page, which this could perhaps be.
If you know there's an article on any of the tricks linked to in the "Techniques" section, please edit the text so it goes to the right article. Thanks!
"I for one will not understand it without some help with basic definitions. Here are the ones I don't know: non-central section (or any kind of section for that matter), derived subgroup. I sort of half remember what a characteristic subgroup is but wouldn't mind being reminded."
Let G be a group.
A SECTION of G is simply a homomorphic image of a subgroup of G (or phrased differently if there is a subgroup H
I just added to the geometry front page, and (as I wrote in a comment there) I took the view-point that topology is a part of geometry. (I hadn't seen this comment thread at that time, but I seem to have acted in accord with its sentiment in any case.)
I added a brief description of geometry. I was motivated here by the somewhat standard tripartite division of mathematics into algebra/analysis/geometry, so my conception of geometry includes topology, for example (i.e. not just metric geometry). This seemed justified, since 3-manifolds was one of the existing links.
I added a short list of modern methods in geometry, together with some discussion of how geometry can be applied in other areas of mathematics. It would be good to flesh this out by adding more examples, and creating pages that illustrate them, with appropriate links.
I'm happy to do that and I can see that it makes sense. Do others agree?
I think geometry and topology should go together as one subject area - often, the difference is a matter of context.
A quick comment: I see that you put the group theory front page as the parent for this article, but that you didn't put a link in that article. I point this out in case you were expecting that link to be added automatically, since that does not happen. I've added the link myself now.
I had a vague plan to convert an old blog post of mine on solving cubics into a Tricki article. If so, then this Method 5 would fit in naturally there.
For now, the title of the method suggests a rather general idea, but I wonder whether it really is. Perhaps it should be given a more specific title such as "Think of a good substitution." In this case, the basic idea is to think of the two roots as and and solve for and which turns out to be easy. And one can motivate the idea by the observation that in specific instances we observe that the solutions are of the form
Added later: that article now exists
I feel as though if I thought about this for long enough and checked all the calculations then I would eventually see why you chose this particular substitution. But it would be nice if you could add a little here, to say something like, "We are trying to do such-and-such, so we want the to be in such-and-such a form. This gives us the following equations ... and solving those equations we arrive at the following substitution." Is it possible to present a fully motivated account along these lines? By the way, this is a great trick to have on the site: starting with something that looks terrifying and showing that in fact it isn't.
Am I right in thinking that the words "are linearly independent" are missing at the end of this sentence?
Actually the problem was that I'd put a full stop inside the link. I've been discovering quite a few dead-looking links to articles that do in fact exist, and often clicking on the link actually works. It's very helpful to have these pointed out, or just quietly corrected.
If you have to show something for all morphisms (of type P), factorise then (e.g. into mono/epi: abelian category, fibration/cofibration: model category, closed immersion/projection: projective morphisms -> this strategy is used in the proof of the Grothendieck-Riemann-Roch theorem) and show it for all the factorisations and that it is preserved by composition.
Edit: Misread the given condition . I had proved that there exists an element with order the smallest prime (which I had misread as ). I am working on it..
Just a note: there seems to be problems with apostrophes. In particular, the `How to use Zorn's lemma' article exists but is listed here as nonexistent.
A strategy in a similar spirit:
Reduction to a univeral example. If there exists a classifying object (a moduli space) M with a universal object (universal family) X for objects (P) on it, i.e. every object with the property (P) is the pullback of X by a morphism to M, and if one wants to show something about all objects with (P) that is preserved by pullbacks, it suffices to show it for X/M.
Example: For the construction of Chern classes, it suffices to calculate the cohomology of infinite Grassmannians.
I like the following proof of the existence part of Sylow best:
1. Show that if there exists a p-Sylow subgroup for G, there exists one for every subgroup H of G (by using double cosets).
2. Show that there exists a p-Sylow subgroup of GL_n(F_p) (upper triangular matrices with 1 on diagonal).
3. GL_n(F_p) are cogenerators in the category of finite groups. (Every finite group can be embedded into a GL_n(F_p) for n = |G|.)
This shows also a general algebraic trick: If there is a class of generators of an (at the best abelian) category, show the result for the (simpler) generators, and then for all quotients of them. (Here, use the dual strategy.)
I just added method 5 but maybe it should put somewhere else in the article letting the last method be the quadratic formula.
Is that method really in the spirit of this Tricki? Or it should be somewhere else?
I just remembered this method from my reflections about the quadratic equation and felt it was a good idea.
Thanks for this great site everybody.
One of the best examples of a proof using group actions in a non-obvious way is Wielandt's proof of Sylow's theorem. (Other proofs use group actions, but it seems Wielandt's proof is the most popular.) It's a fairly well known example and can be found on the Wikipedia page for Sylow's theorem, but seeing as it's probably the easiest known proof of the most important theorem of finite group theory, one which was seen as an extremely difficult result when originally obtained, it warrants an extensive exposition here. I'll write something about it on this page at some point unless someone else wants to.
I've rewritten the article in a way that hopefully assumes less background of the reader, and also included Terence's example. This is indeed a very good example of what I am talking about. Is it a bad thing to have it written up here and in Tim's article, or should there just be a link to the most detailed instance of an example (Tim's in this case)?
I see that you too have written up the example that Terence Tao suggested. I think this is absolutely fine: the example can now be found in two genuinely distinct ways: one by somebody who wants to find a normal subgroup of small index (who will naturally arrive at this article) and the other by somebody who wants to understand why people are interested in group actions and how they manage to use them to solve problems that are not ostensibly about actions (who will be led to the article on proving results by letting a group act on a finite set. In general, I am all in favour of the same example popping up in more than one place, and wouldn't even be unduly disturbed if it was simply cut and pasted from one article to another, though that has not happened in this case.
Hmm, that's not quite what I meant. If you have a look at most subject-matter front pages (the combinatorics one is a bit of an exception) then you will see that they tend to consist of a lot of links and not much text. So what you now have as "Techniques for counting" is in that kind of Front Page style, and what you have as "Enumerative combinatorics front page" is not really in that style.
What I want to encourage is two routes to a typical article: one by means of gradual zeroing in on its subject matter, and the other by means of classifying the type of problem that the article can help you solve. So I would imagine an article entitled "Techniques for counting" as doing something like starting with a general description of what counting is, giving some kind of classification of counting problems into different types, trying to explain briefly which techniques are good for which types of problems, and giving links to fuller articles about those techniques. (For example, in the Princeton Companion to Mathematics, Doron Zeilberger gives a nice explanation of when you use straight generating functions and when you use exponential generating functions.) It seems to me that we'd get a better approximation to all this if we exchanged the titles of your two articles!
I've gone ahead and written up this example as a way of trying to get started an article on proving results by letting a group act on a finite set. I think it's a nice one. (I came across it last year when giving supervisions on a first Cambridge course in group theory: the lecturer had set it as a question.)
For me, it seems unnatural to say this without mentioning the symmetric group. I would give the proof as follows. The action of on is the same thing as a homomorphism . If is the kernel of this homomorphism then is isomorphic to a subgroup of and so of order dividing . It's clear that is contained in (indeed, it's precisely the intersection of with its conjugates).
I think you're right. I took what was the introduction and made it the Enumerative Combinatorics front page (still needs some tuning up) and re-wrote 'techniques for counting'. Presumably the distinction is that 'techniques for counting' should link not only to the combinatorial methods, but also to articles or proofs in non-combinatorial fields that make use of combinatorial techniques. That is, it should link to the answer to the question: what does [technique A] look like when applied in [field W]?
I can add a link for the probabilistic method, and will do so in a moment. We should perhaps think about whether this is the most appropriate title for the article – it would also be naturally described as "Enumerative Combinatorics Front Page," except perhaps for the brief mentions of probabilistic and extremal combinatorics. It's genuinely not obvious to me what the right way to organize things is here. One possibility might be to have a rather short navigation page that describes what the word "Counting" means to mathematicians (called "Techniques for counting") which could then link to the combinatorics front page, which is waiting for an enumerative combinatorics front page, which this could perhaps be.
If you know there's an article on any of the tricks linked to in the "Techniques" section, please edit the text so it goes to the right article. Thanks!
Tim wrote:
"I for one will not understand it without some help with basic definitions. Here are the ones I don't know: non-central section (or any kind of section for that matter), derived subgroup. I sort of half remember what a characteristic subgroup is but wouldn't mind being reminded."
Let G be a group.
A SECTION of G is simply a homomorphic image of a subgroup of G (or phrased differently if there is a subgroup H