This article should not be a stub. It is infact the best suggestion I have come across here. To see how it can lead to some 'mathematically interesting information',I suggest this webpage- http://disquisitionesmathematicae.wordpress.com
I have a couple of suggestions. One is to do a Gaussian computation in this article. A natural example is to take a bivariate normal random variable and create out of and two linear combinations and that are linearly independent. That should perhaps come after my second suggestion, which is simply to explain how to diagonalize a quadratic form (by giving a few examples). If I have a spare moment, I may add these examples.
The second one is one I care about because a lot of people I teach seem to have the impression that the only way of diagonalizing a quadratic form is to diagonalize the associated symmetric matrix.
Incidentally, I'm all in favour of using the same example in more than one article, so there might be a case for having the Fourier transform of a Gaussian done here as well.
I've amalgamated the two articles, and kept the nicer name. A few thoughts that I'd be interested in comments on:
1. Perhaps this is now overloaded with examples about free groups. A new page might be appropriate.
2. A good complement might be a theorem about the fundamental group of a surface. Almost anything would do. One could prove the analogue of the Scheier Index Formula for a closed surface, for instance.
3. Did you have a specific Serre-style example in mind, Matthew? Of course, that stuff is a very nice application of these ideas, and one day I hope to write Use actions on trees to study graphs of groups or whatever, but it seems quite hard to develop in the space available on this page.
Thanks for your reply. I think its helpful to explain our various points of view. I'm going to write a little more about mine; it is a little redundant, given what I've already written, but perhaps it will put my thoughts in a larger context than just the probability page.
I think that one thing that is helpful to think about in preparing tricki pages, and especially the top-level pages, is who the possible audience will be, and what they should be getting out of their tricki experience. (My impression is that this is every much in Tim's mind as he writes his articles and comments.)
There are some areas of math which I would say are almost purely technical, and where it is reasonable to expect a certain level of mathematical competence of the reader. For example, How to use spectral sequences (a page that I hope to eventually write, if someone doesn't beat me to it) is unlikely to be studied by anyone who doesn't have a solid undergraduate background in algebra, and probably some graduate training in algebra and or topology as well. Spectral sequences are really an intrinsically technical tool. They are important (and so I think they merit a tricki page, especially because many grad students suffer in learning to use them, and any additional help and explanation will, I expect, be welcomed by many), but are not a fundamental topic in mathematics, of interest to those who are not working mathematicians, or aspiring to become such.
On the other hand, one can expect that the Diophantine equations front page, when it is written, to be of much broader interest. Diophantine equations are a central topic in mathematics, and of interest to a broad range of people: amateurs, students involved in the maths olympiad or similar competitions, undergraduates, and of course graduate students and professional mathematicians. There is a lot that one can say technically about Diophantine equations: one can talk about Galois cohomology, and how this plays a key role; also modular, or more generally automorphic forms, and the Langlands program; the Hasse principle, -adic numbers and adeles; - and -functions; the circle method; and so on. These are all central ideas in the modern theory of Diophantine equations. But I don't think that they should be the focus of the Diophantine equations front page; they will be outside the technical range of many visitors to the page, and there will be much of interest that can be said about Diophantine equations that will be accessible to these visitors — but only if it is couched in less technical language. Furthermore, writing things in as accessible way as possible will help professional mathematicians as well: no doubt there will be mathematicians reading the Diophantine equations front page, once it exists, whose own area of expertise will be far from number theory; using the bare minimum of technical language required for any particular example or technique will make it much easier for them to extract utility from the tricki.
Probability is similar: it is a fundamental topic in mathematics. It is not a purely technical subject, but rather is of interest to a very wide range of people, from those interested in elementary calculations, to those interested in substantial applications (to the real world, or perhaps to problems in other parts of mathematics), to those interested in the theory itself for its own sake.
We will want the tricki page to be as accessible to as many of these people as possible; not all of them will know measure theory, and there will be many examples that they can understand despite this, if we write them in an a certain way.
I don't think that every example should be requried to use the absolutely minimal amount of machinery necessary to make it tick; indeed, some examples will have as their point the illustration of how to use a particular piece of machinery or advanced technique in a simple situation (as part of an explanation of how to use that piece of machinery or technique). But I don't believe that we want every example to be burdened by machinery, especially in broad topics of wide interest such as probability or number theory. And a logical consequence of this belief is that the top level introductions to these topics shouldn't themselves be entirely enmeshed in machinery and technical language. They should allow for multiple viewpoints on their topic, both elementary and advanced.
I will close by emphasizing that I am very far from an expert in probability, so I don't think that I will actually be contributing much to the probability branches of the tricki tree. On the other hand, I do hope to learn from it as it grows (and perhaps improve my understanding of measure theory in the process!).
Thanks for catching this — when I wrote this I wasn't mentally distinguishing between homotopy equivalence and homeomorphism.
With regard to your comment on combining pages:
I hadn't looked carefully at Actions on topological spaces; I rather just saw the comment about this result on subgroups of free groups, and thought that it would be a good thing to put into the tricki.
Feel free to combine the articles as you think is best. One thing that might be good would be to follow Tim's encouragement to have imperative titles. (You can see that I followed it in naming this article.) But since you've probably put more work into the actions page than I have into this page, you should do what you think makes the most sense.
Being terrified by the word measure is I think as reasonable as being terrified by the words `group' or `ideal.'
But the fact remains that it is possible to ask questions about probability without knowing what a measure is. And this is the probability front page.
If you go to the Group Theory front page, you'll see that it starts with a (brief) attempt to explain that the notion of group formalizes the notion of symmetry. So anyone who has a question about symmetry will perhaps be able to figure out that they need to start thinking about groups. Something similar should apply here.
> Now one can try to study and answer as many questions as possible just in terms
> of , and my impression is that many results in probability theory proceed along
> these lines: one has a (or perhaps a sequence) of random variables satisfying
> certain restrictions on their distribution, and one draws some conclusion about
> the random variable, or about some kind of normalized limit of the sequence. (I
> am thinking say of the central limit theorem.) In these theorems the particular
> sample space doesn't really play any role, as far as I know; one can just argue
> with the distributions (or more abstractly, and perhaps more generally,
> with the push-forward measures attached to the random variables).
> It is, I would think, an important point to know that often one doesn't really > have to understand the full details of the sample space, but rather, just the
> distribution of the random variable one is studying.
Once again, we are in agreement, but let's be careful about what the random variables and their distributions are.
Let's look at the example you gave, the central limit theorem. What is this theorem about?
Suppose is a probability measure on with unit second moment, i.e., and zero mean, i.e., . Here is the borel sigma field. Let be the product measure on . Let be the coordinate maps. Let be the measure defined on by the map. The central limit theorem states that
This explicit way of writing things makes it clear that the central limit theorem is a statement not about real random variables and their distributions but about product probability measures on and the transformation of such measures under summation and scaling. As you said, there could have been even a larger space on which were defined. In that case one first pushes the measure forward onto using the map .
Added later: Another point of view, for this problem, is to think only about the pair and a measure on this pair with the above mentioned properties. We then define , where is the convolution of with itself times. The CLT says . . This point of view is also the one that quickly suggests the most well known proof of this result: take Fourier transforms and study their limits. Most generalizations though will involve measures on or larger sets.
I guess, I made my point thoroughly :) I find it extremely useful to state the sets and the measures I am using and focusing on these and thinking about methods in terms of these. I am completely open to other ways of thinking, especially from the point of view of writing tricki articles. Sorry if I took too long in expressing my thoughts. Many thanks again for your thoughts!
Thanks for the comments. My goal in writing the things I wrote here was a succint classification of the central problems that people study in probability theory. I thought such a classification would allow someone to easily locate a technique of interest.
I think we all agree that the front page can be made more lively and less technical.
Here are few points on some of your comments:
The perception measure theory as an abstract tool in probability which is to be avoided as much as possible, especially in simple cases, is an unfortunate situation in today's mathematics. I think eventually this perception will change. A measure can be on a finite set; it can be the uniform distribution on a finite set and then computing with it means exactly counting.
I think the terms `discrete' and `continuous' probability distributions are informal and not very clear. Currently, many people use these terms and therefore I do understand if we would like to use them in tricki. But I do think that much better terminology is possible and available.
Being terrified by the word measure is I think as reasonable as being terrified by the words `group' or `ideal.' I think these are standard mathematical terminology, and my feeling is let's get used to them.
Randomness is one thing that we model with probability measures. For example, wikipedia begins with a statement about chance and randomness. This is one important aspect probability theory but it is not the only one. For example, another important use is counting. We can also use probability to model our beliefs, or an allocation of a resource. I tried to reflect this point of view in my first attempt and perhaps this could also be kept in mind as we rewrite the initial page.
Yes. My first choice is titles like "To prove that a small group is simple, look at the sizes of the conjugacy classes," but that won't work for more general pages that discuss or link to several techniques. So then titles such as you suggest (perhaps I'd go for "problems" rather than "a problem" but that's a very minor quibble!) would come into play.
Also, this is false - it's homotopy equivalent to a bouquet, but is never a bouquet itself unless it's a homeomorphism. Again, this is already discussed in Actions on topological spaces.
I suppose you're saying that you'd rather have pages with names like "How to solve a problem about finite groups", right? I'm all for that. I'll make the change at some point.
I think that Tim's suggestions make good sense. The measure-theoretic foundations for probability are needed for studying sophisticated questions, but there are lots of elementary quesitons in probability theory that are basically questions of counting, and it would for a reader to the tricki to be able to reach them in the tree structure without having to go through measure theory first.
Related to an earlier comment: here is my guess as to what it might mean to avoid referring to the sample space:
If is a probability space and a random variable,
then we can push-forward the meausre via , to get a meausre on :
the push-forward is defined by Now one frequently
finds that is absolutely continuous with respect to Lebesque measure, and so can be written in the form , where is a function and denotes Lebesgue measure.
I guess is the probability distribution of the random variable .
Now one can try to study and answer as many questions as possible just in terms of , and my impression is that many results in probability theory proceed along these lines: one has a (or perhaps a sequence) of random variables satisfying certain restrictions on their distribution, and one draws some conclusion about the random variable, or about some kind of normalized limit of the sequence. (I am thinking say of the central limit theorem.) In these theorems the particular sample space doesn't really play any role, as far as I know; one can just argue with the distributions (or more abstractly, and perhaps more generally, with the push-forward measures attached to the random variables).
It is, I would think, an important point to know that often one doesn't really have to understand the full details of the sample space, but rather, just the distribution of the random variable one is studying. (In practical terms, the latter is at least in principle capable of being measured, while the sample space might be completely inaccessible.)
One suggestion is to begin with a more intuitive discussion of probability on the front page. One could define a random variable as being a quantity, that varies randomly, and that one wants to study (e.g. sum of the faces of two rolled dice, height of a random American, ...). It would be good to emphasize that random variables can be discrete (the dice example) or continuous (the height example).
We could then explain that a random variable has a distribution, which basically governs its behaviour. (And again this could be illustrated with the examples.) In general, emphasizing that there are some very common and important distributions, e.g. binomial, normal, Poisson, would be good.
Next could come the discussion of how this is mathematically modelled, with the idea being that simple models suffice for easy discrete questions like rolling a pair of dice, that more analysis is needed for studying continuous random variables, and that the general foundations rely on measure theory, etc.
Then we could begin with links to various subpages, with appropriate commentary. One page could be rigorous foundations of probability, which is where the current discussion of measure spaces, random variables as meausrable functions, etc., would live.
The question here is not what the correct way is of setting up a rigorous theory of probability (either for the purposes of research or for the purposes of giving a lecture course on the subject) but rather what is the best thing to put on a probability front page that is intended to help people find ways of solving probabilistic problems.
Suppose that I have a problem in discrete probability: to prove, say, that if I toss a coin 100 times then I have a very high probability of getting at least 30 heads. And suppose I come to the Tricki for help and visit the probability front page. As it is written at the moment, I would be terrified: if I needed help with that kind of problem, then I probably wouldn't know any measure theory, but the clear impression I would get from the page was that measure theory was an essential prerequisite.
I do not object to the content, but merely to its place in the Tricki hierarchy. I think a more obvious organization would be as follows. One begins with an informal description of probability theory, mentioning simple problems about calculating and estimating probabilities. One then explains that the intuitive view of probability has important limitations, and that a modern theory of probability makes heavy use of measure theory. And one then has links to more specialized pages such as elementary probability (essentially just counting problems), discrete probability, and general measure-theoretic probability. That way, the reader interested in problems about particular kinds of convergence of random variables would go to the advanced page, and the reader interested in the probability of getting all four aces in a bridge hand would go to the elementary page, and nobody would be frightened off.
A question that would have to be considered is how much of the basic theory of probability actually belongs on the Tricki. The default would be not much: you assume that the reader knows the basics and the Tricki articles are there to help him/her solve problems.
Added later: an interesting comparison can be drawn with the Wikipedia introduction to probability, which is more in the style that I would expect for a Tricki front page (though it wouldn't be perfect because it is not focused on problem-solving in quite the right way).
This page seems to me to be shaping up nicely, but I'd like to raise a question about all the dead links in these three paragraphs. The implication seems to be that there will be pages called "Abelian groups front page", "Finite groups front page", "Lie groups front page", "Profinite groups front page", "Combinatorial group theory front page" and "Geometric group theory front page". Is that the idea?
If it is (and it seems sensible), then at some stage I would probably want to rewrite this section slightly, in order to avoid the impression it currently gives that the Tricki focuses on subject matter (things like the definition of an Abelian group) rather than proof techniques. So I'd probably make those dead links into Wikipedia links and then have a paragraph saying something like "These special classes of groups have their own particular techniques, which are discussed and linked to from the following pages."
I had noticed that too, and considered merging them. I may yet do so, but I want to wait until I have thought of good examples for each. Somehow, with this article the focus is on generalization via abstraction, and thereby obtaining a question that gets more to the heart of the problem, whereas with the other one you may not be generalizing at all, because you may be using an abstract description that completely characterizes the original concrete object. (As an example of the latter, it is very useful to think of the reals as a complete ordered field rather than say as a set of Dedekind cuts, even though there is only one complete ordered field. And it is sometimes helpful to think of the exponential function just as "a" function that satisfies and .)
If we do end up with two articles, there's no doubt that they will be closely related, and that this should be highlighted.
It seems to me that they are getting toward a similar point. Of course both articles are still stubs; if the intention is to explain different tricks, perhaps someone could explain in the quick descriptions how they are different. Just my two cents.
As an example, how about proving that a functor that preserves exactness in sequences of the form necessarily preserves exactness in sequences of the form ? One could then apply this to tensor products with a flat module.
"The proof of Theorem 5 brings out an important point, namely that if W is a representation of G\times H, and U is an irreducible G-representation, then"
That sounds sensible. I don't object at all to using the phrase Peirce decomposition to describe this procedure, as long as we don't rely on it as the only means of navigating to this page. I think in general that it is better to avoid using names (especially people's names) for things if one can avoid it, unless the name is absolutely universally known or absolutely unavoidable.
Thus ``Prove that your holomorphic function is bounded" (or some variant of this) would be better than ``Apply Liouville's theorem" as an instruction related to trying to show that a holomorphic functions is constant. After all, there is always the possibility that someone doesn't know or has forgotten what Liouville's theorem says, and then the name carries absolutely no hint about the method. (As a contrast, ``Apply the maximum modulus theorem'' is probably not quite as bad in this respect, because it contains in its name at least some semblance of what it is about.)
As a possible compromise, what about a link to this article on the "How to use" page, called "How to use the Peirce decomposition"? The title of the article would be different, but there's no problem with that. There could be a hidden-text quick description that said that the Peirce decomposition was a decomposition of rings using idempotents.
The two types of arguments that this article lists are very common in probability theory. They are also of different nature and they usually appear in different contexts.
The first one, the one that goes to the subsubsequences is a soft analysis method, I have seen it used to prove weak convergence in function spaces very often (for example see the large deviations book by Dupuis and Ellis). In this setup, you have an obvious limit process. To establish convergence you first prove total boundedness (in the context i just mentioned this involves the use of the relative entropy function or sometimes the boundedness of the increments). This automatically guarantees convergence along subsequences (and this is a consequence of a general Bolzano-Weirstrass theorem). It remains to show that each subsequence has a further subsubsequence that converge to the candidate limit process. This is usually clear intuitively and can be proved as a consequence of a general law of large numbers type result.
The second one, finding a fast converging subsequence which approximate the whole sequence well: this I would call a hard analysis method. You first have to find your fast convergence sequence: this means to do computation to show that the subsequence you picked does converge at a certain rate. Then you have to also show, via hard calculation, that the whole sequence doesn't wander off too much from this subsequence. Almost all `convergence almost surely' arguments that I know that don't use martingales are of this form. For example: law of the iterated logarithm, the strong law of large numbers. I would say this is one of the main methods of proving almost sure convergence (of random variables, processes).
Because of these differences I think it would be logical to have separate articles for them. Whether to have a further page like the one i proposed? My personal take is I always like classification, if there is an obvious way to classify arguments then I would do. These are different types of arguments typically used for different goals and appearing in different contexts, but they both use subsequences. There may be other methods using subsequences and to me it would be nice to have a page that lists them all together.
Based on Tim's suggestion, let me second his proposal of ``Decompose your ring using idempotents''.
``Peirce decomposition'' may be standard in some ring theory circles, but it is not at all standard among algebraic geometers/commutative algebraists, nor among representation-theorists, both of whom would be interested in this article. As I said, I had never before heard of it (the name, that is, not the concept), and I have been studying these fields for close to fifteen years. I'm not saying this to pull rank, but rather to point out that this name is not nearly as well-known as the concept, and so there will be many people interested in the contents of this page who wouldn't have any idea about its contents under the present name.
This article should not be a stub. It is infact the best suggestion I have come across here. To see how it can lead to some 'mathematically interesting information',I suggest this webpage- http://disquisitionesmathematicae.wordpress.com
I have a couple of suggestions. One is to do a Gaussian computation in this article. A natural example is to take a bivariate normal random variable and create out of and two linear combinations and that are linearly independent. That should perhaps come after my second suggestion, which is simply to explain how to diagonalize a quadratic form (by giving a few examples). If I have a spare moment, I may add these examples.
The second one is one I care about because a lot of people I teach seem to have the impression that the only way of diagonalizing a quadratic form is to diagonalize the associated symmetric matrix.
Incidentally, I'm all in favour of using the same example in more than one article, so there might be a case for having the Fourier transform of a Gaussian done here as well.
I've amalgamated the two articles, and kept the nicer name. A few thoughts that I'd be interested in comments on:
1. Perhaps this is now overloaded with examples about free groups. A new page might be appropriate.
2. A good complement might be a theorem about the fundamental group of a surface. Almost anything would do. One could prove the analogue of the Scheier Index Formula for a closed surface, for instance.
3. Did you have a specific Serre-style example in mind, Matthew? Of course, that stuff is a very nice application of these ideas, and one day I hope to write Use actions on trees to study graphs of groups or whatever, but it seems quite hard to develop in the space available on this page.
Thanks for your reply. I think its helpful to explain our various points of view. I'm going to write a little more about mine; it is a little redundant, given what I've already written, but perhaps it will put my thoughts in a larger context than just the probability page.
I think that one thing that is helpful to think about in preparing tricki pages, and especially the top-level pages, is who the possible audience will be, and what they should be getting out of their tricki experience. (My impression is that this is every much in Tim's mind as he writes his articles and comments.)
There are some areas of math which I would say are almost purely technical, and where it is reasonable to expect a certain level of mathematical competence of the reader. For example, How to use spectral sequences (a page that I hope to eventually write, if someone doesn't beat me to it) is unlikely to be studied by anyone who doesn't have a solid undergraduate background in algebra, and probably some graduate training in algebra and or topology as well. Spectral sequences are really an intrinsically technical tool. They are important (and so I think they merit a tricki page, especially because many grad students suffer in learning to use them, and any additional help and explanation will, I expect, be welcomed by many), but are not a fundamental topic in mathematics, of interest to those who are not working mathematicians, or aspiring to become such.
On the other hand, one can expect that the Diophantine equations front page, when it is written, to be of much broader interest. Diophantine equations are a central topic in mathematics, and of interest to a broad range of people: amateurs, students involved in the maths olympiad or similar competitions, undergraduates, and of course graduate students and professional mathematicians. There is a lot that one can say technically about Diophantine equations: one can talk about Galois cohomology, and how this plays a key role; also modular, or more generally automorphic forms, and the Langlands program; the Hasse principle, -adic numbers and adeles; - and -functions; the circle method; and so on. These are all central ideas in the modern theory of Diophantine equations. But I don't think that they should be the focus of the Diophantine equations front page; they will be outside the technical range of many visitors to the page, and there will be much of interest that can be said about Diophantine equations that will be accessible to these visitors — but only if it is couched in less technical language. Furthermore, writing things in as accessible way as possible will help professional mathematicians as well: no doubt there will be mathematicians reading the Diophantine equations front page, once it exists, whose own area of expertise will be far from number theory; using the bare minimum of technical language required for any particular example or technique will make it much easier for them to extract utility from the tricki.
Probability is similar: it is a fundamental topic in mathematics. It is not a purely technical subject, but rather is of interest to a very wide range of people, from those interested in elementary calculations, to those interested in substantial applications (to the real world, or perhaps to problems in other parts of mathematics), to those interested in the theory itself for its own sake.
We will want the tricki page to be as accessible to as many of these people as possible; not all of them will know measure theory, and there will be many examples that they can understand despite this, if we write them in an a certain way.
I don't think that every example should be requried to use the absolutely minimal amount of machinery necessary to make it tick; indeed, some examples will have as their point the illustration of how to use a particular piece of machinery or advanced technique in a simple situation (as part of an explanation of how to use that piece of machinery or technique). But I don't believe that we want every example to be burdened by machinery, especially in broad topics of wide interest such as probability or number theory. And a logical consequence of this belief is that the top level introductions to these topics shouldn't themselves be entirely enmeshed in machinery and technical language. They should allow for multiple viewpoints on their topic, both elementary and advanced.
I will close by emphasizing that I am very far from an expert in probability, so I don't think that I will actually be contributing much to the probability branches of the tricki tree. On the other hand, I do hope to learn from it as it grows (and perhaps improve my understanding of measure theory in the process!).
Thanks for catching this — when I wrote this I wasn't mentally distinguishing between homotopy equivalence and homeomorphism.
With regard to your comment on combining pages:
I hadn't looked carefully at Actions on topological spaces; I rather just saw the comment about this result on subgroups of free groups, and thought that it would be a good thing to put into the tricki.
Feel free to combine the articles as you think is best. One thing that might be good would be to follow Tim's encouragement to have imperative titles. (You can see that I followed it in naming this article.) But since you've probably put more work into the actions page than I have into this page, you should do what you think makes the most sense.
Being terrified by the word measure is I think as reasonable as being terrified by the words `group' or `ideal.'
But the fact remains that it is possible to ask questions about probability without knowing what a measure is. And this is the probability front page.
If you go to the Group Theory front page, you'll see that it starts with a (brief) attempt to explain that the notion of group formalizes the notion of symmetry. So anyone who has a question about symmetry will perhaps be able to figure out that they need to start thinking about groups. Something similar should apply here.
> Now one can try to study and answer as many questions as possible just in terms
> of , and my impression is that many results in probability theory proceed along
> these lines: one has a (or perhaps a sequence) of random variables satisfying
> certain restrictions on their distribution, and one draws some conclusion about
> the random variable, or about some kind of normalized limit of the sequence. (I
> am thinking say of the central limit theorem.) In these theorems the particular
> sample space doesn't really play any role, as far as I know; one can just argue
> with the distributions (or more abstractly, and perhaps more generally,
> with the push-forward measures attached to the random variables).
> It is, I would think, an important point to know that often one doesn't really > have to understand the full details of the sample space, but rather, just the
> distribution of the random variable one is studying.
Once again, we are in agreement, but let's be careful about what the random variables and their distributions are.
Let's look at the example you gave, the central limit theorem. What is this theorem about?
Suppose is a probability measure on with unit second moment, i.e., and zero mean, i.e., . Here is the borel sigma field. Let be the product measure on . Let be the coordinate maps. Let be the measure defined on by the map. The central limit theorem states that
This explicit way of writing things makes it clear that the central limit theorem is a statement not about real random variables and their distributions but about product probability measures on and the transformation of such measures under summation and scaling. As you said, there could have been even a larger space on which were defined. In that case one first pushes the measure forward onto using the map .
Added later: Another point of view, for this problem, is to think only about the pair and a measure on this pair with the above mentioned properties. We then define , where is the convolution of with itself times. The CLT says . . This point of view is also the one that quickly suggests the most well known proof of this result: take Fourier transforms and study their limits. Most generalizations though will involve measures on or larger sets.
I guess, I made my point thoroughly :) I find it extremely useful to state the sets and the measures I am using and focusing on these and thinking about methods in terms of these. I am completely open to other ways of thinking, especially from the point of view of writing tricki articles. Sorry if I took too long in expressing my thoughts. Many thanks again for your thoughts!
Thanks for the comments. My goal in writing the things I wrote here was a succint classification of the central problems that people study in probability theory. I thought such a classification would allow someone to easily locate a technique of interest.
I think we all agree that the front page can be made more lively and less technical.
Here are few points on some of your comments:
Yes. My first choice is titles like "To prove that a small group is simple, look at the sizes of the conjugacy classes," but that won't work for more general pages that discuss or link to several techniques. So then titles such as you suggest (perhaps I'd go for "problems" rather than "a problem" but that's a very minor quibble!) would come into play.
Also, this is false - it's homotopy equivalent to a bouquet, but is never a bouquet itself unless it's a homeomorphism. Again, this is already discussed in Actions on topological spaces.
I suppose you're saying that you'd rather have pages with names like "How to solve a problem about finite groups", right? I'm all for that. I'll make the change at some point.
This article is very similar to Actions on topological spaces. Perhaps they should be combined?
I think that Tim's suggestions make good sense. The measure-theoretic foundations for probability are needed for studying sophisticated questions, but there are lots of elementary quesitons in probability theory that are basically questions of counting, and it would for a reader to the tricki to be able to reach them in the tree structure without having to go through measure theory first.
Related to an earlier comment: here is my guess as to what it might mean to avoid referring to the sample space:
If is a probability space and a random variable,
then we can push-forward the meausre via , to get a meausre on :
the push-forward is defined by Now one frequently
finds that is absolutely continuous with respect to Lebesque measure, and so can be written in the form , where is a function and denotes Lebesgue measure.
I guess is the probability distribution of the random variable .
Now one can try to study and answer as many questions as possible just in terms of , and my impression is that many results in probability theory proceed along these lines: one has a (or perhaps a sequence) of random variables satisfying certain restrictions on their distribution, and one draws some conclusion about the random variable, or about some kind of normalized limit of the sequence. (I am thinking say of the central limit theorem.) In these theorems the particular sample space doesn't really play any role, as far as I know; one can just argue with the distributions (or more abstractly, and perhaps more generally, with the push-forward measures attached to the random variables).
It is, I would think, an important point to know that often one doesn't really have to understand the full details of the sample space, but rather, just the distribution of the random variable one is studying. (In practical terms, the latter is at least in principle capable of being measured, while the sample space might be completely inaccessible.)
One suggestion is to begin with a more intuitive discussion of probability on the front page. One could define a random variable as being a quantity, that varies randomly, and that one wants to study (e.g. sum of the faces of two rolled dice, height of a random American, ...). It would be good to emphasize that random variables can be discrete (the dice example) or continuous (the height example).
We could then explain that a random variable has a distribution, which basically governs its behaviour. (And again this could be illustrated with the examples.) In general, emphasizing that there are some very common and important distributions, e.g. binomial, normal, Poisson, would be good.
Next could come the discussion of how this is mathematically modelled, with the idea being that simple models suffice for easy discrete questions like rolling a pair of dice, that more analysis is needed for studying continuous random variables, and that the general foundations rely on measure theory, etc.
Then we could begin with links to various subpages, with appropriate commentary. One page could be rigorous foundations of probability, which is where the current discussion of measure spaces, random variables as meausrable functions, etc., would live.
The question here is not what the correct way is of setting up a rigorous theory of probability (either for the purposes of research or for the purposes of giving a lecture course on the subject) but rather what is the best thing to put on a probability front page that is intended to help people find ways of solving probabilistic problems.
Suppose that I have a problem in discrete probability: to prove, say, that if I toss a coin 100 times then I have a very high probability of getting at least 30 heads. And suppose I come to the Tricki for help and visit the probability front page. As it is written at the moment, I would be terrified: if I needed help with that kind of problem, then I probably wouldn't know any measure theory, but the clear impression I would get from the page was that measure theory was an essential prerequisite.
I do not object to the content, but merely to its place in the Tricki hierarchy. I think a more obvious organization would be as follows. One begins with an informal description of probability theory, mentioning simple problems about calculating and estimating probabilities. One then explains that the intuitive view of probability has important limitations, and that a modern theory of probability makes heavy use of measure theory. And one then has links to more specialized pages such as elementary probability (essentially just counting problems), discrete probability, and general measure-theoretic probability. That way, the reader interested in problems about particular kinds of convergence of random variables would go to the advanced page, and the reader interested in the probability of getting all four aces in a bridge hand would go to the elementary page, and nobody would be frightened off.
A question that would have to be considered is how much of the basic theory of probability actually belongs on the Tricki. The default would be not much: you assume that the reader knows the basics and the Tricki articles are there to help him/her solve problems.
Added later: an interesting comparison can be drawn with the Wikipedia introduction to probability, which is more in the style that I would expect for a Tricki front page (though it wouldn't be perfect because it is not focused on problem-solving in quite the right way).
This page seems to me to be shaping up nicely, but I'd like to raise a question about all the dead links in these three paragraphs. The implication seems to be that there will be pages called "Abelian groups front page", "Finite groups front page", "Lie groups front page", "Profinite groups front page", "Combinatorial group theory front page" and "Geometric group theory front page". Is that the idea?
If it is (and it seems sensible), then at some stage I would probably want to rewrite this section slightly, in order to avoid the impression it currently gives that the Tricki focuses on subject matter (things like the definition of an Abelian group) rather than proof techniques. So I'd probably make those dead links into Wikipedia links and then have a paragraph saying something like "These special classes of groups have their own particular techniques, which are discussed and linked to from the following pages."
I had noticed that too, and considered merging them. I may yet do so, but I want to wait until I have thought of good examples for each. Somehow, with this article the focus is on generalization via abstraction, and thereby obtaining a question that gets more to the heart of the problem, whereas with the other one you may not be generalizing at all, because you may be using an abstract description that completely characterizes the original concrete object. (As an example of the latter, it is very useful to think of the reals as a complete ordered field rather than say as a set of Dedekind cuts, even though there is only one complete ordered field. And it is sometimes helpful to think of the exponential function just as "a" function that satisfies and .)
If we do end up with two articles, there's no doubt that they will be closely related, and that this should be highlighted.
Based on the quick descriptions of this article, and the following article:
http://www.tricki.org/article/Think_axiomatically_even_about_concrete_objects
It seems to me that they are getting toward a similar point. Of course both articles are still stubs; if the intention is to explain different tricks, perhaps someone could explain in the quick descriptions how they are different. Just my two cents.
You are correct, but then one probably shouldn't use the term "multiplicity space" to describe . I will make an edit that reflects your comment.
As an example, how about proving that a functor that preserves exactness in sequences of the form necessarily preserves exactness in sequences of the form ? One could then apply this to tensor products with a flat module.
"The proof of Theorem 5 brings out an important point, namely that if W is a representation of G\times H, and U is an irreducible G-representation, then"
That sounds sensible. I don't object at all to using the phrase Peirce decomposition to describe this procedure, as long as we don't rely on it as the only means of navigating to this page. I think in general that it is better to avoid using names (especially people's names) for things if one can avoid it, unless the name is absolutely universally known or absolutely unavoidable.
Thus ``Prove that your holomorphic function is bounded" (or some variant of this) would be better than ``Apply Liouville's theorem" as an instruction related to trying to show that a holomorphic functions is constant. After all, there is always the possibility that someone doesn't know or has forgotten what Liouville's theorem says, and then the name carries absolutely no hint about the method. (As a contrast, ``Apply the maximum modulus theorem'' is probably not quite as bad in this respect, because it contains in its name at least some semblance of what it is about.)
As a possible compromise, what about a link to this article on the "How to use" page, called "How to use the Peirce decomposition"? The title of the article would be different, but there's no problem with that. There could be a hidden-text quick description that said that the Peirce decomposition was a decomposition of rings using idempotents.
OK, why not have a look at what I've done and see what you think. I don't insist on keeping things like this.
The two types of arguments that this article lists are very common in probability theory. They are also of different nature and they usually appear in different contexts.
The first one, the one that goes to the subsubsequences is a soft analysis method, I have seen it used to prove weak convergence in function spaces very often (for example see the large deviations book by Dupuis and Ellis). In this setup, you have an obvious limit process. To establish convergence you first prove total boundedness (in the context i just mentioned this involves the use of the relative entropy function or sometimes the boundedness of the increments). This automatically guarantees convergence along subsequences (and this is a consequence of a general Bolzano-Weirstrass theorem). It remains to show that each subsequence has a further subsubsequence that converge to the candidate limit process. This is usually clear intuitively and can be proved as a consequence of a general law of large numbers type result.
The second one, finding a fast converging subsequence which approximate the whole sequence well: this I would call a hard analysis method. You first have to find your fast convergence sequence: this means to do computation to show that the subsequence you picked does converge at a certain rate. Then you have to also show, via hard calculation, that the whole sequence doesn't wander off too much from this subsequence. Almost all `convergence almost surely' arguments that I know that don't use martingales are of this form. For example: law of the iterated logarithm, the strong law of large numbers. I would say this is one of the main methods of proving almost sure convergence (of random variables, processes).
Because of these differences I think it would be logical to have separate articles for them. Whether to have a further page like the one i proposed? My personal take is I always like classification, if there is an obvious way to classify arguments then I would do. These are different types of arguments typically used for different goals and appearing in different contexts, but they both use subsequences. There may be other methods using subsequences and to me it would be nice to have a page that lists them all together.
Based on Tim's suggestion, let me second his proposal of ``Decompose your ring using idempotents''.
``Peirce decomposition'' may be standard in some ring theory circles, but it is not at all standard among algebraic geometers/commutative algebraists, nor among representation-theorists, both of whom would be interested in this article. As I said, I had never before heard of it (the name, that is, not the concept), and I have been studying these fields for close to fifteen years. I'm not saying this to pull rank, but rather to point out that this name is not nearly as well-known as the concept, and so there will be many people interested in the contents of this page who wouldn't have any idea about its contents under the present name.