Thanks for the comment and sorry for the confusion. What I meant was "sorted". We can use `unordered', if "sorted" is confusing/ not standard. Thanks again.
For the constant in van der Corput's lemma we have . The fact that this is best possible up to numerical constants can be seen by testing against the function .
Although that's true, I'm not completely in favour of mentioning it, because the point of this particular article is to encourage the reader to use the result with "finite". For example, to prove that is countable, I prefer defining to defining and using induction, even though both proofs are correct.
Actually, a weaker requirement is that the preimages be countable (i.e. they could be countably infinite). This works because as mentioned earlier in the article, the countable union of countable sets is countable.
I started editing the article, but couldn't phrase it well without making it sound recursive. We're trying to show that A is countable, and I'm saying the pre-image of every n is countable. Strictly speaking, I'm not being recursive or circular, but I'd like someone to write it in a way that won't confuse any one.
Excellent – I look forward to the results, and to making heavy use of links to these articles when I come to write the I have a problem about integration page.
The article was getting rather unwieldy, and I was always relabeling the references between one method and another. It will lead to ten stubby articles rather than one overly long one, but this is probably better for the longer term development of these pages.
A quick question about this: is it your intention to chop up the existing "bounding integrals" article into a fairly large number of subarticles, as this suggests? I myself think that would be a good idea – each technique could then be illustrated by more than one example, and each article could be devoted to a single technique (which, though not an absolute requirement, is in general preferable I think).
As I also wrote in a comment on the geometry front page, one aspect of hyperbolic, projective, etc. geometries is that they are associated to symmetric spaces, and so have strong ties to group theory (especially Lie groups) and related topics. So I think that there will be a lot to say about them tricki-wise.
I might try to write an article discussing this aspect of things sometime soon. (Something that explains how the group theoretic view-point can be helpful in certain contexts.)
The reason your indentation didn't work was that you were leaving a line between the items in the list. It seems that the asterisks are interpreted in a relative rather than an absolute way and that once you start a new paragraph everything starts again.
I've made some additions to the list, but am running into an awkward problem. I put in "hyperbolic geometry", for example, but that would normally be thought of not so much as a branch of geometry but as a tool that is very useful in branches of geometry such as 3-manifolds. Nevertheless, it seems to me that there are useful tricks for how to answer questions in hyperbolic geometry and that it makes sense to have a front page for it. Maybe this isn't a serious problem: there is no reason for the Tricki classification of articles to be the same as a typical classification of mathematics in to research areas. (To give another example, one would clearly want a linear algebra front page, even though one would think of linear algebra as part of the background of all mathematicians rather than as a research area in itself – even if it is possible to come up with interesting problems about matrices and things).
As a possible answer to the question this note raises, I will suggest that we should have an (or potentially more than one) article on symmetric spaces, which include all these highly symmetric geometries. These are central to a whole host of fields (number theory and automorphic forms, representation theory, geometric group theory, and many others), and there is a lot that could and should be said about them.
Perhaps the first such article could explain that these ``highly symmetric geometries" have a natural group-theoretic origin, which one can exploit when studying them.
I have updated this page somewhat in line with the discussion on the ``Front pages" page. I also added links, matching those on the ``Front pages" page. (For some reason, I couldn't achieve the nice indentation that is on that page. If someone can fix my formatting, please do so.)
Based on Henry's reply, and the sentiments of Daniel's comment, I will go ahead and change the title to ``Geometry and topology front page'', and perhaps also edit the blurb a little to reflect this. (Thank you, too, for the kind words, Henry. But if I run out of steam, as is quite possible, then others should please go ahead and make their own edits in the same direction — or in any other direction that seems appropriate, for that matter!)
I think it should be called "Geometry and topology". Clearly some subfields are distinctly one or the other, but there's a continuum between them: the use of hyperbolic geometry in 3-manifold topology is a good example.
Matthew's done a beautiful job starting the geometry front page, and I didn't want to meddle, but I'd be in favour of changing the title to "Geometry and topology front page".
Geometry can mean many things, clearly. It can mean metric geometry (with roots in Euclidean geometry, going all the way through to Riemannian geometry), but also projective geometry, say (which is one precursor to modern algebraic geometry), symplectic geometry, etc.
In 3-manifold topology, hyperbolic geometry can often play a big role (as well as the other Thurston geometries).
Of course, as you wrote, other quite different methods can also play a role.
It might make sense to change the title of this front page to "Geometry and topology" (which is in any case how currently it is listed on the Front pages for different areas of mathematics page. One could then write a slightly more elaborate blurb in the quick description, explaining some of what I wrote above.
As Tim wrote, general topology has a different, more analytic flavour, in comparison to the kind of topology we are likely to be linking to from here (whether it be low-dimensional, algebraic, differential, symplectic, or whatever), so we could include a remark to this effect, with a link to the general topology page.
What do other people think about this possible name change?
Historically, topology was geometria situs...
However, most low-dimensional topology seems to have little to do with geometry (it might sometimes even be math.QA). I certainly don't know much geometry.
Maybe with further subdivisions it makes more sense:
Geometry and Topology having "Low-dimensional topology" as a subcategory, which in turn has "4-manifolds", "3-manifolds", "knots and links", and "surfaces" as sub-discplines... what do you think?
That's not quite what I'm doing. If we accept the dictum that a statement is obvious if a proof instantly springs to mind, then this statement is certainly obvious modulo Lebesgue measure: that is, the proof instantly springs to mind that the measure of the resulting union is too small. But the consistency of the definition of Lebesgue measure is not itself obvious. What I am arguing here is that there is no simple proof from first principles. I think the title of the section makes it fairly clear that we start from the position that Lebesgue measure is not yet established.
In the light of that, why don't we go ahead and fuse the pages? It seems to me that the best approach is to make an algebraic topology front page a child page of the geometry front page. I had originally imagined that Topology would split into General Topology and Algebraic Topology, but from the point of view of the Tricki, with its focus on types of problems, it makes much more sense to make general topology a subbranch of analysis.
I'd do the reorganization myself, but I'd rather leave it to one of you two, since you know more about the area and may not agree with everything I've just said.
You're assuming here that one is not yet convinced that measure theory is correct. This is an odd assumption, and it would be better to state it in the beginning of the problem. As it is, the "this is obvious" reaction is quite right and can be made rigorous with straightforward use of the Lebesgue measure. (rationals are a whole different kettle of fish of course, being a null set)
Thanks for the comment and sorry for the confusion. What I meant was "sorted". We can use `unordered', if "sorted" is confusing/ not standard. Thanks again.
Surely you are counting unordered strings here?
Just a note: There is an extra / at the end of the ``circle problem" link.
Thanks – now corrected.
For the constant in van der Corput's lemma we have . The fact that this is best possible up to numerical constants can be seen by testing against the function .
Anonymous:
Usually, all those are different algebraic structures:
represents the ring of polynomials on one variable () with coefficients on , i.e., $K[x]=\{\sum_{i=0}^n k_i
Although that's true, I'm not completely in favour of mentioning it, because the point of this particular article is to encourage the reader to use the result with "finite". For example, to prove that is countable, I prefer defining to defining and using induction, even though both proofs are correct.
Actually, a weaker requirement is that the preimages be countable (i.e. they could be countably infinite). This works because as mentioned earlier in the article, the countable union of countable sets is countable.
I started editing the article, but couldn't phrase it well without making it sound recursive. We're trying to show that A is countable, and I'm saying the pre-image of every n is countable. Strictly speaking, I'm not being recursive or circular, but I'd like someone to write it in a way that won't confuse any one.
Excellent – I look forward to the results, and to making heavy use of links to these articles when I come to write the I have a problem about integration page.
The article was getting rather unwieldy, and I was always relabeling the references between one method and another. It will lead to ten stubby articles rather than one overly long one, but this is probably better for the longer term development of these pages.
A quick question about this: is it your intention to chop up the existing "bounding integrals" article into a fairly large number of subarticles, as this suggests? I myself think that would be a good idea – each technique could then be illustrated by more than one example, and each article could be devoted to a single technique (which, though not an absolute requirement, is in general preferable I think).
Oops – thanks! I've added that in now.
As I also wrote in a comment on the geometry front page, one aspect of hyperbolic, projective, etc. geometries is that they are associated to symmetric spaces, and so have strong ties to group theory (especially Lie groups) and related topics. So I think that there will be a lot to say about them tricki-wise.
I might try to write an article discussing this aspect of things sometime soon. (Something that explains how the group theoretic view-point can be helpful in certain contexts.)
The reason your indentation didn't work was that you were leaving a line between the items in the list. It seems that the asterisks are interpreted in a relative rather than an absolute way and that once you start a new paragraph everything starts again.
I've made some additions to the list, but am running into an awkward problem. I put in "hyperbolic geometry", for example, but that would normally be thought of not so much as a branch of geometry but as a tool that is very useful in branches of geometry such as 3-manifolds. Nevertheless, it seems to me that there are useful tricks for how to answer questions in hyperbolic geometry and that it makes sense to have a front page for it. Maybe this isn't a serious problem: there is no reason for the Tricki classification of articles to be the same as a typical classification of mathematics in to research areas. (To give another example, one would clearly want a linear algebra front page, even though one would think of linear algebra as part of the background of all mathematicians rather than as a research area in itself – even if it is possible to come up with interesting problems about matrices and things).
As a possible answer to the question this note raises, I will suggest that we should have an (or potentially more than one) article on symmetric spaces, which include all these highly symmetric geometries. These are central to a whole host of fields (number theory and automorphic forms, representation theory, geometric group theory, and many others), and there is a lot that could and should be said about them.
Perhaps the first such article could explain that these ``highly symmetric geometries" have a natural group-theoretic origin, which one can exploit when studying them.
I have updated this page somewhat in line with the discussion on the ``Front pages" page. I also added links, matching those on the ``Front pages" page. (For some reason, I couldn't achieve the nice indentation that is on that page. If someone can fix my formatting, please do so.)
It's to make the differences as simple as possible, as these are what the polynomials are built up out of. I added that into the text- thanks!!!
Based on Henry's reply, and the sentiments of Daniel's comment, I will go ahead and change the title to ``Geometry and topology front page'', and perhaps also edit the blurb a little to reflect this. (Thank you, too, for the kind words, Henry. But if I run out of steam, as is quite possible, then others should please go ahead and make their own edits in the same direction — or in any other direction that seems appropriate, for that matter!)
This isn't too significant, but I think it would help to mention that you're working in base 3 here.
I think it should be called "Geometry and topology". Clearly some subfields are distinctly one or the other, but there's a continuum between them: the use of hyperbolic geometry in 3-manifold topology is a good example.
Matthew's done a beautiful job starting the geometry front page, and I didn't want to meddle, but I'd be in favour of changing the title to "Geometry and topology front page".
Geometry can mean many things, clearly. It can mean metric geometry (with roots in Euclidean geometry, going all the way through to Riemannian geometry), but also projective geometry, say (which is one precursor to modern algebraic geometry), symplectic geometry, etc.
In 3-manifold topology, hyperbolic geometry can often play a big role (as well as the other Thurston geometries).
Of course, as you wrote, other quite different methods can also play a role.
It might make sense to change the title of this front page to "Geometry and topology" (which is in any case how currently it is listed on the Front pages for different areas of mathematics page. One could then write a slightly more elaborate blurb in the quick description, explaining some of what I wrote above.
As Tim wrote, general topology has a different, more analytic flavour, in comparison to the kind of topology we are likely to be linking to from here (whether it be low-dimensional, algebraic, differential, symplectic, or whatever), so we could include a remark to this effect, with a link to the general topology page.
What do other people think about this possible name change?
Historically, topology was geometria situs...
However, most low-dimensional topology seems to have little to do with geometry (it might sometimes even be math.QA). I certainly don't know much geometry.
Maybe with further subdivisions it makes more sense:
Geometry and Topology having "Low-dimensional topology" as a subcategory, which in turn has "4-manifolds", "3-manifolds", "knots and links", and "surfaces" as sub-discplines... what do you think?
That's not quite what I'm doing. If we accept the dictum that a statement is obvious if a proof instantly springs to mind, then this statement is certainly obvious modulo Lebesgue measure: that is, the proof instantly springs to mind that the measure of the resulting union is too small. But the consistency of the definition of Lebesgue measure is not itself obvious. What I am arguing here is that there is no simple proof from first principles. I think the title of the section makes it fairly clear that we start from the position that Lebesgue measure is not yet established.
In the light of that, why don't we go ahead and fuse the pages? It seems to me that the best approach is to make an algebraic topology front page a child page of the geometry front page. I had originally imagined that Topology would split into General Topology and Algebraic Topology, but from the point of view of the Tricki, with its focus on types of problems, it makes much more sense to make general topology a subbranch of analysis.
I'd do the reorganization myself, but I'd rather leave it to one of you two, since you know more about the area and may not agree with everything I've just said.
You're assuming here that one is not yet convinced that measure theory is correct. This is an odd assumption, and it would be better to state it in the beginning of the problem. As it is, the "this is obvious" reaction is quite right and can be made rigorous with straightforward use of the Lebesgue measure. (rationals are a whole different kettle of fish of course, being a null set)