Hi folks,

Here are the latest problems from the 20 Questions seminar for the query-hungry :)

====1 Sune====

Is it possible to explicitly define (without AC) a total ordering on a set with cardinality greater than ?

====2 Yuhao====

Let be the function field of an algebraic curve . Then by the

birational nature of the genus, it determines the genus of the

curve. How can we see directly from the field?

(One way is to PICK a function in and we get a ramified morphism , then use Riemann-Hurwitz to compute the genus; However, I

think there should be a more “intrinsic” way to see that, i.e. without

picking a function in an arbitrary manner. The genus is a well-defined

invariant for any extension of the ground field of tr.deg. 1, right?)

====3 Anon====

Determine explicitly a partition of the plane into two sets A and B such that neither A nor B contains (the image of) a non-constant continuous curve.

====4 An H====

Let denote the cyclotomic field obtained by adjoining th roots of unity to , where is prime. Let be a prime element in the ring of

integers of which is coprime to . Let denote the Kummer extension of

by adjoining a th root of to .

Question: is it always true that divides the exponent of the

prime ideal in the relative discriminant ? (as usual,

denotes a primitive pth root of unity)

====5 Critch====

In the categories Set and AbGrp (and I believe any topos), finte limits (e.g. kernels, products, equalizers…) commute with directed colimits (aka direct limits). Give an example category where this fails.

====6 Anton====

Can a coequalizer in the category of Schemes fail to be surjective? (Note: it must hit all closed points in the target, because otherwise the closed point could be removed to make the coequalizer smaller.)

See the mathoverflow.net question: [http://mathoverflow.net/questions/63/can-a-coequalizer-of-schemes-fail-to-be-surjective]

====7 Darsh====

Let be rational.

In what generalized notions of convergence does the sequence converge?

====8 Harold====

In the category of smooth manifolds, when does the fibre product of two maps exist?

A) Necessary conditions

B) Sufficient conditions (e.g. that each map is a fibration).

====9 James T====

Which bounded linear maps on a Hilbert space are the exponentials of other maps? I.e., what is the image of the map ?

====10 Pablo====

Give a simple example of a (necessarily infinte dimensional) Lie algebra that is not the lie algebra of any (necessarily infinite dimensional) lie group.

====11 Theo====

[http://mathoverflow.net/questions/134 See mathoverflow.net].

****

Cheers!

-Critch

Apologies for the gripe, but this is getting a tad confusing. We have these questions here, and there’ve been answers in the threads, the questions also have their own wiki, and answers appear there, plus the questions get posted on mathoverflow and answers appear there! Is there a canonical site? Personally, I’d prefer mathoverflow since then by answering them I can get a bit more reputation and over take Scott Morrison again (Ben’s rapidly getting out of my league, but I reckon you lot have a cartel to post algebraic geometry questions for each other to answer, leaving us poor topologists floundering in your wake).

I agree with Andrew that this is becoming confusing. My preference would be to have the discussions here, and keep a record of final solutions on the wiki.

And, to start some discussion, it seems to me that the answers to #1 on the wiki are quite wrong. Any subset of a well ordered set is well ordered. So, if there were a well ordered set with cardinality greater than or equal to that of , then there would be a well ordering of , which is something that can not be constructed without AC. Am I missing something dumb here?

The references on the wiki do explain very nicely how to create a well ordered set that is not less than or equal to the cardinality of .

Also, does anyone have any good ideas about how to attack 9? The solution on the wiki doesn’t make any sense to me; it seems to me that the same argument would suggest that the exponential map is surjective, which is not true.

Sigh — I’m dumb on 1; I misread the question. Still want to know about 9, though.

Question 11 seems to be asking whether nondegeneracy is an open condition for a smoothly varying symmetric bilinear form. Is this an adequate translation?

Incidentally, what is the image of exp on sl_n? I’ve been curious about this for a while (but apparently not enough to ask anyone).

I don’t know a concise way to state the answer. Let denote the Jordan block with eigenvalue and size . Then the logarithms of this block (in are of the form . This has trace .

If a matrix in has Jordan decomposition , then the problem is to choose such that . The first sum is always in . So it is always possible to take logs when all the Jodan blocks are of size , or more generally when their sizes have no common factor. The first example where you can’t take a log is .

@Andrew, re: #1. I’m also strongly in favour of having mathoverflow.net be the “canonical” site. At the last 20 questions seminar, we were simultaneously typing up questions for the wiki and for mathoverflow. Now that we’ve created a 20questions user at mathoverflow, I think in future they’ll just be recorded there. You can easily select questions tagged “20-questions”.

So — if you have answers to these questions, please consider putting your answer up over there, too!

Mathoverflow would be my choice, I’ll admit. Not for the spurious reputation reason I gave above, of course. Mainly because there it’s possible to separate out the questions and discuss them. My problem with mathoverflow is that it doesn’t seem possible to get a good discussion going. For example, I put an answer to the B(H) -> gl(H) question there that I thought was correct, but now I see that it has holes in it. Unfortunately, a discussion on that really takes the form of a list of answers (which can get arbitrarily reordered). However, I prefer that to here were it can be very difficult to follow since there are potentially 20 discussions going on simultaneously. In the first lot of 20 questions then I completely missed the answer to what was the only interesting question there because it was swamped by a load of algebraic geometry nonsense. (Sorry, my warped sense of humour getting the better of me there.)

Maybe the answer is this “community wiki” part of mathoverflow. I have no idea what that is, but if it is what it sounds like then maybe there should be the convention that 20 questions answers are “community wiki”fied.

Some questions on mathoverflow are easy to give a short answer (such as contractibility of the infinite sphere). Others, though, will take a little work. I would like to be able to work towards an answer, polymath-style, with the possibility that without any one person doing too much then an answer appears. If the “community wiki” is what enables that, then I’ll start using it a lot more!

Another argument for mathoverflow is that if it works, then it will be a lot easier to search and find answers on than a dedicated wiki or even this blog. So if people want their answers to be recorded in a place where they can be easily found, then mathoverflow seems the most logical place.

Just to reiterate my view, I think that we have the best forum for discussion. Once there are complete solutions, I agree that it is nice to have them recorded someplace clean like mathoverflow or the wiki.

I will point out that the mathoverflow FAQ states, quite correctly in my opinion: “This is not a discussion board.”

Since everyone else likes mathoverflow better, I have put some notes on Question 9 over there.

UpdateI also added a few more thoughts to the wiki on Question 2, and am heading over to mathoverflow to copy my answer there.> I will point out that the mathoverflow FAQ states, quite correctly in my opinion: “This is not a discussion board.”

You mean we’re supposed to take some notice of that?

Seriously, whilst it’s useful to have answers to the kind of questions that can be answered quickly, it’s not always possible to know that your question is that type of question when you ask it. So putting the “this is not a discussion board” notice at the front door either means I won’t go in, or I’ll go in and ignore the notice because I don’t know how to take it into account.

But I do find discussions in blog comments extremely difficult to follow. Threading is a mild improvement, but there’s still the problem that only a handful can start a discussion, and searching is problematic. One needs different systems for different parts of the discussion (and if anyone mentions Google Wave then I’ll threaten them with a pointed set). But until we’ve got all that, mathoverflow is the best of what we do have.

Chipping in with the general commentary on Q9 (logs in B(H)) – my impression is that this one of those functional analysis questions that are easy to state but which people as a whole have drifted away from. The Conway-Morrel paper seems to have been the last substantial effort to get anywhere on this question.

Yes, my position on mathoverflow.net is that it does one thing much better than existing math blogs, namely that any mathematician can register and post a question; but otherwise it is too Twitter-like and I would rather have something more blog-like.

Regarding Q1, just consider the constructible universe L. It’s a well known fact that L has a definable well-ordering (see Jech or Kunen), so I guess that it settles this question.

Greg-

That seems like a very strange description of mathoverflow. Other than fitting into some generic rubric of Web 2.0, I can’t see what it has in common with twitter. You’re right that mathoverflow and blogs have different strengths, but as far as I can tell, mathoverflow is *much* more conducive to actual mathematics happening. Look at the content that’s appear on mathoverflow in a week, and think about how it compares with what we’ve produced on this blog in 2 and half years.

I think “Twitter like” is not correct. The ideal use of mathoverflow is to ask well focused questions for which there is probably some expert who knows the answer. I would compare it to calling up a car mechanic — you want to know what is making that funny noise and how you can stop it.

The ideal use of a blog is to have discussions of ideas that are not yet fully fleshed out; either because the argument is flawed, or because the goal is not completely clear. I would compare it to going to the departmental tea; where I can pull someone aside and say “So, have you ever seen anything like…”

As far as I can tell, the ideal use of Twitter is to shout “Look at me! I know something interesting!” But I don’t Tweet, so I may be missing something here.

Greg-

I’m also a little surprised that you don’t like mathoverflow, in light of our conversation a couple years ago about making a centralized math blog site. I think the democratic format of mathoverflow plays to the strengths of centralization, while blogging wouldn’t benefit much (caveat: I don’t have any evidence to back up this argument).

I expect to see more in-depth answers on mathoverflow as the site matures. For example, I’ve been going back to old questions and revising my answers for clarity, adding links to references, etc.