Tag Archives: math overflow

Why Laba is not on Math Overflow

A thoughtful post from harmonic analyst Izabella Laba about why she isn’t participating in Math Overflow (and, by extension, why other women in math might not be.)  The comments are good too.

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The different does not have a canonical square root

Just wanted to draw attention to this very nice exchange on Math Overflow.   Matt Emerton remarks that the different of a number field is always a square in the ideal class group, and asks:  is there a canonical square root of the ideal class of the different?

What grabs me about this question is that the word “canonical” is a very hard one to define precisely.   Joe Harris used to give a lecture called, “The only canonical divisor is the canonical divisor.”  The difficulty around the word “canonical” is what gives the title its piquancy.

Usually we tell students that something is “canonical” if it is “defined without making any arbitrary choices.”  But this seems to place a lot of weight on the non-mathematical word “arbitrary.”

Here’s one way to go:  you can say a construction is canonical if it is invariant under automorphisms.  For instance, the abelianization of a group is a canonical construction; if f: G_1 -> G_2 is an isomorphism, then f induces an isomorphism between the abelianizations.

It is in this sense that MathOverflow user “Frictionless Jellyfish” gives a nice proof that there is no canonical square root of the different; the slick cnidarian exhibits a Galois extension K/Q, with Galois group G = Z/4Z, such that the ideal class of the different of K has a square root (as it must) but none of its square roots are fixed by the action of G (as they would have to be, in order to be called “canonical.”)  The different itself is canonical and as such is fixed by G.

But this doesn’t seem to me to capture the whole sense of the word.  After all, in many contexts there are no automorphisms!  (E.G. in the Joe Harris lecture, “canonical” means something a bit different.)

Here’s a sample question that bothers me.  Ever since Gauss we’ve known that there’s a bijection between the set of proper isomorphism classes of primitive positive definite binary quadratic forms of discriminant d and the ideal class group of a quadratic imaginary field.

Do you think this bijection is “canonical” or not?  Why?

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Why Math Overflow works, and why it might not

I spent a bunch of time yesterday playing with Math Overflow, the new math Q&A website launched last week by Berkeley grad students David Brown and Anton Gerashchenko. The site is built on the popular Stack Exchange platform, giving users the power not only to ask and answer questions but to vote on other people’s answers, giving those users “reputation points” which they can use to unlock more features of the site.

I was chatting with Tim Gowers last month, in the context of PolyMath, about what made a website “sticky,” or, to put it more pungently, “addictive” — what makes users willing to go back to the same site multiple times a day, and keep it up for weeks or months?  Math Overflow seems to have this quality in a particularly pure form.  Unlike PolyMath — where showing up half a day late might well give you no chance of catching up and making a contribution — Math Overflow offers a constantly changing array of new questions.  Questions to which you might know the answer right off the top of your head — or at least if you take ten minutes to think about it, or just a quick half-hour to scan through some references or…

Now at this point you might say “I could answer this, but I don’t really have the time right now.”  But then somebody else would answer it first! And then you wouldn’t get that warm feeling of helping somebody out!

I think this quality of rightnowness is what’s kind of great about Math Overflow, the thing that will get a lot of people to look at it consistently and thus make it a useful place to ask questions.  But there’s also something worrisome about it.  It shouldn’t be important to be the first one to answer.  A much more rational response to that “right now” feeling would be:  “I don’t need the warm feeling.  An earnest, hard-working grad student will come along and give the same answer I would have given; except the E,HWGS will spend more time and give a more thorough answer with more details included.”  And maybe giving a terse, dashed-off answer as soon as you see the question will actually prevent that E,HWGS from ever writing the ideal answer!

But then, a terse, dashed-off answer is a lot better than no answer.  At the moment I’m very high on this site.  I hope a lot of people — even earnest, hard-working senior faculty — will put a shoulder to it, and see what happens.

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