Monthly Archives: July 2010

Fill up my cup. Mazel Tov.

It has come to my attention that there are those who will deny the unstoppable pop genius that is the Black Eyed Peas’ international megahit “I Gotta Feeling.”

I want to speak only about the line “Fill up my cup — mazel tov!”  Amazing.  Amazing! You start with “fill up my cup” — which at this moment in the song reads as a cup of beer at a party or maybe even a cup (glass?) of champagne at a sophisticated nightclub.  And then with just two words the song pulls aside the curtain and says — you are at a bar mitzvah. Look to your left — the man dancing there is 75.  Look to your right — the girl dancing there is 12.  Look down at the cup — it’s a kiddush cup.

It’s a moment of incredible self-knowledge and assurance:  the band saying:  We’re here for one reason, and that’s to outdo “Celebration,” and write the greatest wedding/barmitzvah song ever made.  If you’re listening to this at a bar mitzvah, this is your song.  And if you actually are listening to this while drinking champagne at a sophisticated nightclub, guess what — you’re dancing to the same song as the septuagenarians in the Marriott Ballroom 3, and you can’t stop yourself.

I know of nothing like it in contemporary pop music.  Like oh my God.

No need to link to this much-heard song, so here’s Spongebob singing it.

I have not even spoken of the poignancy of the closing “Do it again” refrain — partly because I’m not sure it’s actually supposed to be poignant.  It might just be that it reminds me of the Kinks song:


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Canonical

One more comment on “canonical,” promoted to its own post because the non-mathematicians presumably stopped reading the other one very early on.

It’s common for mathematicians to use the word “canonical” colloquially, to mean something like  “a choice universally or at least generally agreed on.”  For instance:

The clock in Grand Central Station is the canonical place to rendezvous with people in midtown New York City.

I always thought of this as an outgrowth of the mathematical use of the word; but actually, there’s a bit of tension, because I think in this sense “canonical” almost always refers to a choice which is conventionally agreed on, and for which there might be a good reason, but which isn’t really forced upon you the way that canonical things are in mathematics.  The canonical rendezvous might just as well have been the lobby of the Empire State building.

I found a definition of “canonical” in a Hacker Slang dictionary which roughly agrees with this usage:

The usual or standard state or manner of something. This word has a somewhat more technical meaning in mathematics. Two formulas such as 9 + x and x + 9 are said to be equivalent because they mean the same thing, but the second one is in `canonical form’ because it is written in the usual way, with the highest power of x first. Usually there are fixed rules you can use to decide whether something is in canonical form. The jargon meaning, a relaxation of the technical meaning, acquired its present loading in computer-science culture largely through its prominence in Alonzo Church’s work in computation theory and mathematical logic (see Knights of the Lambda Calculus). Compare vanilla…

Anyway.  Non-math readers, would you ever use the word “canonical” in the sense described here?  Math readers, can you give an account of its colloquial usage more articulate than my own?

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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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Green eggs

We don’t eat ham in our house, but CJ got excited about having green eggs anyway.  We had them for lunch today.  And this is how you make them:  get the blender out and puree a bunch of peas, a handful of basil, and some grated parmesan or what have you.  (I used Farmer John’s excellent asiago.)   Put in a little half and half if you swing that way.  You will get something very green.  Mix the greenness with eggs and scramble up as desired.

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Duel at Dawn

Speaking of Galois, my review of Amir Alexander’s Duel at Dawn is up at BN Review today.  The book draws an interesting connection between the Romantic literary area and the invention of the “romantic” mathematical hero, of whom Galois is obviously the sterling example.  But Alexander commendably reaches past the endlessly-repeated Galois story to cover a lot of material less familiar to readers of pop math; I learned a lot about Abel, Bolyai, D’Alembert, and Cauchy (who was constantly getting rebuked by his deans for teaching epsilons and deltas in first-year calculus!)

The uncollected and very worthwhile David Foster Wallace essay “Rhetoric and the Math Melodrama,” which I mention towards the end of the piece, can be found in .pdf here.

Also, writing this review gave me the opportunity to use the word “emo” in print for the first time.  I hope my younger readers will let me know whether my usage is roughly correct.

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Bhargava and Satriano on Galois closures of rings

Manjul Bhargava and Matt Satriano (starting a postdoc at Michigan this fall) posted a nice paper on the arXIv, “On a notion of “Galois closure” for extensions of rings.” The motivation for this work (I’m guessing) comes from Bhargava’s work on parametrizations of number fields.  Bhargava needs to generalize many classical objects of algebraic number theory from the setting of fields to rings.  For instance, the sextic resolvent of a degree 5 polynomial f(x) (developed by Lagrange, Malfatti, and Vandermonde in the late 18th century) is a degree 6 polynomial g(x) which has a rational root if and only if the equation f(x)=0 is solvable in radicals.

In modern Galois-theoretic language, we would describe the sextic resolvent as follows.  If K/Q is the Galois closure of the field generated by a root of f, then the Galois group Gal(K/Q) acts on the 5 roots of f, and is thus identified with a subgroup of S_5.  On the other hand, S_5 has an index-6 subgroup H, the normalizer of a 5-cycle.  So the action of Gal(K/Q) on S_5 / H gives rise to an extension of Q of degree 6 (maybe a field, maybe a product of fields) and this is Q[x] / g(x).

So far, so good.  But Bhargava needs to define a sextic resolvent which is a rank-6 free Z-module, not a 6-dimensional Q-vector space.  The sticking point is the notion of “Galois closure.”  What is the Galois closure of a rank-5 algebra over Z?  Or, for that matter, over a general commutative ring?

Bhargava gets around this question in his paper on quintic fields by using a concrete construction particular to the case of quintics.  But in the new paper, he and Satriano propose a very nice (“very nice” means “functorial”) completely general construction of a “Galois closure” G(A/B) for any extension of rings A/B such that A is a locally free B-module of rank n.  G(A/B) is a locally free B-module, endowed with an S_n-action, as you might want, and it agrees with the usual definition when A/B is an extension of fields.

But there are surprises — for instance, the Galois closure of the rank 4 algebra C[x,y,z]/(x,y,z)^2 over C is 32-dimensional!  In fact, the authors show that there is no definition of Galois closure which is functorial and for which G(A/B) always has the “expected” rank n! over B.  This might explain why no one has written down this definition before, and I think it is what gives the paper a sort of offbeat charm.  It illustrates a useful point:  you’ve got to know when it’s time to mulch an axiom.

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I am cranky about people who are cranky about the tenure system

Via Deane Yang’s Facebook feed, this New York Times round table on the question of tenure, featuring weigh-ins from faculty members in education, English, religion, education again, and economics.  Notice anything missing?  Like, say, science, engineering, law, and medicine?  I said this before but I’m cranky about this piece so I’ll say it again.  The reason we need tenure in these fields is not because we’re worried about getting fired for teaching an anti-establishment line on epsilons and deltas.*  It’s because universities have to compete with private employers for scientists, mathematicians, engineers, lawyers and doctors.  Mark Taylor writes:

If you were the C.E.O. of a company and the board of directors said: “We want this to be the best company of its kind in the world. Hire the best people you can find and pay them whatever is required.” Would you offer anybody a contract with these terms: lifetime employment, no possibility of dismissal, regardless of performance? If you did, your company would fail and you would be looking for a new job. Why should academia be any different from every other profession?

Maybe because academia pays a lot less than many other professions?  Does Taylor have any suggestions as to what alternative benefit we should offer candidates in order to make an academic job worth their while?  Does he really think that, absent tenure, our board of trustees would tell our chancellor, “Hire the best people you can find and pay them whatever is required?”

Right now, tenure is what universities have instead of money.  I don’t see a lot of money coming our way soon.  So I think we’d better hold on to tenure.

* Although this actually happened to Cauchy!

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Nick Markakis is the new Mark Grudzielanek

Nick Markakis is leading the American League in doubles with 31, and has just 31 RBI.  Doesn’t it seem like it would be hard to hit with enough power to lead the league in doubles, and have so few RBI?  I thought maybe he was en route to a historic feat, but per Baseball Reference it’s not actually that rare to have more doubles than RBI.  The champ in this department is Mark Grudzielanek, who led the NL in 1997 with 54 doubles, and managed just 51 RBI.

It’s pretty clear that achieving this feat involves a serious commitment to not hit home runs.  But check out Ryan Freel’s 2006, in which he hit 30 doubles, 2 triples, and a respectable 8 homers, for a slugging percentage of .399, and still managed to drive in only 27 runs.  How did this happen?

Markakis, for that matter, has 6 home runs himself, and is slugging a respectable .455.  As far as I can tell, the highest slugging percentage turned in by a full-time player who ended the season with more doubles than RBI is .419, recorded by Frank Baumholtz in 1953.  So maybe Nick is on his way to the record books after all.

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Pizzayola

Full disclosure:  a few days after I posted this, a nice PR person from Ian’s mailed me a thank-you Post-it and a coupon for two free slices.

Reader, I used it.

Feel free to ignore my opinions about squid pizza from now on if you feel my integrity is now suspect.  I know I can’t be bought.

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Up yours, Cook’s Illustrated

1.  For signing me up for a “preview” program when I ordered one of your books — no doubt because I neglected to toggle a checkmark in some tiny opt-out box on your website, thereby “signing up” for this “service” — and then sending me a new book, which I have to mail back if I don’t want to pay you for it.

2.  For including an invoice that says I have to pay for the return postage, not revealing until I called to complain that I can print out a postage-paid label from your website.

I’m going to the post office now with their book and their label to mail this back.  A person more vindictive than I would include a brick.

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