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Cohen-Lenstra as categorification

Followup on my earlier claim to have been categorified.

The Cohen-Lenstra conjectures govern the p-adic variation of class numbers of quadratic imaginary fields.  At first glance they look very strange.  If you ask the conjectures what proportion of quadratic imaginary class numbers are indivisible by 3, you don’t get 2/3, as you might expect if “class numbers were random numbers,” but rather the infinite product

(1-1/3)(1-1/9)(1-1/27)….

It gets worse — if you ask for the probability that the class number is indivisible by 27, you get the same infinite product multiplied by some completely meaningless-looking rational function in 1/3.

And you ask:  how could anyone ever come up with these conjectures?

The answer, of course, is that they didn’t think about class numbers at all — they thought about class groups.  And in that language, their conjecture is clean to the point of being self-explanatory:  each finite abelian p-group G appears as the p-primary part of a class group with probability inversely proportional to |Aut(G)|.

Numbers are great; but in this context they are merely the Grothendieck K_0 of the category of finite abelian groups.  If life gives you finite abelian groups, use them.  To pass to the Grothendieck group is to decategorify.  And to decategorify is to tempt fate.

Active clustering

Just a note — the paper “Active Clustering: Robust and Efficient Hierarchical Clustering using Adaptively Selected Similarities,” by Eriksson, Dasarathy, Singh, and Nowak, describing the algorithm which I (but not they) refer to as “compressed clustering,” is now on the arXiv.

Somebody out there liked my book.

In fact, she singles out for praise a single sentence.  And the sad truth is:  I have no memory of having written this sentence. I guess I’d imagined her favorite sentence would be something I, too, would have singled out in my mind.  But no.

Anyway, here it is:

My father, a mild man, dedicated to prudent consistency, demurred.

I’ll stand by this sentence.  I think the long part (“dedicated to prudent consistency”) is a bit too chunky in the mouth — too many palatal consonants.  I like the faintly comic tang you get from delaying the verb to the end — I stole this trick from somewhere, I don’t remember where.  (It might have just been the German language in general.)

Anyway, I have a favorite sentence in the book, but I don’t care to reveal it.  Instead, here are a couple of my very favorites from other people’s books.

One from Michael Chabon’s The Mysteries of Pittsburgh, I have quoted here before:

…the library, the dead core of my education, the white, silent kernel of every empty Sunday I had spent trying to ravish the faint charms of economics, my sad and cynical major.

And, in another register, from Richard Brautigan’s Trout Fishing in America:

The sun was like a huge fifty-cent piece that someone had poured kerosene on and then lit with a match and said ‘Here, hold this while I go get a newspaper,’ and put the coin in my hand but never came back.

I like the way this sentence is not a sentence, but reads as one.

My favorite Wisconsin labor tweet

“One way or another, when this Walker experiment is over, one of the Koch brothers is going to owe the other exactly one dollar.”

In which John Tierney annoys me regarding women in science, part — wait, I’ve lost count

Similarly, Larry Summers, then president of Harvard, was ostracized in 2005 for wondering publicly whether the preponderance of male professors in some top math and science departments might be due partly to the larger variance in I.Q. scores among men (meaning there are more men at the very high and very low ends). “This was not a permissible hypothesis,” Dr. Haidt said. “It blamed the victims rather than the powerful. The outrage ultimately led to his resignation. We psychologists should have been outraged by the outrage. We should have defended his right to think freely.”

Instead, the taboo against discussing sex differences was reinforced, so universities and the National Science Foundation went on spending tens of millions of dollars on research and programs based on the assumption that female scientists faced discrimination and various forms of unconscious bias.

Here’s a Google Scholar search for “gender differences in cognition.” The first page of results includes the 1995 paper “Magnitude of sex differences in spatial abilities: A meta-analysis and consideration of critical variables,” by Voyer, Voyer, and Bryden, which has been cited 791 times.

Camilla Benbow’s paper Sex differences in mathematical reasoning ability in intellectually talented preadolescents: Their nature, effects, and possible causes” has been cited over 300 times:  the abstract concludes “It is therefore proposed that the sex difference in SAT-M scores among intellectually talented students, which may be related to greater male variability, results from both environmental and biological factors.”

Here’s a selection of papers from the Organization for the Study of Sex Differences, including “Evidence for sex-specific shifting of neural processes underlying learning and memory following stress,” about cognitive differences between men and women under conditions of stress.  The OSSD’s 2010 annual meeting was funded by the National Science Foundation.

All I can say is, this is some really crappy taboo enforcement.  Politically correct mandarins of academia, get on the stick!

Four things people mostly aren’t saying about the Wisconsin collective bargaining protests

I try to keep politics out of the blog, mostly.  But the current protests are occupying a big share of our attention here, and  it would be weird to let it pass without comment.  A huge amount has already been written — my own political stance carries no special weight and has in any case been ably explained by others who share it.  So for the blog I’ll try to say a few things that I think are right, but which haven’t been tweeted and multiply retweeted.

• Lots of people are asking “Why did the protests happen and why was everyone so taken by surprise?”  It’s not clear to me a better answer than “It’s not usually 50 degrees in February in Wisconsin” is needed.
• Where is Tommy Thompson?  He is still, I think, the most popular Republican in the state.  As governor, he negotiated with state unions, and he’s generally thought of as a moderate; on the other hand, he was the keynote speaker at a Tea Party rally last spring.  If he supports the removal of collective bargaining for state employees, why isn’t he coming out and saying it?  Why hasn’t anyone asked him?
• Underreported part of the story:  the right to collective bargaining is guaranteed, by federal law, to every non-governmental employee in the state.  The public employees are asking to play by the same rules as private employees, not special ones.
• Underreported part of the story 2:  let’s suppose the Senate reaches a compromise:  taxpayers in the public employ get a bigger chunk of the paycheck taken by the state, to offset their pension and healthcare costs, but they retain the same collective bargaining rights that private employees have.  Unions claim victory for keeping bargaining, Senate R’s claim victory for trimming the budget.  The D’s return from Rockford, the Senate passes a bill.  But what about the Frankenveto?  Let me emphasize for those not from Wisconsin that what I’m about to say is actually true:  the Governor has the power to veto individual sentences and even words out of bills, thus enacting into law assertions entirely of the executive’s creation.  (Until 1990, the Governor could veto individual letters and digits!)   Scott Walker won’t hesitate to slice and dice what the legislature sends him until it says what he wants it to say — and it’s hard to imagine him paying much of a political price.  Crazy as this is, it’s a Wisconsin tradition.

Do I have any idea what the result of this fracas will be, for workers in Wisconsin or for Scott Walker’s political future?  I do not.

Dan Sharfstein, “The Invisible Line”

My friend and former It’s Academic teammate Dan Sharfstein has a new book out, The Invisible Line.  I haven’t read it but it’s surely terrific.  Not to mention well-blurbed:

The Invisible Line offers a trilogy of remarkable tales brimming with risk taking, camouflage, irony, narrow escapes, misgivings, regret, delight, and full-scale human drama. Excellent histories have been published about the Great Migration of twentieth-century African Americans from the rural South to the urban North, but, until now, no authoritative and cumulative work has looked at this preceding and overlapping social movement of race changing. One by one, or family by family, since the dawn of American history, individuals have slipped through the loopholes of racial identity. This book overthrows nearly everything Americans thought they knew about race.”
—Melissa Fay Greene, author of Praying for Sheetrock and There Is No Me Without You: One Woman’s Odyssey to Rescue Her Country’s Children

Highly recommended.

Are the Orioles the most improved team in baseball?

The CAIRO projection system has the Orioles winning an average of 77 games this year; that’s 11 games better than their 2010 record (biggest projected jump in the majors) and 7 over the 2011 projection for the team as constituted prior the winter meetings, also baseball’s best figure.  I don’t think the Orioles are 11 games better than last year’s squad, but I think the 2010 team was better than their record.  So 77 wins sounds about right.

I don’t think this is even the biggest lineup upgrade the Orioles have seen lately — between 2003 and 2004 we replaced Brook Fordyce with Javy Lopez and Deivi Cruz with Miguel Tejada.  What I never realized is that the terrible Fordyce wasn’t even the second-worst hitter on that team; that would be Tony Batista, who despite hitting 26 home runs managed to rack up an OBP of .270.  According to Play Index, Batista’s 2003 was the worst offensive season of all time by a guy with at least 25 home runs!  Anyway, he was replaced in 2004 by Melvin Mora’s career year.  And still the Orioles only improved by seven games!

Making a bad team good is hard.  Even making a bad team substantially less bad is hard.

In which I stop by NYU and get categorified by Kazhdan

I had the good luck to be in New York on Friday when David Kazhdan gave an unscheduled lecture at NYU about categorification and representations of finite groups.  For people like me, who spend most of our days dismally uncategorified, the talk was a beautiful advertisement for categorification.

Actually, the first twenty minutes of the talk were apparently a beautiful advertisement for the Langlands program, but I got lost coming from the train and missed these.  As a result, I don’t know whether the results described below are due to Kazhdan, Kazhdan + collaborators, or someone else entirely.  And I missed some definitions — but I think I can transmit Kazhdan’s point even without knowing them.  You be the judge.

It went something like this:

Let G be a reductive split group over a finite field k and B a Borel. Then C[G(k)/B(k)] is a representation of G(k), each of whose irreducible constituents is a unipotent representation of G(k).  (Note:  the definition of “unipotent representation” is one that I missed but it comes from Deligne-Lusztig theory.)

When G = GL_n, all unipotent representations of G appear in C[G(k)/B(k)], so this procedure gives a very clean classification of unipotent representations — they are precisely the constituents of C[G(k)/B(k)].  Equivalently, they are the direct summands of the center of the Hecke algebra C[B(k) \G(k) / B(k)].  For more general G (e.g. Sp_6, E_8) this isn’t the case.  Some unipotent representations are missing from C[G(k)/B(k)]!

Where are they?

One category-level up, naturally.

(see what I did there?)

OK, so:  instead of C[B(k)\G(k)/B(k)], which is the algebra of B(k)-invariant functions on G(k)/B(k), let’s consider H, the category of B-invariant perverse l-adic sheaves on G/B.  (Update:  Ben Webster explained that I didn’t need to say “derived” here — Kazhdan was literally talking about the abelian category of perverse sheaves.)  This is supposed to be an algebra (H is for “Hecke”) and indeed we have a convolution, which makes H into a monoidal category.

Now all we have to do is compute the center of the category H.   And what we should mean by this is the Drinfeld center Z(H).  Just as the center of an algebra has more structure than the algebra structure — it is a commutative algebra! — the Drinfeld center of a monoidal category has more structure than a monoidal category — it is a braided monoidal category.  It’s Grothendieck Group K_0(Z(H)) (if you like, its decategorification) is just a plain old commutative algebra.

Now you might think that if you categorify C[B(k)\G(k)/B(k)], and then take the (Drinfeld) center, and then decategorify, you would get back the center of C[B(k)\G(k)/B(k)].

But you don’t!  You get something bigger — and the bigger algebra breaks up into direct summands which are naturally identified with the whole set of unipotent representations of G(k).

How can we get irreducible characters of G(k) out of Z(H)?  This is the function-sheaf correspondence —  for each object F of Z(H), and each point x of G(k), you get a number by evaluating the trace of Frobenius on the stalk of F at x.  This evidently yields a map from the Grothendieck group K_0(Z(H)) to characters of G(k).

To sum up:  the natural representation C[G(k)/B(k)] sometimes sees the whole unipotent representation theory of G(k), but sometimes doesn’t. When it doesn’t, it’s somewhat confusing to understand which representations it misses, and why.  But in Kazhdan’s view this is an artifact of working in the Grothendieck group of the thing instead of the thing itself, the monoidal category H, which, from its higher categorical perch, sees everything.

(I feel like the recent paper of Ben-Zvi, Francis and Nadler must have something to do with this post — experts?)