The Viscount Deligne’s coat of arms

I had no idea, until somebody told me at lunch yesterday, that Pierre Deligne had been granted a viscounty by the Queen of Belgium; and that, as part of his ennoblement, he had to devise for himself a coat of arms.  And here it is:

Lieblich’s counter counterexample example

Max Lieblich gave a great talk at WAGS yesterday about something that looks like a counterexample to the Hasse principle, but secretly isn’t!  All mistakes in this summary are my own.  For a more authoritative take on the material below, see Max’s recent arXiv preprint.

The counterexample Max countered is the central simple algebra A over the field Q(t) obtained as the tensor product of the two generalized quaternion algebras $(17,t)$ and $(13,6(t-1)(t-11))$.  This algebra has index 4, which is to say (if I understand correctly) that its Brauer class isn’t the cup product of two elements in Q(t)^*.  On the other hand, it turns out that $A \otimes \mathbf{Q}_v$ has index 1 or 2 — that is, it’s either trivial or a cup product — for all places v of Q.

This seems like an example of a situation where there’s no Hasse principle; A fails to be a cup product despite the fact that A is a cup product over every completion of Q.  But the truth, as Max explained, is more complicated.

We know (and if we don’t, we read A Course in Arithmetic to remember) that the quaternion algebra (a,b) is trivial (i.e., has index 1) over a field k precisely when the conic $x^2 - ay^2 - bz^2$ has a k-rational point.  You might ask whether there’s a similar criterion for the tensor product of two quaternion algebras to have index 2.  It turns out that (at least over Q(t)) there is indeed a variety that does this trick — it’s a coarse moduli space M parametrizing certain twisted vector bundles on — well, not quite P^1/Q, but a certain orbifold version of P^1 with a bunch of stacky points with inertia Z/2Z.  And the assertion that A has index 4 is equivalent to the assertion that M(Q) is empty.

To really talk about what M is would take us too far afield; I just want to record Max’s observation that the definition of M is truly global, in the sense that the scheme $M_{\mathbf{Q}_v}$ is not determined by $A \otimes \mathbf{Q}_v$.  In particular, the fact that $A \otimes \mathbf{Q}_v$ has index 2 doesn’t imply that M has a $\mathbf{Q}_v$-point.  And indeed, in the case at hand, M has no points over $\mathbf{Q}_{17}$.  So there is, after all, a local obstruction to A having index 2; but it’s a local obstruction which, it seems, can’t be seen except in this rather intricate geometric way.

It makes you wonder what should actually be meant by “Hasse principle.”  Suppose, for instance, you had some class C of varieties X/Q, and suppose you had some construction which attached to each X in C a variety Y/Q such that X(Q) is empty if and only if Y(Q) is empty.   Now one way to prove X(Q) empty would be to prove that Y(Q) was empty, which you could in turn prove if you knew that Y(Q_v) was empty for some place v.  Do you consider this a local obstruction to rational points on X?

Don Henley Must Drive

We were driving back from Milwaukee Airport the other day and got stuck for a while behind an old Civic with a license plate reading DON HNLY.  It puzzled me.  Are there really people, in 2009, whose commitment to Don Henley is so strong as to demand a personalized license plate?  EAGLES, all right, I can see, they still sell out arenas.  But solo Don?

Then I started to wonder — what if it is Don Henley?  I didn’t have a fully worked-out theory for why Don Henley would be driving an old Civic on westbound I-94, but the idea was appealing.  If I were Don Henley, I could totally imagine buying a modest vehicle, registering it with a personalized license plate advertising myself, then enjoying the reaction whenever I climbed out at a rest stop — the magnificent triple-take from “That guy has a Don Henley license plate!” to “That guy with the Don Henley license plate is a ringer for Don Henley!” to “Can it really be….”

So I cruised along behind the Civic, in no hurry, enjoying my fantasy, until everything was ruined by the Impala in the passing lane with the MENTORS plate.

More MALBEC: Niyogi on geometry of data, Coen on abstract nonsense

Tuesday, April 21 — tomorrow! — brings the third lecture in the MALBEC series:  Michael Coen, of computer sciences and biostat, talks on “Toward Formalizing “Abstract Nonsense”,” in Computer Sciences 1221 at 4pm.  Here’s the abstract:

The idea of a category — a set of objects sharing common properties
— is a fundamental concept in many fields, including mathematics,
artificial intelligence, and cognitive and neuroscience.  Numerous
frameworks, for example, in machine learning and linguistics, rest
upon the simple presumption that categories are well-defined.  This is
slightly worrisome, as the many attempts formalizing categories have
met with equally many attempts shooting them down.

Instead of approaching this issue head on, I derive a robust theory of
“similarity,” from a biologically-inspired approach to perception in
animals.  The very idea of creating categories assumes some implicit
notion of similarity, but it is rarely examined in isolation.
However, doing so is worthwhile, and I demonstrate the theory’s
applicability to a variety of natural and artificial learning
problems.  Even when faced with Watanabe’s “Ugly Duckling” theorem or
Wolpert’s stingy cafeteria (serving the famous “No Free Lunch”
theorems), one can make significant progress toward formalizing a
theory of categories by examining their often unstated properties.

I demonstrate practical applications of this work in several domains,
including unsupervised machine learning, ensemble clustering, image
segmentation, human acquisition of language, and cognitive
neuroscience.

(Joint work with M.H.Ansari)

Delicious food will follow the talk, as if this delicious abstract isn’t enough!

On Friday,  Partha Niyogi gave a beautiful talk on “Geometry, Perception, and Learning.”  His work fits into a really exciting movement in data analysis that one might call “use the submanifold.”  Namely:  data is often given to you as a set of points in some massively high-dimensional space.  For instance, a set of images from a digital camera can be thought of as a sequence of points in R^N, where N is the number of pixels in your camera, a number in the millions, and the value of the Nth coordinate is the brightness of the Nth pixel.  A guy like Niyogi might want to train a computer to distinguish between pictures of horses and pictures of fish.  Now one way to do this is to try to draw a hyperplane across R^N with all the horses are on one side and all the fish on the other.  But the dimension of the space is so high that this is essentially impossible to do well.

But there’s another way — you can take the view that the N-dimensionality of the space of all images is an illusion, and that the images you might be interested in — for instance, some class of images including all horses and all fish — might lie on some submanifold of drastically smaller dimension.

If you believe that manifold is linear, you’re in business:   statisticians have tons of tools, essentially souped-up versions of linear regression, for fitting a linear subspace to a bunch of data.  But linearity is probably too much to ask for.  If you superimpose a picture of a trout on a picture of a walleye, you don’t get a picture of a fish; which is to say, the space of fish isn’t linear.

So it becomes crucial to figure out things about the mystery “fish manifold” from which all pictures of fish are sampled; what are its connected components, or more generally its homology?  What can we say about its curvature?  How well can we interpolate on it to generate novel fish-pictures from the ones in the input?  The work of Carlsson, Diaconis, Ghrist, etc. that I mentioned here is part of the same project.

And in some sense the work of Candes, Tao, and a million others on compressed sensing (well-explained on Terry’s blog) has a similar flavor.  For Niyogi, you have a bunch of given points in R^N and a mystery manifold which is supposed to contain, or at least be close to, those points.  In compressed sensing, the manifold is known — it’s just a union of low-dimensional linear subspaces parametrizing vectors which are sparse in a suitable basis — but the points are not!

Despite the dispiriting sweep we just endured at the hands of the Red Sox, the Orioles are for the first time in recent memory a team whose future seems kind of interesting — so it’s an opportune time for Tom’s reminiscence, via Joe Posnanski, of the Orioles’ last Turn Ahead The Clock Day.  At the time, the future of the franchise was supposed to include a lot of Albert Belle:

I saw Albert Belle try to turn down a HBP once. It was Turn Ahead the Clock day, and the Orioles were wearing billowing trash-bag “futuristic” uniforms. Belle was 4-for-4 with a walk and 3 home runs already, including a two-out game-tying shot in the bottom of the ninth. And he had driven in 6 of the O’s 7 runs. So when the ball ran in on his floppy outfit in the bottom of the 11th, with a man aboard, he waved off the ump and tried to stay in the box.

My friend and I at the game had absolutely no doubt that had he gotten away with it, he would have hit his fourth homer. Belle felt the same, evidently. But eventually they ordered him along to first base, and Cal Ripken singled in the winning run three batters later.

I, the “friend” above, recounted the same game in my list of Underappreciated Orioles, on which Belle appears at #5.  Only I forgot the two most interesting details, the future jerseys and Belle’s attemped snub of the free base!  Which is why Tom is a professional sportswriter and I’m just a guy who complains about the Orioles on the Internet.

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Gehry/Serra

Back from a very brief trip to Princeton. The much-maligned new science library, built by Frank Gehry, is now up. I like it — it’s very much like the Stata Center, but more humble. Here’s a small piece of it viewed from inside the Richard Serra sculpture “Fox and Hedgehog,” also much-maligned, and which I also like.

Note: this is my first attempt to blog by iPhone. I’m sitting in my car, in the back of which CJ has dozed off, and I thought I’d let him sleep instead of continuing immediately with the shopping as planned.

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A little more about random configurations

In my previous post about the configuration space of hard discs in a box, I neglected to say anything about the main point of Persi’s article!  It’s the following — even though  you don’t know anything about the topology of the space of configurations, you can still do an excellent job of drawing a configuration at random from the natural distribution using Monte Carlo techniques.  And if you’re a physicist trying to model the behavior of a gas or a fluid, you might be more interested in what a random configuration looks like than whether the space of configurations is connected — it might not be so relevant that you can get from something that looks like tight packing 1 to something that looks like tight packing 2, if the probability of doing so is vanishingly small.  Or to put it another way — if the space looks like a bunch of big blobs connected by extremely narrow paths, then from the point of view of physics it might as well be disconnected.

Still, as a topologist, you might ask:  if I can do a good random sample of points from a mystery manifold M, can I compute topological invariants of M with high confidence?  Can you guess whether M is connected?  More generally, can you guess the homology groups of M?

You might think of this as a massive geometric generalization of the age-old problem of “cluster analysis.”  Let’s say you have a bunch of people, and for each person you measure N variables — let’s say, height, weight, and shoe size.  So you have a bunch of points in R^N — in this case R^3.

Maybe you hope that these points are well-modeled as a sample from some multivariate normal distribution.  But under some circumstances, this is a really bad hope!  For instance, if your sample isn’t segregated by gender, you’re going to see two big clusters — one cluster of women where the mean height is around 5’6″, one cluster of men where the mean height is around 5’9″.  You’re not really sampling from a normal distribution — you might be sampling from a superimposition of two different normal distributions with different centers.  Or, alternately, you could think of yourself as sampling points from a manifold in R^3 consisting of just two points — “the ideal woman and the ideal man” — where your measurements are subject to some error that’s distributed normally.

My impression is that statisticians are pretty good at distinguishing between a normal distribution and a superimposition of some small finite set of normal distributions.  But I think it’s much harder to look at a giant cloud of points in R^100 and say “aha — this is actually a random sample from a normal distribution centered on the union of a surface of genus 2 sitting over here, and these ten disjoint circles sitting over there.”

If you were wondering about this while reading the Diaconis article in the Bulletin, you’d be in luck, because flipping forward a few pages you’d get to Gunnar Carlsson’s long survey article on precisely this genre of problem!  More on “topological statistics” once I’ve read Carlsson’s article, but let me point out now that if you’re a young mathematician interested in these matters you might consider going to the CBMS summer school this August, centered on a lecture series by Ghrist.

Of all the crops, true peace is tops

I took CJ to the terrific UW Science Expeditions last weekend — he had a great time, petting the stuffed badger, looking through a microscope for the first time, filling (and almost breaking) a pipette, and holding a caterpillar provided by the Department of Entomology.  He declined, as did I, the opportunity to handle a Madagascar hissing cockroach, but apparently we were unusual because there was a line to handle the Madagascar hissing cockroach.  Lots of the exhibits were presented by the departments in the Agriculture School — at one of these I learned that Wisconsin has an official state soil, established by statute in 1983.  But not only that — our official state soil has an official state soil song!  And it gets better — the composer of the Antigo Silt Loam song, Francis D. Hole, isn’t just a one-hit wonder but a prolific musical interpreter of soil. I give you “Some Think That Soil is Dirt:”

Some think that soil is dirt and quite disgusting.
This is not true.

Some think that soil is dirt and quite disgusting.
This is not true.

Some think it makes the air all brown and dusting.
Good dust’s in me!

Some think it makes the air all brown and dusting.
Good dust’s in you!

Praise Mother earth she is our earthly Mother

Praise ground, the holy ground that’s softly under

Vigor, Vigor from the soil does flow;
Roots and life are teeming down below

No wonder that the land’s so green,
the ferns and flowers so fresh and clean!

Soil is everywhere;
from it sweet blessings gently flow.

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In which my friends write things

• We’re already on to the next Jewish holiday, but Jay Michaelson’s piece on the financial crisis as Purimspiel is still well worth reading.
• Opening Day has come at last, and the Orioles rang it in by drubbing the Yankees 10-5, in a weird-all-over kind of game.  I can’t describe it better than Tom does.
• Alison Buckholtz, a year ahead of me in high school, has a book out. Alison’s husband is a Navy pilot, and Standing By is about the world of military spouses, who occupy a funny boundary space:  of the military but not officially in it.  Alison’s little brother Charlie was a year below me — his essay about surving a car accident, “Catheterized!” has stayed with me for twenty years. He has a book too, about the mysterious death of the guy who wrote the “Lady in the Radiator” song.  I think he should write a novelization of “Catheterized!”
• And yet another book, from Cary Chugh, a.k.a. What Went Wrong‘s husband:  Don’t Swear With Your Mouth Full!, which offers an approach to parental discipline based on behavioral science.  Endorsed by Dr. Mrs. Q!   CJ, of course, is always instantly and cheerfully compliant, and would never, for example, insist to the point of hysteria that “getting dressed for school” requires putting his underwear on his head, his sweatshirt over his legs, and one sock on his genitals; so I have no need for this book, but maybe it will be useful for some of you.

Harvard beats Yale 29-29, both beat Princeton

I’m sorry to say that I only made it to one movie at the Wisconsin Film Festival this year.  But I picked a good one.  I went to see Kevin Rafferty’s Harvard Beats Yale 29-29, a documentary about the most thrilling Harvard-Yale game ever played between two of the best teams Harvard and Yale ever fielded, just because I sincerely like Ivy League football.  But in fact it’s an authentically good documentary whether or not you care about Harvard or college football — though it might be hard going if e.g. you don’t know what “pass interference” means.

I won’t say too much, to avoid spoiling it.  But it’s particularly remarkable how Rafferty manages to develop Yale defenseman Mike Bouscaren, over about five minutes of total screen time, from a comic caricature to a sincerely terrifying villain (drawing hisses and gasps from the packed house) to a thoughtful and even remorseful ex-combatant.  There’s a good interview with Rafferty at the New York Times college sports blog.

In less inspiring Ivy news, Princeton’s admission rate bumped up a half a percentage point and disgruntled seniors went nuts on the Princetonian comment page, decrying the current administration and everything associated with it.  One of the enjoyable things about teaching at Princeton was getting the Princeton Alumni Weekly — that’s right, their alumni magazine is a weekly! — and reading the three pages of cranky letters from alumni with something on their chest about how they do things nowadays. The Princetonian comments are a great opportunity to hear from the cranky alumni of tomorrow.

At the moment, the CAoT are upset about “grade deflation” and the “war on Fun.”  Both got started while I was teaching at Princeton.  The former policy was aimed at the fact that grades in science and engineering classes were about a half-point lower, on average, than those in humanities; so that students who were planning grade-sensitive careers in law or medicine had a weird incentive not to major in science.  The “war on Fun,” refers, I think, to the establishment and promotion of a four-year residential college as an alternative to the eating clubs — and more generally a sense that the administration is hostile to the clubs.  When I was teaching at Princeton, about a quarter of the students weren’t in clubs, and life seemed to be sort of logistically annoying for them.  It’s hard for me to get my head around the idea that club members object to a nice cafeteria for the students who didn’t bicker in.

Oh, and I think “the war on Fun” also includes some kind of rule about registering dorm parties in advance with your RA.  Harvard introduced this policy when I was an undergraduate, and people grumbled about it then, too.  And you know what?  People still had parties.  Message to all undergraduates everywhere — your university is not conspiring to keep you from drinking beer in groups. I promise!

Anyway, the comment thread got linked from lots of places and so there’s some question how many of the posts are from authentic Princeton undergrads.  This, for instance, can’t be real — can it?

I am a triple legacy and I feel I have a more outstanding right to be here than a lot of the so-called Academic 1’s. Princeton used to stand for something, not just be a humorless grade factory. What’s next, bed checks? The Street is a shadow of what it was in my parents’ day and the current workload is just ridiculous.