Monthly Archives: April 2010

The census will be wrong. We could fix it.

I have an op/ed in tomorrow’s Washington Post about statistical sampling and the census.  It boils down to the claim that by failing to use the best statistical techniques we have to enumerate the population accurately, we’re getting the answer wrong on purpose in order to avoid getting it wrong by accident, and possibly violating the Constitution as a result.  And that estimating an unmeasured quantity to be zero is a really bad estimate.

The book Who Counts?  The Politics of Census-Taking In Contemporary America, by Margo Anderson and Stephen Fienberg, was an invaluable resource for the piece — highly recommended.

One argument I cut for space involves Kyllo vs. US, in which the Supreme Court ruled, in an opinion written by Antonin Scalia, that the use of a thermal imaging device to detect heat coming off the exterior wall of a house, and thus to infer the presence of a drug operation inside, can constitute a “search” for Fourth Amendment purposes.  On the other hand, Scalia questions the constitutionality of statistical adjustment of the census, expressing doubt that such a procedure would still be an “actual enumeration” as required by the Constitution.  So, for Scalia:

  • “Search,” in 2010, includes a scenario in which something of interest inside the house is not seen or otherwise sensed by any person or people, but is inferred by means of a scientific instrument that didn’t exist in Constitutional times.
  • But “enumeration,” in 2010, does NOT include a scenario in which the population is not counted one by one by any person or people, but is inferred by means of a statistical instrument that didn’t exist in Constitutional times.

Is that a problem?

Update (4 May):  It turns out I’m not the first arithmetic geometer to weigh in on census adjustment.  Brian Conrad in the New York Times, August 1998 does in three sentences what took me 1000 words:

Human intelligence plus a little brute force is often far more efficient and accurate than brute force alone. This is why statistical sampling is the superior way to carry out an ”actual enumeration” of a large population. Just ask any Republican who relies on a poll or who takes a blood test rather than drain every drop from his body.

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What would the proposed immigration reform law mean for mathematics?

According to the conceptual proposal released by Senators Menendez, Reid, and Schumer yesterday, the immigration reform to be taken up by Congress would require that

a green card will be immediately available to foreign students with an advanced degree from a United States institution of higher education in a field of science, technology, engineering, or mathematics, and who possess an offer of employment from a United States employer in a field related to their degree. Foreign students will be permitted to enter the United States with immigrant intent if they are a bona fide student so long as they pursue a full course of study at an institution of higher education in a field of science, technology, engineering or mathematics. To address the fact that workers from some countries face unreasonably long backlogs that have no responsiveness to America’s economic needs, this proposal eliminates the per-country employment immigration caps.

How does this affect math?  Does it change the visa status of our Ph.D. students?  Is a postdoc an “offer of employment,” and if so, will non-U.S. graduate students be eligible to receive NSF postdocs, given that they’d become permanent residents upon taking up their position?

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Quaker Steak and Lube

Another letter of the alphabet, another night out with Eating in Madison A to Z.  Perfectly adequate chicken wings (though judging from what I saw around the table, their definition of “wing” is “any breaded piece of chicken longer than it is wide.”)  Various hacked-open and truncated vehicles hanging from the ceiling, which delighted CJ and led him to call this “the funny restaurant.”  If you were going to have dinner at a giant chain restaurant in outer Middleton, QS&L would be a respectable choice.

The case of XXXXXX XXXXXX

Update: At the request of third parties, and with the agreement of the people involved, I have anonymized this post to remove the name of the people and universities involved.

I don’t like to wander into controversy on the blog, but I do want to share what I know about our postdoc XXXXX’s job search this year, in order to counteract some incorrect impressions I’ve heard about.

  • XXXX interviewed at AAAA, and got an early offer of an assistant professorship, with a deadline in February.  She had other interviews already scheduled, and asked for an extension on the deadline.  They didn’t give her one.
  • XXXX accepted the AAAA  job, while on an interview visit to BBBB.
  • Later, XXXX was offered an assistant professorship at BBBB as well.  BBBB, understanding that XXXX had already accepted a position at AAAA, agreed to make the offer effective in Fall 2011 if she so chose.
  • XXXX  told AAAA about her situation, making clear that she had no intention of reneging on her acceptance of the position, and that she was honestly not sure which department was the better home for her.  She asked for a year of unpaid leave for 2011-2012 so that she could visit BBBB after one year at AAAA and make an informed decision.
  • This request, too, was denied.  At this point, the chair at AAAA told her that she had to make up her mind now which job she wanted to take; she was released from her commitment to AAAA  and told that she should immediately start whichever of the two positions she chose.  At this point, XXXX chose the job at BBBB.

As far as I can see, no one acted unethically here.  At every stage, XXXX was upfront with everyone involved, and never considered not showing up at AAAA until the chair there explicitly authorized it.  BBBB made an offer to someone who already had a job, yes:  but I see no difference between making her an offer in March 2010 for Fall 2011, and making her the same offer in October 2010, which would obviously be OK.  As for AAAA, they ran their hiring process in a somewhat nonstandard and maybe suboptimal way — in particular, by denying XXXX the unpaid leave and releasing her to go to BBBB next fall instead, it seems to me they denied themselves the opportunity to convince XXXX that AAAA was the right department for her.  (But I’m told that, at some departments, unpaid leave is not routinely granted as it is at UW.)

In case you hear someone say “XXXX accepted a job at AAAA and then reneged,” please let them know that the story is more complicated.

Update: Timeline above corrected to clarify that XXXX’s AAAA deadline coincided with her interview at BBBB; she didn’t interview at BBBB after already having accepted AAAA, as the original version suggested.

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Pseudo-Anosov puzzle 2: homology rank and dilatation

In fact, following on what I wrote about the two Farb-Leininger-Margalit theorems below, one might ask the following.  Is there an absolute constant c such that, if f is a pseudo-Anosov mapping class on a genus g surface, and the f-invariant subspace of H_1(S) has dimension at least d, then

log λ(f) >= c (d+1)  / g?

This would “interpolate” between Penner’s theorem (the case d=0) and the F-L-M theorem about Torelli (the case d=2g).

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Pseudo-Anosovs with low dilatation: Farb-Leininger-Margalit, and a puzzle

I spent a very enjoyable weekend learning about the dilatation of pseudo-Anosov mapping classes at a workshop organized by Jean-Luc Thiffeault and myself.  The fact that a number theorist and a fluid dynamicist would organize a workshop about an area in low-dimensional topology should indicate, I hope, that the topic is of broad interest!

There are lots of ways to define dilatation, which is a kind of measure of “complexity” of a mapping class.  Here’s the simplest.  Let f be a diffeomorphism from a genus-g Riemann surface S to itself, which is pseudo-Anosov.  Loosely speaking, this means the dynamics of  f are “irreducible” on the surface; for instance, no power of f acts trivially on any subsurface.  (“Most” diffeomorphisms, in any reasonable sense, are pA.)  For any two curves a,b on S, let i(a,b) be the minimal number of intersection points between a and any curve isotopic to b.  (Note that this is typically a lot bigger than the intersection of the homology classes of a and b; the latter measures the number of intersection points counted with sign, which doesn’t change when you isotop the curves.)  It turns out that the quantity

(1/k) log i(f^k(a),b)

approaches a limit as k grows, which strictly exceeds 1;  this limit is called λ(f), the dilatation of f.  It’s invariant under deformation of f; in other words, it depends only on the class of f in the mapping class group of S.  That this limit exists is exciting enough; better still, and indicative of lots of structure I’m passing over in silence, is that λ(f) is an algebraic integer!

(I just remembered that I gave a different description of the dilatation on the blog last year, in connection with an analogy to Galois groups.)

The subject of the conference was pseudo-Anosovs with low dilatation.  The dilatations of pAs in a given genus g are known to form a discrete subset of the interval (1,infinity); thus it makes sense to ask what the smallest dilatation in genus g is.  Lots of progress on this problem has been made in recent years; Joan Birman, Eriko Hironaka, Chia-Yen Tsai, and Ji-Young Ham all talked about results in this vein.  But for general g the answer remains unknown.

A theorem of Penner guarantees that, for any pseudo-Anosov f on a surface of genus g, we have λ(f) > c^(1/g) for some constant c.  So one might call a family f_1, f_2,…. of pAs of varying genera g_1, g_2, …  “low-dilatation” if the quantity λ(f_i)^g_i is bounded.  (One such family, constructed by Hironaka and Eiko Kin, appeared in many of the lectures.)

In this connection, let me advertise the extremely satisfying theorem of Benson Farb, Chris Leininger, and Dan Margalit.  Here’s a natural construction you can do with a pA diffeomorphism f on a surface S.  The diffeo has an invariant foliation which is stretched by f; this foliation has a finite set of singularities.  Remove this to get a punctured surface S^0.  Since the singularities are preserved setwise by f, we have that f restricts to a diffeomorphism of S^0, which is again pA, and which we again call f.  Now we can make a 3-manifold M^0_f by starting with S^0 x [0,1] and gluing S^0  x 0 to S^0 x 1 via f.  By a theorem of Thurston, this will be a hyperbolic 3-manifold; because of the punctures, it’s not compact, but its ends are shaped like tori.

Now here’s the theorem:  suppose f_1, f_2, … is a sequence of pAs which has low dilatation in the sense above.  Then the sequence of 3-manifolds M^0_{f_i} actually consists of only finitely many distinct hyperbolic 3-manifolds.

This has all kinds of marvelous consequences; it tells us that the low-dilatation pAs are in some sense “all alike.”  (For more on the “in some sense” I would need to talk about the Thurston norm and fibered faces and etc. — maybe another post.)  For instance, it immediately implies that in a low-dilatation family of pseudos, the dimension of the subspace of H_1(S_i) fixed by f_i is bounded.

If you’ve read this far, maybe you’d like to see the promised puzzle.  Here it is.  Suppose f_1, f_2, … is a family of pseudos which lie in the Torelli group — that is, f_i acts trivially on H_1(S_i).  Then by the above remark this family can’t be low-dilatation.  Indeed, an earlier theorem of Farb, Leininger, and Margalit tells us that for Torellis we have an absolute lower bound

λ(f) > c

where the constant doesn’t depend on g.

Puzzle: Suppose f_1, f_2, … is a sequence of pseudos in Torelli which has bounded dilatation; this is as strong a notion of “low-dilatation family” as one can ask for.  Is there a “structure theorem” for f_1, f_2, …. as in the general case?  I.E., is there any “closed-form description” of this family?

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Quantifying Orioles late-game dispirit

In innings 1-6, the Orioles have scored 26 runs and allowed 37.

After the 6th, the Orioles have scored 8 runs and allowed 26.

Something is wrong with this team and it isn’t (just) Mike Gonzalez.

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Sleepytime Gorilla Museum: Origin Story

The cast of Cats is locked in a room.  They are there for a long time.  Deprived of an audience their theater becomes steadily more vivid, gestural, and non-referential.  To a certain degree it ferments. Thus it acquires a rotten taste but also a depth and richness it lacked previously.

The actors and singers begin to believe they are receiving messages.  Maybe from the outside, maybe from each other.  They are required to retransmit these messages:  sometimes in the form of a roar, sometimes as a jerk or spasm.

Some members of the troupe are slow to learn the new theater.  They drop beats and misreport their lines.

The strong consume the weak until only five are left.

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“The rest of the world was uncertain what had been proven and what not.”

Until a minute ago I had never heard of Mizar, a project to record as much mathematics as possible in computer-readable form.  The pieces of this project are published in the Journal of Formalized Mathematics.  Here, for instance, is the paper “Non-negative Real Numbers, part I.”

I learned about Mizar when glancing through the publicly available archive of QED, a mailing list from the early 90s devoted to the formalization of mathematics.  It’s interesting to be reminded just how excited people were about the prospects of computerizing large precincts of mathematical practice, an ambition which as far as I can tell has now receded almost entirely from view.

I found the QED archive, in turn, via Math Overflow, which quoted these pointed remarks of Mumford about algebraic geometry in the Italian style:

The best known case is the Italian school of algebraic geometry, which produced extremely good and deep results for some 50 years, but then went to pieces. There are 3 key names here — Castelnuovo, Enriques and Severi. C was earliest and was totally rigorous, a splendid mathematician. E came next and, as far as I know, never published anything that was false, though he openly acknowledged that some of his proofs didn’t cover every possible case (there were often special highly singular cases which later turned out to be central to understanding a situation). He used to talk about posing “critical doubts”. He had his own standards and was happy to reexamine a “proof” and make it more nearly complete. Unfortunately Severi, the last in the line, a fascist with a dictatorial temperament, really killed the whole school because, although he started off with brilliant and correct discoveries, later published books full of garbage (this was in the 30’s and 40’s). The rest of the world was uncertain what had been proven and what not. He gave a keynote speech at the first Int Congress after the war in 1950, but his mistakes were becoming clearer and clearer. It took the efforts of 2 great men, Zariski and Weil, to clean up the mess in the 40’s and 50’s although dredging this morass for its correct results continues occasionally to this day.

Readers with good memories will recall that this is the second time I’ve quoted uncomplimentary remarks about Severi.

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McMullen on dilatation in finite covers

Last year I blogged about a nice paper of Thomas Koberda, which shows that every pseudo-Anosov diffeomorphism of a Riemann surface X acts nontrivially on the homology of some characteristic cover of X with nilpotent Galois group.  (This statement is false with “nilpotent” replaced by “abelian.”)  The paper contains a question which Koberda ascribes to McMullen:

Is the dilatation λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

That question has now been answered by McMullen himself, in the negative, in a preprint released last month.  In fact, he shows that either λ(f) is detected on the homology of a double cover of Σ, or it is not detected by any finite cover at all!

The supremum of the spectral radius of f on the Σ’ is then an invariant of f, which most of the time is strictly bigger than the spectral radius of f on Σ and strictly smaller than λ(f).  Is this invariant interesting?  Are there any circumstances under which it can be computed?

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