Monthly Archives: March 2010

Author photos

Books used to have more awesome author photos than they do now.

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Fritz Grunewald, RIP

Fritz Grunewald died unexpectedly this week, just before his 61st birthday.  I never met him but have always been an admirer of his work, and I’d been meaning to post about his lovely paper with Lubotzky, “Linear representations of the automorphism group of the free group.” I’m sorry it takes such a sad event to spur me to get around to this, but here goes.

Let F_n be the free group of rank n, and Aut(F_n) its automorphism group.  How to understand what this group is like?  A natural approach is to study its representation theory.  But it’s actually not so easy to get a handle on representations of this group.  Aut(F_n) acts on F_n^ab = Z^n, so you get one n-dimensional representation; but what else can you find?

The insight of Grunewald and Lubotzky is to consider the action of Aut(F_n) on the homology of interesting finite-index subgroups of F_n.  Here’s a simple example:  let R be the kernel of a surjection F_n -> Z/2Z.  Then R^ab is a free Z-module of rank 2n – 1, and the -1 eigenspace of R^ab has rank n-1.  Now F_n may not act on R, but some finite-index subgroup H of F_n does (because there are only finitely many homomorphisms F_n -> Z/2Z, and we can take H to be the stabilizer of the one in question.)  So H acquires an action on R^ab; in particular, there is a homomorphism from H to GL_{n-1}(Z).  Grunewald and Lubotzky show that this homomorphism has image of finite index in GL_{n-1}(Z).  In particular, when n = 3, this shows that a finite-index subgroup of Aut(F_3) surjects onto a finite-index subgroup of GL_2(Z).  Thus Aut(F_3) is “large” (it virtually surjects onto a non-abelian free group), and in particular it does not have property T.  Whether Aut(F_n) has property T for n>3 is still, as far as I know, unknown.

Grunewald and Lubotzky construct maps from Aut(F_n) to various arithmetic groups via “Prym constructions” like the one above (with Z/2Z replaced by an arbitrary finite group G), and prove under relatively mild conditions that these maps have image of finite index in some specified arithmetic lattice.  Of course, it is natural to ask what one can learn from this method about interesting subgroups of Aut(F_n), like the mapping class groups of punctured surfaces.  The authors indicate in the introduction that they will return to the question of representations of mapping class groups in a subsequent paper.  I very much hope that Lubotzky and others will continue the story that Prof. Grunewald helped to begin.

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Pinch run, grand slam

From Baseball-Reference Stat of the Day, the 17 times in baseball history a player has entered the game as a pinch runner and hit a grand slam.  Three of these were Orioles:  David Segui, John Shelby, and Chris Hoiles.  That’s right, Chris Hoiles pinch ran!  Chris Hoiles was 5 for 12 lifetime as a base stealer and had 2 career triples.  Who could he possibly have pinch run for? Go ahead, think about the 1991 Orioles and try to guess.  Then go to the scoresheet.  Yep — Sam Horn.

Horn, by the way, played 389 games and never stole a base.  He had one triple, in 1992.  I’d have liked to see that.  I wonder if he ever pinch ran?  Calvin Pickering came up too late.

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Earth-M bracketology

What if every game in the NCAA men’s basketball tournament were won by the school with the better math department?  A group of us put together the alternate-universe bracket above to find out.  Note that the rules of the game forbade us from looking up anything on the Internet, so there are doubtless some matchups where our judgments are questionable or outright wrong.  If there’s a choice that really offends you, please be assured that it was the responsibility of McReynolds.

I couldn’t figure out how to make the image look nice:  if you can’t read the above, here’s a cleaner version.

Update: After one day of play, our bracket is at the 2.9th percentile of the 4 million entrants to ESPN’s bracket contest.  The plan is to make back lots of points when Cal beats Duke.

Update: Back up to 20.6% — thanks, Cornell!

Update: Northern Iowa knocks out Kansas, who we tossed in the first round, and Washington cruises past #3 seed New Mexico to make the Sweet Sixteen, just as we predicted; and we stand at 46.2%.

Update: Guh.  Three of my final four are out.  Back to 3.9%.  It’s all up to you now, Ohio State.

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In which Tucson, AZ is the miniature golf capital of these United States

At least that’s how I felt as a kid, when every visit here included a trip to Green Acres or Magic Carpet.  Miniature golf courses are just bigger and better and awesomer here than anywhere else.  Today I took CJ to Golf n Stuff, where they not only have a solid golf course but go-carts, a batting cage, and bumper boats.  You might think bumper boats would just be a slower, less fun version of bumper cars.  But you would quickly change your tune when I told you that bumper boats have water cannons mounted on them.

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Kids aw hawd to ustan

CJ, like most kids his age, pronounces “r” as “w.”  I was investigating this a bit and found the following interesting page which lists the various phonological shifts young kids undergo as they learn to speak.  A lot of these are things CJ does and which I’d never consciously noticed!  (e.g. “weak syllable deletion”, in which “telephone” becomes “teffone.”)  The r/w issue is called “gliding of  liquids.”  Did you know that people typically can’t pronounce “th” as in “thing” until they’re eight and a half years old?

This kind of thing always reminds me of a bit of comic business in The Mosquito Coast about a guy in Massachusetts with a speech impediment and a wife named Cheryl, who he calls “Shovel.”  The Mosquito Coast is a great, great, great novel, by the way!  Not sure everyone is aware of this.

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Recommendation inflation

The paper submitted by ***** bears conclusive evidence of the profound and penetrating studies of the author in the area to which the topic dealt with belongs, of a diligent, genuinely mathematical spirit of research, and of a laudable and productive independence. The work is concise and, in part, even elegant: yet the majority of readers might well wish in some parts a still greater transparency of presentation. The whole is a worthy and valuable work, not only meeting the requisite standards which are commonly expected from doctoral dissertations, but surpassing them by far.

Sounds like a letter you’d read nowadays about a quite promising but perhaps not absolutely first-rate new Ph.D., right?

It’s what Gauss wrote about Riemann on the occasion of his thesis defense in 1851.

(Via this very interesting expose by Remmert.)

By the way, here’s what Gauss wrote for Dedekind’s thesis:

“The paper submitted by Mr. Dedekind [published in Dedekind’s Werke I, pp. 1-26] deals with problems in calculus which are by no means commonplace. The author not only shows very good knowledge in this field but also an independence which indicates favorable promise for his future achievements. As paper for admission to the examination this text is fully sufficient”.

Tough audience.

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March math link dump

  • If blood found at a crime scene contains a series of genetic markers found in about 1 in a million people, and if you search a database of genetic material from 300,000 people and find just one match, person X, for the blood at the scene, what is the probability that person X is innocent of the crime?  If you said “1 in a million” you might be a prosecutor.  If you said “1 in a million, and I’m barring any expert testimony that says otherwise” you might be a judge.
  • Good article in the New York Times about the challenge of teaching teachers to teach.  Deborah Ball of Michigan talks about what math teachers need:
  • Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

    The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)

    Ball also administered a similar test to a group of mathematicians, 60 percent of whom bombed on the same few key questions.

  • Thurston teams up with the House of Miyake for a Paris runway show loosely based on the fundamental 3-manifold geometries.  Thurston talks fashion:
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Distinguished lectures this week: Gunnar Carlsson on persistent homology

Fun week coming up:  Gunnar Carlsson of Stanford will be giving this semester’s Distinguished Lecture Series at Wisconsin.  The talks:

Monday, March 8 and Tuesday, March 9, 4pm, Van Vleck B239:

“Topology and Data”

There is a growing need for mathematical methodologies which can provide understanding of high dimensional data sets. These methods also need certain kinds of robustness, so that they should not be too sensitive to changes of scale and to noise, and they should be applicable to various kinds of unstructured data. In these talks we will discuss methods for adapting idealized notions coming from algebraic topology and homotopy theory to the world of point clouds, and show numerous examples of applications of these methods.

Wednesday, March 10, 1:30pm, 1209 Engineering Hall:

“Functoriality, Generalized Persistence, And Structural Signatures”

See the computational topology at Stanford home page for a good overview of the topics of Gunnar’s lectures.  Nigel Boston, Rob Nowak, and I have been running a learning seminar on the topic:  I’ll try to post again this week about some data that Laura Balzano and I messed around with with persistent homology in mind, and what we learned thereby.

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