## The conformal modulus of a mapping class

(Warning — this post concerns math I don’t know well and is all questions, no answers.)

Suppose you have a holomorphic map from C^* to M_g,n, the moduli space of curves.  Then you get a map on fundamental groups from $\pi_1(\mathbf{C}^*)$ (otherwise known as Z) to $\pi_1(\mathcal{M}_{g,n})$ (otherwise known as the mapping class group) — in other words, you get a mapping class.

But not just any mapping class;  this one, which we’ll call u, is the monodromy of a holomorphic family of marked curves around a degenerate point.  So, for example, the image of u on homology has to be potentially unipotent.  I’m not sure (but I presume others know) which mapping classes u can arise in this way; does some power of u have to be a product of commuting Dehn twists, or is that too much to ask?

In any event, there are lots of mapping classes which you are not going to see.  Let m be your favorite one.  Now you can still represent m by a smooth loop in M_g,n.  And you can deform this loop to be a real-analytic function

$f: \{z: |z| = 1\} \rightarrow \mathcal{M}_{g,n}$

Finally — while you can’t extend f to all of C^*, you can extend it to some annulus with outer radius R and inner radius r.

Definition:  The conformal modulus of a mapping class x is the supremum, over all such f and all annuli, of (1/2 pi) log(R/r).

So you can think of this as some kind of measurement of “how complicated of a path do you have to draw on M_{g,n} in order to represent x?”  The modulus is infinite exactly when the mapping class is represented by a holomorphic degeneration.  In particular, I imagine that a pseudo-Anosov mapping class must have finite conformal modulus.  That is:  positive entropy (aka dilatation) implies finite conformal modulus.   Which leads Jöricke to ask:  what is the relation more generally between conformal modulus and (log of) dilatation?  When (g,n) = (0,3) she has shown that the two are inverse to each other.  In this case, the group is more or less PSL_2(Z) so it’s not so surprising that any two measures of complexity are tightly bound together.

Actually, I should be honest and say that Jöricke raised this only for g = 0, so maybe there’s some reason it’s a bad idea to go beyond braids; but the question still seems to me to make sense.  For that matter, one could even ask the same question with M_g replaced by A_g, right?  What is the conformal modulus of a symplectic matrix which is not potentially unipotent?  Is it always tightly related to the size of the largest eigenvalue?

## Put the second law of thermodynamics down and slowly step away, New York Times

Yet given her professional background, Dr. Oakley couldn’t help doubting altruism’s exalted reputation. “I’m not looking at altruism as a sacred thing from on high,” she said. “I’m looking at it as an engineer.”

And by the first rule of engineering, she said, “there is no such thing as a free lunch; there are always trade-offs.” If you increase order in one place, you must decrease it somewhere else.

Moreover, the laws of thermodynamics dictate that the transfer of energy will itself exact a tax, which means that the overall disorder churned up by the transaction will be slightly greater than the new orderliness created. None of which is to argue against good deeds, Dr. Oakley said, but rather to adopt a bit of an engineer’s mind-set, and be prepared for energy losses and your own limitations.

Stop hurting physics!

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## Happy birthday, Dick Gross

Just returned from Dick Gross’s 60th birthday conference, which functioned as a sort of gathering of the tribe for every number theorist who’s ever passed through Harvard, and a few more besides.  A few highlights (not to slight any other of the interesting talks):

• Curt McMullen talked about Salem numbers and the topological entropy of automorphisms of algebraic surfaces (essentially the material discussed in his 2007 Arbeitstagung writeup.)  In particular, he discussed the fact that the logarithm of Lehmer’s number — conjecturally the “simplest” algebraic integer — is in fact the smallest possible positive entropy for an automorphism of a compact complex surface.  Here’s a question that occurred to me after his talk.  If f is a Cremona transformation, i.e. a birational automorphism of P^2, then there’s a way to define the “algebraic entropy” of f, as follows:   the nth iterate of f is given by two rational functions (R_n(x,y),S_n(x,y)), you let d_n be the maximal degree of R_n and S_n, and you define the entropy to be the limit of (1/n) log d_n.  Question:  do we know how to classify the Cremona transformations with zero entropy?  The elements of PGL_3 are in here, as are the finite-order Cremona transformations (which are themselves no joke to classify, see e.g. work of Dolgachev.)  Are there others?
• Serre spoke about characters of groups taking few values, or taking the value 0 quite a lot — this comes up when you want, e.g., to be sure that two varieties have the same number of points over F_p for all but finitely many p, supposing that they have the same number of points for 99.99% of all p.  The talk included the amusing fact that a character taking only the values -1,0,1 is either constant or a quadratic character.  (But, Serre said, there are lots of characters taking only the values 0,3 — what are they, I wonder?)
• Bhargava talked about his new results with Arul Shankar on average sizes of 2-Selmer groups.  It’s quite nice — at this point, the machine, once restricted to counting orbits of groups acting on the integral points of prehomogenous vector spaces, is far more general:  it seems that the group of people around Manjul is getting a pretty good grasp on the general problem of counting orbits of bounded height of the action of G(Z) on V(Z), where G is a group over Z (even a non-reductive group!) and V is some affine space on which G acts.  With the general counting machine in place, the question is:  how to interpret these orbits?  Manjul showed a list of 70 representations to which the current version of the orbit-counting machine applies; each one, hopefully, corresponds to some interesting arithmetic enumeration problem.  It must be nice to know what your next 70 Ph.D. students are going to do…

Dick has a lot of friends — the open mike at the banquet lasted an hour and a half!  My own banquet story was from my college years at Harvard, where Dick was my first-year advisor.  One time I asked him, in innocence, whether he and Mazur had been in graduate school together.  He fixed me with a very stern look.

“Jordan,” he said, “as you can see, I am a very old man.  But I am not as old as Barry Mazur.