## Spec is representable

Saw Matt Baker at the Joint Meetings and he told me about this crazy paper he just posted, “Matroids over Hyperfields.”   A hyperring is just like a ring except addition is multivalued; given elements x and y of R, x+y is a subset of R which you can think of as “the possible outcomes of summing x and y.”  A hyperfield is a hyperring in which every nonzero element has a multiplicative inverse.

Here’s an example familiar to tropical geometers:  let T be the hyperfield whose elements are $\mathbf{R} \bigcup -\infty$, whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no!  The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the Krasner hyperfield:  just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object.  K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0).  So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable!  I enjoy that.

Update:  David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161.  In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

• I never new that Mark Alan Stamaty, one of my favorite cartoonists, did the cover of the first They Might Be Giants album.
• Hey I keep saying this and now Allison Schrager has written an article about it for Bloomberg.  Tenure is a form of compensation.  If you think tenure is a bad way to pay teachers, and that compensation is best in the form of dollars, that’s fine; but if California pretends that the elimination of tenure isn’t a massive pay cut for teachers, they’re making a basic economic mistake.
• New “hot hand” paper by Brett Green and Jeffrey Zweibel, about the hot hand for batters in baseball.  They say it’s there!  And they echo a point I make in the book (which I learned from Bob Wardrop) — some of the “no such thing as the hot hand” studies are way too low-power to detect a hot hand of any realistic size.
• Matt Baker goes outside the circle of number theory and blogs about real numbers, axioms, and games.  Daring!  Matt also has a very cool new paper with Yao Wang about spanning trees as torsors for the sandpile group; but I want that to have its own blog entry once I’ve actually read it!
• Lyndon Hardy wrote a fantasy series I adored as a kid, Master of the Five Magics.  I didn’t know that, as an undergrad, he was the mastermind of the Great Caltech Rose Bowl Hoax.  Now that is a life well spent.
• Do you know how many players with at least 20 hits in a season have had more than half their hits be home runs?  Just two:  Mark McGwire in 2001 and Frank Thomas in 2005.

## Idle questions: Diophantine approximation and complex dynamics

Laura DeMarco gave a beautiful talk at the Joint Meetings about her work (with Matt Baker) on the postcritically finite locus in the moduli space of polynomial dynamical systems.  (Here are her slides for a similar talk.)

To say only a tiny bit about what that means:  The dynamical systems in question are those coming from a polynomial map f: C -> C.  Like, say,

$f(z) = z^2 + c$

for some complex number c.  The set of c such that the forward orbit of 0 stays bounded is the Mandelbrot set — you know, this guy:

One way an orbit to be bounded is for it to be eventually periodic; when z^2 + c has this property, we say it is postcritically finite, or PCF.  More generally, the postcritically finite polynomials are those whose critical points all have finite forward orbits.  Number theorists like these because they’re the ones whose inverse iterates generate big interesting number fields with finite ramification.  But that’s not what I want to mention now.  DeMarco mentioned the very interesting fact (sorry, I don’t know who proved this or whether I’m stating it correctly) that as you range over PCFs with longer and longer period, the set of PCFs approaches the uniform distribution (with respect to a standard measure called bifurcation measure) on the boundary of M.

The PCFs, DeMarco told us, should be thought of as special points in the space of all polynomials — in this simple case of quadratics, the PCFs are special points in the complex plane.  They’re kind of like CM points on the j-line, or torsion points on an abelian variety.  The main thrust of DeMarco’s work with Baker concerns dynamical analogues of the Andre-Oort conjecture, which aims to classify those subvarieties of the moduli space of dynamical systems which contain a (Zariski-)dense set of PCF points.  Their striking results demonstrate the unexpected ways in which arithmetic dynamics and complex dynamics have truly started to engage with each other, after a fairly long period of separate development.

But that’s also not what I want to mention now; I just wanted to record a simple thought that a number theorist might have while watching DeMarco’s talk.  (Warning:  as usual with math posts, this is not thought through carefully.)

The PCF points are perhaps sort of like torsion points in C^*, which is to say roots of unity; and just as PCFs of larger and larger period converge to uniform distribution on the Julia set, roots of unity of larger and larger order converge to uniform distribution on the unit circle.  Equivalently: rational numbers of bounded denominator look roughly uniformly distributed on R/Z.

But there are lots of more refined questions one can ask about the way in which the rational numbers sit densely in R/Z.  For example, one can ask about Diophantine approximation; given an irrational point alpha on R/Z, we know there are infinitely many “pretty good” rational approximants to alpha; fractions p/q such that

$|p/q - \alpha| < 1/q^2$.

Are there theorems guaranteeing that any point x on the boundary of the Mandelbrot set has infinitely many PCFs which are “pretty good approximations” to x in the above sense?

What is the most badly approximable point on the Mandelbrot boundary — i.e what is the “golden dynamical system” that plays the role of (1/2)(1+sqrt(5))?

Does x have a canonical sequence of PCF approximants which play the role of continued fraction convergents?