Natural logs and products of no primes

The e-mail you get after you write an article about number theory is very interesting.  For one thing, you’re reminded of phrasings which have one meaning among mathematicians, but a slightly different one outside the tribe.

The majority of the e-mail I’ve gotten about the bounded gaps piece concerns two questions of this kind:  I’ll answer them both here, in case other readers are following the link from Slate to the blog.

Q:  You say that the number of primes less than X is about X/log(X), but don’t you mean X/ln(X)?

A:  When mathematicians say “log” we mean the natural log, the thing which in some other contexts (e.g. Google’s search bar calculator) is denoted “ln.”  But mathematicians never say “ln.”  (To be honest, we kind of think the base-10 logarithm should be called “lu.”)

Q:  You say that every positive number is the product of primes, but this is not true for prime numbers themselves, which can’t be expressed as products.

A:  A prime number is indeed the product of prime numbers!  It is the product of just one prime number, itself.

What about 1?  It’s the product of zero prime numbers.

 

Yitang Zhang, bounded gaps, primes as random numbers

In Slate today, I have a piece about Yitang Zhang’s amazing proof of the bounded gaps conjecture.  Actually, very little of the article is about Zhang himself or his proof; I wanted instead to explain why mathematicians believed that bounded gaps (or twin primes) was true in the first place, via Cramér’s heuristic that primes behave like random numbers.

And a lot of twin primes is exactly what number theorists expect to find no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force be pushing the primes apart.

Tantalisingly close to significance

Matthew Hankins and others on Twitter are making fun of scientists who twist themselves up lexically in order to report results that fail the significance test, using phrases like “approached but did not quite achieve significance” and “only just insignificant” and “tantalisingly close to significance.”

But I think this fun-making is somewhat misplaced!  We should instead be jeering at the conventional dichotomy that a result significant at p < .05 is “a real effect” and one that scores at p = .06 is “no effect.”

The lexically twisted scientists are on the side of the angels here, insisting that a statistically insignificant finding is usually much better described as “not enough evidence” than “no evidence,” and should be mentioned, in whatever language the journal allows, not mulched.

 

 

 

 

Elliptic curves with isomorphic cyclic 13-subgroups?

I liked this MathOverflow question, which asks:  are there two non-isogenous elliptic curves over Q, each one of which has a rational cyclic 13-isogeny, and such that the kernels of the two isogenies are isomorphic as Galois modules?

This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E’,C,C’,φ), where E and E’ are elliptic curves, C and C’ are cyclic 13-subgroups, and φ is an isomorphism between C and C’.

S is a quotient of X_1(13) x X_1(13) by the diagonal in the natural (Z/13Z)^* x (Z/13Z)^* action.

Is S general type, rational, what?

 

 

Every Noise At Once

Glenn McDonald is the guy who wrote the amazing, obsessive, beautiful music blog The War Against Silence, now mostly dormant.  I admire him for writing tens of thousands of words about Alanis Morissette, whom he, and I, and maybe nobody else, still consider an important cultural figure.  He’s also a pretty hardcore data analyst.  I’ve often fallen down the rabbit hole of his analysis of the Pazz and Jop ballots.

Now he works for Echo Nest, the Greater Boston music startup that sponsored the Music Hack Day I participated in a couple of years ago.  And his latest project, Every Noise At Once, is a map of all music.  Seriously!  A map of all music!  By which I mean: an embedding of the set of genres tracked by EN into the Euclidean plane, and, for each genre, an embedding of bands tagged in that genre.

Play with it here.

Random squarefree polynomials and random permutations and slightly non-random permutations

Influenced by Granville’s “Anatomy of integers and permutations” (already a play, soon to be a graphic novel) I had always thought as follows:  a polynomial of degree n over a finite field F_q gives rise to a permutation in S_n, at least up to conjugacy; namely, the one induced by Frobenius acting on the roots.  So the distribution of the degrees of irreducible factors of a random polynomial should mimic the distribution of cycle lengths of a random permutation, on some kind of equidistribution grounds.

But it’s not quite right.  For instance, the probability that a permutation is an n-cycle is 1/n, on the nose.

But the probability that a random squarefree polynomial is irreducible is about (1/n)(1-1/q)^{-1}.

The probability that a random polynomial, with no assumption of squarefreeness, is irreducible, is again about 1/n, the “right answer.”  But a random polynomial which may have repeated factors doesn’t really have an action of Frobenius on the roots — or at least it’s the space of squarefree monics, not the space of all monics, that literally has an etale S_n-cover.

Similarly:  a random polynomial has an average of 1 linear factor, just as a random permutation has an average of 1 fixed point, but a random squarefree polynomial has slightly fewer linear factors on average, namely (1+1/q)^{-1}.

Curious!

 

E.O. Wilson does not think math is unnecessary

This piece by E.O. Wilson has been much shared and much griped about in my circles, but I think it’s a case of a provocative headline (“Great Scientist ≠ Good at Math:  discoveries emerge from ideas, not number-crunching”) prepended by the WSJ to an essay that says something much more modest and defensible.  I’d paraphrase Wilson like this.   Being good in math is like being a good writer.  Everyone agrees:

• You can do great science and be a terrible writer;
• Being better at writing is a worthwhile aspiration for any scientist.

The conjunction of these two statements in no way feels like a denigration of writing.  Nor is Wilson denigrating math.

I’ve said this before but it’s important so I’ll keep saying it — when you write an opinion piece for a publication, you don’t write the headline — the editors do, and they’ll put whatever loosely relevant headline will generate the most clicks.

Encouraging!

The introduction to the textbook States of Matter, by David L. Goodstein:

Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand.  Paul Ehrenfest, carrying on the work, died similarly in 1933.  Now it is our turn to study statistical mechanics.

 

Hiring at and from Wisconsin

Happy to report that the UW-Madison math department has added two more terrific young faculty members, both joining us next fall:  Daniel Erman in commutative algebra and algebraic geometry (seen previously on the blog counting smooth members in semiample linear systems over finite fields) and Uri Andrews in model theory.

In other awesome news, my former Ph.D. student Derek Garton will join the department at Portland State (his master’s degree alma mater!) as a tenure-track assistant professor.

Math on Trial, by Leila Schneps and Coralie Colmez

The arithmetic geometer Leila Schneps, who taught me most of what I know about Galois actions on fundamental groups of varieties, has a new book out, Math on Trial:  How Numbers Get Used and Abused in the Courtroom, written with her daughter Coralie Colmez.  Each chapter covers a famous case whose resolution, for better or worse, involved a mathematical argument.  Interspersed among the homicide and vice are short chapters that speak directly to some of the mathematical and statistical issues that arise in legal matters.  One of the cases is the much-publicized prosecution of college student Amanda Knox for a murder in Italy; today in the New York Times, Schneps and Colmez write about some of the mathematical pratfalls in their trial.

I am happy to have played some small part in building their book — I was the one who told Leila about the murder of Diana Sylvester, which turned into a whole chapter of Math on Trial; very satisfying to see the case treated with much more rigor, depth, and care than I gave it on the blog!  I hope it is not a spoiler to say that Schneps and Colmez come down on the side of assigning a probability close to 1 that the right man was convicted (though not nearly so close to 1 as the prosecution claimed, and perhaps to close enough for a jury to have rightfully convicted, depending on how you roll re “reasonable doubt.”)

Anyway — great book!  Buy, read, publicize!