What experimental math taught me about my intuition

The project I’m working on with David Brown and Bryden Cais is the first thing I’ve ever done involving computational experiments as a serious part of the development of the ideas.  We’re trying to formulate a reasonable conjecture about the limiting distribution of some arithmetic objects, and I thought we’d arrived at a pretty good heuristic — call it conjecture A — which fit nicely in a context with other conjectures and theorems about similar objects.

But the data we’d collected didn’t fit conjecture A very well.  In fact, when we looked at the data carefully, it appeared to be pointing strongly at conjecture B, which has a similarly clean formulation but for which we didn’t have any theoretical justification.

So I spent much of the weekend thinking about this, and by the end of it, I felt pretty confident that conjecture B was pretty reasonable — even, in a way, more reasonable than conjecture A — though we still didn’t have a strong justification for it.

Today it turned out that there was a mistake in the data collection, and conjecture A looks good after all.   But it’s a sobering reminder that my intuitions about “what ought to be true,” which I think of as rather rigorous and principled, are in fact quite malleable.

 

 

Franzen blows a joke

Given the weirdly ambivalent best-friendship between Jonathan Franzen and David Foster Wallace, it’s sort of a strange choice to invite Franzen to give this year’s Kenyon College commencement address, the 2005 edition of which seems destined to be the essay of Wallace’s that stands in the popular imagination as a portrait of the man himself.  (Not without reason.  And if you haven’t read it, then maybe do that instead of continuing on with this somewhat small-minded blog post.)

Franzen’s essay is good, but I thought he made a mistake in one place:

If you dedicate your existence to being likable, however, and if you adopt whatever cool persona is necessary to make it happen, it suggests that you’ve despaired of being loved for who you really are. And if you succeed in manipulating other people into liking you, it will be hard not to feel, at some level, contempt for those people, because they’ve fallen for your shtick. You may find yourself becoming depressed, or alcoholic, or, if you’re Donald Trump, running for president (and then quitting).

Surely the joke is much stronger without Trump, or the parenthetical:  “You may find yourself becoming depressed, or alcoholic, or running for president.”  Then, instead of going for Leno-style yuks, he’s actually gently reminding the high-achieving students at a fancy liberal-arts college that an unreflective drive to achieve, and to win, is second cousin to corrosive melancholy.  That would have been a good nod to Wallace.  And it still would have gotten laughs, while gently turning the knife.

Instead, Franzen talks bird-spotting, reiterating the similar material in his much-discussed New Yorker piece on Wallace and solitude.  This part didn’t sway me.  Jonathan Franzen likes birds, we get it.  Not all enthusiasms have a lesson to teach.

Philosophy is about craziness

From John Holbo at Crooked Timber, writing about Green Lantern and philosophy:

Philosophy really is, substantially, about craziness and the peculiarly stable attraction of absurd and apparently stupid ideas and attitudes.

I was surprised to find that I sort of agree with this provocative claim.  I like academic philosophy a lot — particularly that strange sand-shifting-under-you feeling when you finally understand that a question or problem that initially appeared stupid or content-free is, in fact, difficult and perhaps even deep.

How good is Zach Britton?

The Orioles blew another game against the Yankees last night, but rookie Zach Britton was brilliant again, scattering 6 hits over 7 innings and allowing only one run, that unearned.  His ERA of 2.14 is 5th best in the American League, and he has the 3rd highest WAR among AL pitchers.  In the talk about Baltimore’s good young arms, Britton has gotten less ink than Brian Matusz and Chris Tillman; in the first two months of his big-league career, he’s outpitched both.  Could he really be this good?

There are at least two reasons to think not.  First, Britton doesn’t strike out anybody:  just 5 K per 9 IP so far this year.  Last year, the best ERA among pitchers with 5 or fewer strikeouts per 9 innings was Carl Pavano, at 3.75.  It’s hard to pitch effectively without getting strikeouts.  That suggests Britton’s been pretty lucky; and indeed, his BABIP right now is an unsustainably low .228.  In other words, batters are hitting Britton’s pitches, but the balls haven’t been falling in for hits.  That’s the kind of thing that au courant thinkiing tends to place outside a pitcher’s control.

And yet –

There’s another thing a pitcher controls totally, and that is not giving up home runs.  And Zach Britton is very, very good at not giving up home runs.  This year he’s given up just 4 in 60 innings of work.  That’s a small sample size (league average rate would be 6 HR in that many innings) but it’s been his trademark throughout his professional career.  In 103 minor league starts Britton gave up just 27 home runs.

If nobody homers off you, you can get away with a lot of singles and walks.  Britton is not going to maintain a 2.14 ERA but I think there’s every reason to think he’ll be a legitimate front-of-the-rotation starter.

The less said about the end of the Yankee game, the better, but I’ll remark on one more bright spot — Mariano Rivera blew the save, continuing his record of mediocrity against the Orioles.  His career ERA against Baltimore is 3.15, almost a full point higher than his overall ERA, and we’ve beaten him 8 times, more than any other team.

 

Maples on Cohen-Lenstra for matrices with iid entries

Emmanuel Kowalski pointed me to a very interesting recent paper by Kenneth Maples, a grad student at UCLA working under Terry Tao.  One heuristic justification for the Cohen-Lenstra conjectures, due to Friedman and Washington, relies on the remarkable fact that if M is a random nxn matrix in M_n(Z_p), the distribution of coker(M) among finite abelian p-groups approaches a limit as n goes to infinity; so it makes sense to talk about “the cokernel of a large random matrix” without specifying the size.  (There’s a fuller discussion of Friedman-Washington in this old post.)

Maples shows that the requirement that M is random — that is, that the entries of M are independently drawn from Z_p with additive Haar measure –  is much stronger than necessary.  In fact, he shows that when the entries of M are drawn independently from any distribution on Z_p satisfying a mild non-degeneracy condition, the distribution of coker(M) converges to the so-called Cohen-Lenstra distribution, as in Friedman-Washington.  That’s pretty cool!  I don’t know any arithmetic circumstance that would naturally produce exotic distributions of this kind, but the result helps bolster one’s psychological sense that the Cohen-Lenstra distribution provides the only sensible notion of “cokernel of random matrix,” in some robust sense.

Universality of random matrix laws is a very active and fast-moving topic, but Maples’ result is the first universality result for p-adic matrices that I know of.  More generally, I think there’s a lot to be gained by understanding how well the richly developed theory of random large real and complex matrices carries over to the p-adic case.

Future “women in math” threads: a request

The comments on yesterday’s post turned into another boring “feminism sux / feminism roolz” thread.  I get it — some people think it’s worth thinking about gender issues in math, some people don’t, people are going to have that discussion.  Fine.

My request:  if you are going to comment on a “women in math,” thread, or for that matter, any other thread that gets people’s political dander up, try to direct your comment at the material of the particular post.  If your comment would apply equally well to any imaginable post about women in math, then maybe don’t post it.

You must not act surprised when I inform you that dysfunctional relationship is a product of today

Writing a talk about stable cohomology and listening to my “Unplayed” playlist on iTunes, as one does.  Encountered a much-beloved and almost-forgotten piece of collegiana:  “Dysfunctional Relationship,” by Consolidated.  Glum radical politics over beats and scratches, somehow wooden and charming at once.   Was there ever another dance track that asked the backup vocalists to repeatedly sing the word “symptomatic?”  Doubtful.  If your memory of US culture between 1988 and 1992 is in any way fond, you really want to watch the video.

Why Laba is not on Math Overflow

A thoughtful post from harmonic analyst Izabella Laba about why she isn’t participating in Math Overflow (and, by extension, why other women in math might not be.)  The comments are good too.

Stop me before I evoke again!

From Chris Rickert in Saturday’s Wisconsin State Journal, writing on Sun Prairie’s denial of a land use license to the town mosque:

It’s not often you encounter a business park condominium association and Friday call to prayers in the same context.

The first is pure ex-urban Americana, evoking images of identically nondescript buildings, large parking lots and easy access to a major highway.

The second is worlds away, evoking images of rows of dark-skinned men kneeling in unison, Arabic broadcast over bad PA systems, even terrorism.

Reader survey: have you ever clicked on a Facebook ad?

Apparently Facebook is said by some to be valued at $75 billion.  Facebook’s revenue, just under $2b last year, derives mostly from online ads.

I visit Facebook most days and was only dimly aware that it had ads.  I just looked, and yes, they’re there:  not on the main newsfeed page, but on my profile page.  I’m trying to get my head around the fact that buying Facebook ad space is a good investment.

Question:  have you ever clicked on a Facebook ad?  Have you ever clicked through and then purchased?

Secondary (or maybe really prior) question:  Does it matter to advertisers whether Facebook users click?  Or is the point just to put the name of the product in the corner of people’s eyes, for brand awareness?

Anticipated objections:  1.  Yes, I know the readers of this blog are not a good sample of FB users.  2.  No, I don’t think “businesses wouldn’t buy ads if they weren’t effective” is obviously correct, though it does carry some force.

Update:  Reader Rod Carvalho points out that the same question applies to Google — so feel free to answer that too!