Show report: Jens Lekman

I heard a lot of people say Jens Lekman was the new Stephin Merritt or the new Neil Hannon, but now, having seen him tonight at the Old Music Hall, I’m pretty sure he’s the new Jonathan Richman. Songs about whimsy, songs about nostalgia, songs about unexceptional daily events made big by means of deeply felt singing. Audience participation. Little stories, backed with strumming, between and included within the songs.

Physically he’s very different — part of the Jonathan Richman experience is seeing a guy who looks like a tight end sing like an elf. Lekman actually does look like an elf. His band, too — elves in smocks. Except for the DJ, who is extravagantly tall and ungainly, yet dances so smoothly that he appears to be a Twin Peaks character inserted into the scene at the last minute.

First day of a long season

The always great Tom Scocca on the mental state of Oriole Nation as the 2008 campaign gets underway:

Beyond plain categories of optimism and pessimism live those of us who see a sparkling half-glass of water and know for sure that the Orioles are eventually going to take a crap in it.

More Orioles dyspepsia at Tom’s season preview at Deadspin.

My WNYC piece about sabermetrics and Alex Rodriguez (plus a little Orioles dyspepsia for my fellow orange-and-blackers) can now be heard online.

In today’s New York Times, Samuel Arbesman and Steven Strogatz argue that Joe DiMaggio’s streak wasn’t as miraculous as you think. They ran 10,000 Monte Carlo simultations of the history of major league baseball and found that, 42% of the time, someone had a hitting streak 56 games or longer. In every case, there was some player in some season who put together a hitting streak of at least 39 games.

That’s a nice experiment, but I don’t think it quite justifies the headline. The figure below shows that, in the simulation, long hitting streaks were strongly concentrated in the pre-1905 era, when higher batting averages were more common. In 1894 (the big spike in the chart below) the batting average for the entire National League was well over .300. The relevant question is not so much “is it surprising that someone had a 56-game hitting streak?” but “is it surprising that someone playing baseball under modern conditions had a 56-game hitting streak? And how likely is it ever to happen again?” The number I’d like to see is: of the 10,000 simulated seasons, in how many did a player have a 56-game hitting streak after 1941?

Despite my criticism, I’m delighted the NYTimes published this. The main point — that unlikely-seeming events are actually quite likely, as long as you give them enough chances to happen — is a crucial and subtle one, which should be repeated in a loud voice at every possible opportunity.

Arbeson has a blog which is mostly about computational biology and urban planning, not baseball. Strogatz has no blog, but his book Sync: The Emerging Science of Spontaneous Order is surely very good, based on the lectures I’ve seen him deliver.

Happy Grothenday!

Alexander Grothendieck is 80 today. It’s truly surprising that his strange and marvelous biography — to put it all in one sentence, he rewrote much of the foundation of number theory and geometry in an immense burst of energy in the 1960s, then, over time, came to feel that the mathematical establishment had betrayed him and his ideas, and moved to the Pyrenees to be alone and herd sheep — is not better known.

Read the excellent biographical article by Allyn Jackson, “Comme Appelé du Néant — As If Summoned From the Void”: Part I, Part II. And if that doesn’t sate you, skim through the mass of scanned manuscripts, appreciations both technical and non-, and photographs at The Grothendieck Circle.

My own story: in my last year of grad school I came across Grothendieck’s famous late article, “Esquisse d’un programme” (actually an unsuccessful grant application.) My advisor saw me reading it, and, aware of its seductive effect, told me, “I forbid you from reading the Esquisse until your thesis is finished!”

On your radio

I’ll be on the Leonard Lopate Show on WNYC from 1-1:20pm this Friday, March 28, talking about why, from a quantitative standpoint, Yankee fans should get off A-Rod’s back.

Give the people (who like the Kakeya problem) what they want

Wow — make one comment on Terry’s blog and you get a ton of traffic.

Terry encouraged me in the comments to experiment with posting math. So I’m going to experiment!

The background: Zeev Dvir this week presented a beautiful and simple proof of the Kakeya conjecture over finite fields. A Kakeya set in F_q^n is a subset containing a line in every direction. Dvir proved that every Kakeya set has at least c_n q^n elements. (Hat tip to The Accidental Mathematician for alerting me to Dvir’s paper!)

In the special case n=2, Dvir’s method shows that a Kakeya set has at least 1/2 q(q+1) elements. In fact (as I learned from Terry’s blog) the best known lower bound is

1/2 q(q+1) + (5/14)q + O(1)

due to Cooper.

Question: Can Dvir’s method be refined to give a better lower bound?

(Note that there are examples of 2-dimensional Kakeya sets, due to Mockenhaupt and Tao, of size 1/2 q(q+1) + (1/2) q + O(1), so Cooper’s bound can’t be improved very much!)

One might naturally start out as follows. The main idea of Dvir’s proof is to show that a Kakeya set can’t be contained in an affine plane curve of degree q-1.

What if S is a Kakeya set contained in an affine plane curve of degree q? That is, what if F in F_q[x,y] is a polynomial of degree q vanishing on S? This places rather strong conditions on F. In each direction m we have a line L_m (say y = mx+b) such that F vanishes on L_m(F_q); since deg F = q, this implies that

F(x,mx + b) = L_m G + c (x^q – x)

for m = 0,1, … q-1. (Of course, there is one more direction, the infinite one, which you can deal with separately.)

Let V be the space of degree-q polynomials vanishing on S. One might like to bound the dimension of V above; because the dimension of the space of degree-q polynomials vanishing on S is at least (1/2)(q+1)(q+2) – |S|, so

|S| <= (1/2)(q+1)(q+2) – dim V.

We note that V is contained in the intersection of q spaces V_0, … V_{q-1}, where

V_m = span of multiples of L_m and x^q-x.

(Note that if not for the x^q – x, we would be looking for polynomials which were multiples of q different linear forms, which indeed makes V very small! This is what happens in Dvir’s case, where the degree is q-1.)

So far, I’m actually not sure whether any of this does more than restate the problem. But one might try to make an argument along the following lines: we can think of V+[x^q-x] as a linear system of degree-q curves in P^2. Now the Kakeya condition on S shows that a whole lot of these curves have one of the lines L_0, … , L_{q-1} as an irreducible component; in particular, the locus R of reducible curves in this linear system contains q+1 hyperplanes.Does this give an upper bound on dim V? One might, for example, observe that if R contains q+1 hyperplanes, then a general pencil of curves in V + [x^q -x] has q+1 reducible fibers; there is a substantial literature about reducible fibers in pencils of hypersurfaces — particularly relevant here seem to be theorems of Lorenzini (1993) and Vistoli (1993). (Beware: these theorems are usually stated in characteristic 0!)

Evening update:  I spent a few hours thinking about this and don’t immediately see how to push it through — the hard upper bound on the number of reducible fibers from Lorenzini or Vistoli is q^2 – 1, and it’s not clear to me how you could ensure that some pencil in V + [x^q - x] satisfies the more delicate conditions necessary to get that number down below q.  It may be that cutting down to pencils is the wrong thing to do, and one should instead try to show directly that the reducible locus in V + [x^q - x] doesn’t contain q hyperplanes.

Give the people what they want

What do people tend to look at on Quomodocumque? Now that WordPress offers statistics on most-viewed posts over the whole life of the blog, I can tell you. The most popular post, by a factor of three, is my post on the Mathematics Jobs Wiki, presumably because it appears on the first page of the “math jobs wiki” Google search that anxious candidates apparently carry out several times daily.

It turns out that, despite this not being a math blog, the math posts are in general the most popular; 8 of the top 10 posts are math posts. The other two are food posts, and food is a clear second among topics I frequently blog about. In fact, the most popular post about neither math nor food is my report on a German Art Students show, all the way down at #21.

And you know what people really don’t care about? At all? The Orioles. Truly, these are degraded times.

The most popular search leading people to the blog was, as expected, “math jobs wiki.” The next few leaders are variations on that, the name of the blog and my name. After that comes Inka Heritage, the local Peruvian restaurant I wrote about last summer.

Today someone came to the blog via a search for “Jonathan Richman was a Mormon.” Awesome.

Improvised miso soup

I went out this afternoon during CJ’s nap to do a little spectral sequence wrangling in a local coffee shop. On the way, I was happily surprised to find that my local Asian grocery, Lee’s Oriental, was open on Easter Sunday. So I bought a couple of bags of stuff and made the following off-the-cuff soup, which I record here for future reference:

Saute about half a big head of napa cabbage, a bunch of enoki mushrooms, and 6 cloves garlic in the bottom of the soup pot. Pour a gallon of water on top of it and bring to boil. Stir in about 1/4 c miso (I was using Korean miso, which the shopkeeper — Lee himself? — warned me was very strong; usual miso recipes seem to want more miso per unit of water.) Then add 2 lb soba noodles (I used Sukina brand “Japanese vermicelli”) and boil for about 7 minutes. Three minutes before the end of cooking, add a big handful of bean sprouts (about a cup) and a bunch of scallions, coarsely chopped. One minute before the end of cooking, crack three eggs in the soup and stir furiously until scrambled.

At table, the soup can be garnished with pieces of deep-fried tofu (more precisely: ajitsuke inari age) and sesame oil.

Remarks: The bean sprouts and enoki mushrooms were probably unnecessary; neither really asserted themselves in the final soup. I might have used less soba; it absorbs a lot of soup, so that by the time we were done with dinner, the leftovers were really no longer a soup with noodles, but a noodle dish with a thick glutinous sauce. But it will probably be easier for CJ to eat this way. Soup, per se, remains a challenge for him.

Carnival of Mathematics 29

Since I’m an academic mathematician, this Carnival will be a little more heavily weighted towards that side of things than usual.

Let me start by welcoming my colleague Emmanuel Kowalski to the Grand High Council of Blogging Number Theorists — he’s got plenty of math posts up, but the non-technical reader may want to go straight to his series of mystery stories, Les fabuleuses aventures de Schlomo Cohen le Mathématicien détective.

Noah Snyder at Secret Blogging Seminar wonders what we should mean when we ask “how complicated is a finite group?” In order to be really interesting, any question of this kind should have at least three answers, and Noah offers four. Good stuff in the comments, too. While we’re talking finite group theory, Isabel at God Plays Dice asks “are most groups solvable?”

Good Math, Bad Math promises some upcoming group theory posts, but at the moment they’re leading with a thoughful and readable introduction to game theory.

At the Everything Seminar, a great idea for how to make it through a bad seminar talk: Bad Talk Bingo.

The Accidental Mathematician, an analyst at UBC, just started blogging: this month she asks, does it matter if math blogging is a boys’ club?

Rod of Reasonable Deviations explains how the natural problem

Say we have a network of agents where each agent takes measurements and communicates with a few other agents. The agents form a network. Each agent

1. measures a given quantity
2. stores the measurement in its internal state
3. updates its internal state by replacing its own state with a weighted average of its neighboring agents’ states and its own state.

Our objective: we want each agent’s internal state to converge to the arithmetic average of all measurements.

can be boiled down to some nice linear algebra.

Meeyauw provides a link-filled introduction to the Collatz Conjecture (also called the 3x+1 conjecture), and solicits fellow bloggers to join a reading group for the best book-length work of mathematical exposition of all time, Godel, Escher, Bach.

What’s fascinating about the number 17? Find out at MathNotations, where you can also accept his challenge to think of fascinating things about 97 and 153. Maria Anderson at Teaching College Math, on the other hand, thinks numbers aren’t fascinating enough, and suffers from astronomy envy:

I know that Pi-Day is tomorrow, and I should be excited to be celebrating a math holiday, but honestly I feel a little down. Where are the cool “explorer-style” applications for mathematics?

Maybe if you’re not finding pi fascinating enough in its numerical form, you could try listening to pi as a song over at 360.

Let’s Play Math! offers a series of combinatorial puzzles which ought to first confound, then intrigue any geometrically-minded middle-schoolers within reach.

Out in Left Field’s daughter is getting grades of 3 instead of perfect 4 in math.

What exactly are my daughter’s 3s based on? One thing is certain: they don’t reflect her math skills.

Is the Reform Math curriculum to blame?

Elsewhere in the Great Curriculum Kerfuffle, the state of New York is about to introduce a new Euclidean geometry course; JD2718 points out that, since such a course hasn’t been standard there for decades, there may not be enough teachers qualified to handle the material the Regents have asked for.

Does statistics count as math? Last week brought a new issue of the always enlightening Chance News, which brings us statistical tidbits from the last month’s news. Some of the tidbits show off statistical thinking at its best, and some are more like this:

Twenty-six new cases of the inflammatory lung disease sarcoidosis [were seen amongst rescuers] in the first five years after 9/11. Five or fewer rescuers got sarcoidosis anually before 9/11.

Why this is not a math blog

I’m hosting the Carnival of Mathematics tomorrow, which is a little strange for me, since this is not actually a math blog.

Why not? I’m a mathematician, after all, and I spend the majority of my day thinking about math — so why don’t I blog about it very much?

I once had a conversation with a colleague of mine in mathematical physics about people who announce important results and then take a long time writing them up. I was complaining about such people, and my colleague was defending them. There are good arguments on both sides, of course. On one hand, if “everyone knows” that X has proved Y, nobody’s going to work on Y — but people may also be reluctant to prove things that depend on Y, since the proof hasn’t appeared. So a whole area of inquiry can get stuck. On the other hand, when things are rushed into print, there can be mistakes, or just inadequate explanations, and once the paper’s published it may be a long time before the author or someone else writes a really readable version. Anyway, my friend and I went back and forth on this for a while, and finally he said, “But really, why is it so annoying to you to have to wait a month or two to see the proof?”

And that’s when the lightbulb went on. For him, “a long time” meant “a month.” For me, it means “two years.” Physics is fast. Math is slow.

And I like that about math. I like that I don’t have to anxiously scan the new number theory postings on the arXiv each morning for new developments, and I like that I’ll never have to finish a paper by nightfall in order to avoid being scooped. In math we let our ideas simmer a long time before we share them with our colleagues, and even longer before we make them public. We’re contemplatives.

Math blogging, in some ways, works against that. If I blogged about my mathematics as it happened, what you’d see is a lot of first drafts, and a lot of “whoops! I take that back.” Wouldn’t you rather just read my paper, when I carefully, thoughtfully, and eventually, write it?

Two remarks:

• This is really an explanation of why I don’t blog about math I’m working on. It doesn’t apply to blogging of the form “Here’s a recent paper someone just posted on the arXiv, and here’s why I think it’s interesting,” a la Not Even Wrong. This kind of mathblogging is an unmixed good and maybe I’ll start doing more of it.
• An interesting counterargument might be something like: “Blogging about your mathematical ideas makes those ideas available to people outside your circle of elite research universities — since you do talk informally about these ideas with your colleagues inside this circle, it’s undemocratic and bad not to make them available to outsiders.” I haven’t decided whether this counterargument makes sense or not.

Raymond Carver’s Gordon Lish problem, and mine

When my parents visited, they dropped off a stack of old New Yorkers, including this year’s fiction issue, which featured “Beginners,” Raymond Carver’s original version of “What We Talk About When We Talk About Love.” Carver’s version is moving, but also talky and baggy, possessing none of the rigorous terseness that we talk about when we talk about Raymond Carver. Except that maybe we were actually talking about Gordon Lish, who edited the story down to its bones. Or, more precisely, about the collaboration between the two. The New Yorker offers a remarkable chance to see how this kind of collaboration works: they’ve posted Carver’s original version with Lish’s edits superimposed, “Track Changes”-style:

My friend Mel Herb McGinnis, a cardiologist, was talking. Mel McGinnis is a cardiologist, and sometimes that gives him the right. ¶ The four of us were sitting around his kitchen table drinking gin. It was Saturday afternoon. Sunlight filled the kitchen from the big window behind the sink. There were Mel Herb and me I and his second wife, Teresa—Terri, we called her—and my wife, Laura. We lived in Albuquerque, then. But but we were all from somewhere else. ¶ There was an ice bucket on the table. The gin and the tonic water kept going around, and we somehow got on the subject of love. Mel Herb thought real love was nothing less than spiritual love. He said When he was young he’d spent five years in a seminary before quitting to go to medical school. He He’d left the Church at the same time, but he said he still looked back on to those years in the seminary as the most important in his life.

That opening paragraph is actually treated pretty mildly; farther into the story, whole pages get the axe. Carver didn’t take Lish’s edits easily — in an exchange of letters published in the same New Yorker issue, he tries to pull back the stories from publication after seeing what Lish did to them. Carver felt Lish had violated his work — and he was right! But the work was better for being violated.

I know how he feels, a little, because I, too, have been edited by Gordon Lish. Sometime in 1995 I wrote a story called “What Can We Expect From the New Currency?” and submitted it to Lish’s magazine, The Quarterly. Lish called me on the phone to tell me he was accepting the story and that he’d send me a version with some edits by mail. When the package from Lish arrived, I discovered that what he was accepting was about a third of my story: most of the opening, broken up by the insertion of some paragraphs rescued from the mostly deleted latter sections.

And you know what? His version wasn’t really my story — but it was a lot better and cleaner than my story. And he didn’t change my title.

What I didn’t know was that The Quarterly was out of money, and would never publish another issue.