Ungepotch? Yes. But it has that ineffable “it shouldn’t work but it does” that marks really great art.

But here’s something I didn’t know about my home state’s flag:

Despite the antiquity of its design, the Maryland flag is of post-Civil War origin. Throughout the colonial period, only the yellow-and-black Calvert family colors are mentioned in descriptions of the Maryland flag. After independence, the use of the Calvert family colors was discontinued. Various banners were used to represent the state, although none was adopted officially as a state flag. By the Civil War, the most common Maryland flag design probably consisted of the great seal of the state on a blue background. These blue banners were flown at least until the late 1890s….

Reintroduction of the Calvert coat of arms on the great seal of the state [in 1854] was followed by a reappearance at public events of banners in the yellow-and-black Calvert family colors. Called the “Maryland colors” or “Baltimore colors,” these yellow-and-black banners lacked official sanction of the General Assembly, but appear to have quickly become popular with the public as a unique and readily identifiable symbol of Maryland and its long history.

The red-and-white Crossland arms gained popularity in quite a different way. Probably because the yellow-and-black “Maryland colors” were popularly identified with a state which, reluctantly or not, remained in the Union, Marylanders who sympathized with the South adopted the red-and-white of the Crossland arms as their colors. Following Lincoln’s election in 1861, red and white “secession colors” appeared on everything from yarn stockings and cravats to children’s clothing. People displaying these red-and-white symbols of resistance to the Union and to Lincoln’s policies were vigorously prosecuted by Federal authorities.

During the war, Maryland-born Confederate soldiers used both the red-and-white colors and the cross bottony design from the Crossland quadrants of the Calvert coat of arms as a unique way of identifying their place of birth. Pins in the cross bottony shape were worn on uniforms, and the headquarters flag of the Maryland-born Confederate general Bradley T. Johnson was a red cross bottony on a white field.

By the end of the Civil War, therefore, both the yellow-and-black Calvert arms and the red-and-white colors and bottony cross design of the Crossland arms were clearly identified with Maryland, although they represented opposing sides in the conflict.

In 4th grade, in Maryland history, right after having to memorize the names of the counties, we learned about the flag’s origin in the Calvert coat of arms

but not about the symbolic meaning of the flag’s adoption, as an explicit gesture of reconciliation between Confederate sympathizers and Union loyalists sharing power in a post-war border state.

The Howard County flag is based on the Crossland arms. (There’s also a sheaf of wheat and a silhouette of Howard County nosing its way through a golden triangle.) The city of Baltimore, on the other hand, uses the Calvert yellow-and-black only.

Oh, and there’s one more flag:

That’s the flag of the Republic of Maryland, an independent country in West Africa settled mostly by free black Marylanders. It existed only from 1854 to 1857, when it was absorbed into Liberia, of which it’s still a part, called Maryland County. The county flag still has Lord Baltimore’s yellow, but not the black.

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On the good math days, you have an idea, you try it, it doesn’t work, you click over to the next idea, you get over the obstacle that was blocking you, then you’re stuck again, you ask your mind “What’s the next thing to do?” you get the next idea, you take another step, and you just keep going.

You don’t feel smarter on the good math days. It’s not even momentum, exactly, because it’s not a feeling of speed. More like: the feeling of being a big, heavy, not very fast vehicle, with very large tires, that’s just going to keep on traveling, over a bump, across a ditch, through a river, continually and inexorably moving in a roughly fixed direction.

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Missed most of Camp Friends, who were billed as experimental but in fact played genial, not-real-tight college indie. Singer took his shirt off.

Omni, though — this is the real thing. Everyone says it sounds like 1981 (specifically: 1981), and they’re right, but it rather wonderfully doesn’t sound like any particular *thing* in 1981. There’s the herky-jerky-shoutiness and clipped chords (but on some songs that sounds like Devo and on others like Joe Jackson) and the jazz chords high on the neck (the Fall? The Police?) and weird little technical guitar runs that sound like Genesis learning to play new wave guitar on *Abacab* and arpeggios that sound like Peter Buck learning to play guitar in the first place (these guys are from Georgia, after all.) What I kind of love about young people is this. To me, all these sounds are separate styles; to a kid picking up these records now, they’re just 1981, they’re all material to work from, you can put them together and something kind of great comes out of it.

You see a lot of bands with a frontman but not that many which, like Omni, have a frontman and a backman. Philip Frobos sings and plays bass and mugs and talks to the audience. Frankie Broyles, the guitar player, is a slight guy who looks like a librarian and stands still and almost expressionless while he plays his tight little runs. Then, every once in a while, he unleashes an absolute storm of noise. But still doesn’t grimace, still doesn’t move! Amazing. Penn and Teller is the only analogue I can think of.

Omni plays “Jungle Jenny,” live in Atlanta:

And here’s “Wire,” to give a sense of their more-dance-less-rock side:

Both songs are on Omni’s debut album, *Deluxe*, listenable at Bandcamp.

Best show I’ve seen at the Terrace in a long time. Good job, WUD.

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2006: 27

2007: 19

2008: 22

2009: 30

2010: 23

2011: 19

2012: 27

2013: 35

2014: 31

2015: 38

2016: 29

Don’t quite know what to make of this. I’m sort of surprised there’s so much variation! I’d have thought I’d have read less when my kids were infants, or when I was writing my own book, but it seems pretty random. I do see that I’ve been clearly reading more books the last few years than I did in 2012 and before.

Lists, as always, are here (2011 on) and here (2006-2010.)

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*Confusions.”*

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Imagine a room full of 100 people with 100 dollars each. With every tick of the clock, every person with money gives a dollar to one randomly chosen other person. After some time progresses, how will the money be distributed?

People often expect the distribution to be close to uniform. But this isn’t right; the simulations in the post show clearly that inequality of wealth rapidly appears and then persists (though each individual person bobs up and down from rich to poor.) What’s going on? Why would this utterly fair and random process generate winners and losers?

Here’s one way to think about it. The possible states of the system are the sets of nonnegative integers (m_1, .. m_100) summing to 10,000; if you like, the lattice points inside a simplex. (From now on, let’s write N for 100 because who cares if it’s 100?)

The process is a random walk on a graph G, whose vertices are these states and where two vertices are connected if you can get from one to the other by taking a dollar from one person and giving it to another. We are asking: when you run the random walk for a long time, where are you on this graph? Well, we know what the stationary distribution for random walk on an undirected graph is; it gives each vertex a probability proportional to its degree. On a regular graph, you get uniform distribution.

Our state graph G isn’t regular, but it almost is; most nodes have degree N, where by “most” I mean “about 1-1/e”; since the number of states is

and, of these, the ones with degree N are exactly those in which nobody’s out of money; if each person has a dollar, the number of ways to distribute the remaining N^2 – N dollars is

and so the proportion of states where someone’s out of money is about

.

So, apart from those states where somebody’s broke, in the long run *every possible state is equally likely;* we are just as likely to see $9,901 in one person’s hands and everybody else with $1 as we are to see exact equidistribution again.

What is a random lattice point in this simplex like? Good question! An argument just like the one above shows that the probability nobody goes below $c is on order e^-c, at least when c is small relative to N; in other words, it’s highly likely that somebody’s very nearly out of money.

If X is the maximal amount of money held by any player, what’s the distribution of X? I didn’t immediately see how to figure this out. You might consider the continuous version, where you pick a point at random from the *real* simplex

.

Equivalently; break a stick at N-1 randomly chosen points; what is the length of the longest piece? This is a well-studied problem; the mean size of the longest piece is about N log N. So I guess I think maybe that’s the expected value of the net worth of the richest player?

But it’s not obvious to me whether you can safely approximate the finite problem by its continuous limit (which corresponds to the case where we keep the number of players at N but reduce the step size so that each player can give each other a cent, or a picocent, or whatever.)

What happens if you give each of the N players just one dollar? Now the uniformity really breaks down, because it’s incredibly unlikely that nobody’s broke. The probability distribution on the set of (m_1, .. m_N) summing to N assigns each vector a probability proportional to the size of its support (i.e. the number of m_i that are nonzero.) That must be a well-known distribution, right? What does the corresponding distribution on partitions of N look like?

**Update**: Kenny Easwaran points out that this is basically the same computation physicists do when they compute the Boltzmann distribution, which was new to me.

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It’s not far from Madison. You drive southwest through the Driftless Zone, where the glaciers somehow looped around and missed a spot while they were grinding the rest of the Midwest flat.

At the exit to Platteville there was a sign for a “Mining Museum.” We had about six seconds to decide whether we all wanted to go to a mining museum but that was plenty of time because obviously we all totally wanted to go to a mining museum. And it was great! Almost the platonic ideal of a small-town museum. Our guide took us down into the old lead mine from the 1850s, now with electric lights and a lot of mannequins caught in the act of blasting holes in the rock. (One of the mannequins was black; our guide told us that there were African-American miners in southwestern Wisconsin, but not that some of them were enslaved.)

This museum did a great job of conveying the working conditions of those miners; ankle-deep in water, darkness broken only by the candle wired to the front of their hat, the hammers on the rock so loud you couldn’t talk, and had to communicate by hand signals. Riding up and down to the surface with one leg in the bucket and one leg out so more men could fit in one load, just hoping the bucket didn’t swing wrong and crush your leg against the rock wall. There’s nothing like an industrial museum to remind you that everything you buy in a store has hours of difficult, dangerous labor built into it. But it was also labor people traveled miles to get the chance to do!

Only twenty miles further to the Mississippi, my daughter’s first time seeing the river, and across it Dubuque. Which has a pretty great Op-Art flag:

Our main goal was the National Mississippi River Museum; slick where the Platteville museum was homespun, up-to-date where the Plateville Museum was old-fashioned. The kids really liked both. I wanted fewer interactive screens, more actual weird river creatures.

The museum is on the Riverwalk; Dubuque, like just about every city on a body of water, is reinventing its shoreline as a tourist hub. Every harbor a Harborplace. OK, I snark, but it was a lovely walk; lots of handsome bridges in view, all different, an old-timey band playing in the gazebo, Illinois and Wisconsin and Iowa invisibly meeting across the water….

Only disappointment of the afternoon; the famous funicular railway was closed. Maybe they could have posted that on their website or something. But in a way it’s good they didn’t; if I’d known it was closed, I probably would have decided to put off the trip, and who knows if we’d ever have gone?

On the way back we stopped in Dickeyville to get gas but missed the Dickeyville Grotto; would have stopped there for sure if I’d known about it. Dinner in Dodgeville at Culver’s, the Midwest’s superior version of In-N-Out, where I got my free Father’s Day turtle. I like cheese curds and brats as much as the next guy, but I gotta say, I think the turtle is my favorite of the many foods I’d never heard of before I moved to Wisconsin.

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**Update:** “Related posts” reminds me that the last time I went to a conference in Toronto, I learned a lot of interesting math from Julia Wolf, and the same was true this time!

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We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points. Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.

But wait, don’t we already know that, by Faltings? Of course we do. So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim. And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.

Chabauty’s theorem tells you that if X/K is a curve over a global field, and the rank of J(X)(K) is strictly smaller than g(X), then X(K) is finite. That theorem statement is superseded by Faltings, but the *method* is not; and it’s been the subject of active interest recently, in part because Chabauty gives you more control over the number of points. (See e.g. this much-lauded recent paper of Katz, Rabinoff, and Zureick-Brown.)

But of course the Chabauty condition on the Mordell-Weil rank of J(X) isn’t always satisfied. So what do you do with curves outside the Chabauty range? A natural and popular idea is to exploit an etale cover Y -> X. All the rational points of X are covered by those of a finite list of twists of Y; so if you can show all the relevant twists of Y satisfy Chabauty, you get finiteness of X(K). And that seems pretty reasonable; there’s no reason two distinct points of X should lie on the same twist Y_e of Y, and once a twist Y_e has only one rational point, the stuff in J(Y) that’s not in J(X) doesn’t have any reason to have large Mordell-Weil rank. Even if J(X) has pretty big rank, it should get “diluted” by all that empty Prym-like space.

About 15 years ago (amazing how projects that old still feel like “things I was recently thinking about”) I spent a lot of time working on this. What I wanted to show was something like this: if X is a curve (let’s say with a given rational basepoint), there’s a natural etale cover X_n obtained by mapping X to its Jacobian and then pulling back the multiplication map [p^n]:J(X) -> J(X). In fact, the X_n fit together into a tower of curves over X. Is it the case that

for n large enough? And what about the twists of X_n? The idea was to work at the “top” of the tower, where you have actions of lots of p-adic groups; the action of T_p J(X) by deck transformations, the action of some symplectic quotient of Gal(K) on T_p J(X), making the Selmer group at the top a great big non-abelian Iwasawa module. I still think that makes sense! But I ended up not being able to prove anything good about it. (This preprint was as far as I got.)

That project was about abelian covers (the Jacobians) of abelian covers of X; so it was really about the *metabelian quotient *of the fundamental group of X. What Minhyong Kim did was quite different, working with the much smaller *nilpotent* quotient; and it turns out that here you can indeed show that, for certain X, some version of Chabauty applies with Jac(X) replaced by a “unipotent Albanese,” which is to the quotient of pi_1(X) by some term of the lower central series as Jac(X) is to the Jacobian. Very very awesome.

What are “certain X”? Well, Kim’s method applies to all curves whose Jacobians have CM, as proved in this paper by him and Coates.

Most hyperelliptic curves don’t have CM Jacobians. But now the etale cover trick comes to the rescue, because, by an offbeat result of Bogomolov and Tschinkel I have always admired, every hyperelliptic curve X has an etale cover Y which geometrically dominates the CM genus 2 curve with model y^2 = x^6 – 1! So the main point of our paper is to generalize the argument of Coates and Kim to apply to curves whose Jacobian has a nontrivial CM part. (This is familiar from Chabauty; you don’t actually need the Jacobian to have small rank, it’s enough for just one chunk of it to have small rank relative to its dimension.) Having done this, we get finiteness for Y and all its twists, whence for X.

There are further results by Poonen of Bogomolov-Tschinkel flavor; these allow us to go from hyperelliptic curves to a more general class of curves, including all curves which are solvable covers of P^1. But of course here’s the natural question:

**Question:** Does every algebraic curve over Q admit an etale cover which geometrically dominates a curve Y whose Jacobian has CM?

An affirmative answer would extend the reach of non-abelian Chabauty to all curves over Q, which would be cool!

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Let X be a projective variety over a global field K, which is *Fano* — that is, its anticanonical bundle is ample. Then we expect, and in lots of cases know, that X has lots of rational points over K. We can put these points together into a *height zeta function*

where H(x) is the height of x with respect to the given projective embedding. The height zeta function organizes information about the distribution of the rational points of X, and which in favorable circumstances (e.g. if X is a homogeneous space) has the handsome analytic properties we have come to expect from something called a zeta function. (Nice survey by Chambert-Loir.)

What if X is a variety with two (or more) natural ample line bundles, e.g. a variety that sits inside P^m x P^n? Then there are two natural height functions H_1 and H_2 on X(K), and we can form a “multiple height zeta function”

There is a whole story of “multiple Dirichlet series” which studies functions like

where denotes the Legendre symbol. These often have interesting analytic properties that you wouldn’t see if you fixed one variable and let the other move; for instance, they sometimes have finite groups of functional equations that commingle the s and the t!

So I just wonder: are there situations where the multiple height zeta function is an “analytically interesting” multiple Dirichlet series?

Here’s a case to consider: what if X is the subvariety of P^2 x P^2 cut out by the equation

This has something to do with Eisenstein series on GL_3 but I am a bit confused about what exactly to say.

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