## Trace test

Jose Rodriguez gave a great seminar here yesterday about his work on the trace test, a numerical way of identifying irreducible components of varieties.  In Jose’s world, you do a lot of work with homotopy; if a variety X intersects a linear subspace V in points p1, p2, .. pk, you can move V a little bit and numerically follow those k points around.  If you move V more than a little bit — say in a nice long path in the Grassmannian that loops around and returns to its starting point — you’ll come back to p1, p2, .. pk, but maybe in a different order.  In this way you can compute the monodromy of those points; if it’s transitive, and if you’re careful about avoiding some kind of discriminant locus, you’ve proven that p1,p2…pk are all on the same component of V.

But the trace test is another thing; it’s about short paths, not long paths.  For somebody like me, who never thinks about numerical methods, this means “oh we should work in the local ring.”  And then it says something interesting!  It comes down to this.  Suppose F(x,y) is a form (not necessarily homogenous) of degree at most d over a field k.  Hitting it with a linear transformation if need be, we can assume the x^d term is nonzero.  Now think of F as an element of k((y))[x]:  namely

$F = x^d + a_1(y) x^{d-1} + \ldots + a_d(y).$

Letting K be the algebraic closure of k((y)), we can then factor F as (x-r_1) … (x-r_d).  Each of these roots can be computed as explicitly to any desired precision by Hensel’s lemma.  While the r_i may be power series in k((y)) (or in some algebraic extension), the sum of the r_i is -a_1(y), which is a linear function A+by.

Suppose you are wondering whether F factors in k[x,y], and whether, for instance, r_1 and r_2 are the roots of an irreducible factor of F.  For that to be true, r_1 + r_2 must be a linear function of y!  (In Jose’s world, you grab a set of points, you homotopy them around, and observe that if they lie on an irreducible component, their centroid moves linearly as you translate the plane V.)

Anyway, you can falsify this easily; it’s enough for e.g. the quadratic term of r_1 + r_2 to be nonzero.  If you want to prove F is irreducible, you just check that every proper subset of the r_i sums to something nonlinear.

1.  Is this something I already know in another guise?
2.  Is there a nice way to express the condition (which implies irreducibility) that no proper subset of the r_i sums to something with zero quadratic term?

## Game report: Cubs 5, Brewers 0

• I guess the most dominant pitching performance I’ve seen in person?  Quintana never seemed dominant.  The Brewers hit a lot of balls hard.  But a 3-hit complete game shutout is a 3-hit complete game shutout.
• A lot of Cubs fans. A lot a lot.  My kids both agreed there were more Cubs than Brewers fans there, in a game that probably mattered more to Milwaukee.
• For Cubs fans to boo Ryan Braun in Wrigley Field is OK, I guess.  To come to Miller Park and boo Ryan Braun is classless.  Some of those people were wearing Sammy Sosa jerseys!
• This is the first time I’ve sat high up in the outfield.  And the view was great, as it’s been from every other seat I’ve ever occupied there.  A really nice design.  If only the food were better.
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My kids both wanted to see the eclipse and I said “that sounds fun but it’s too far” and I kept thinking about it and thinking about it and finally, Saturday night, I looked inward and asked myself is there really a reason we can’t do this? And the answer was no.  Or rather the answer was “it might be the case that it’s totally impossible to find a place to sleep in the totality zone within 24 hours for a non-insane amount of money, and that would be a reason” so I said, if I can get a room, we’re going.  Hotel Tonight did the rest.  (Not the first time this last-minute hotel app has saved my bacon, by the way.  I don’t use it a lot, but when I need it, it gets the job done.)

Notes on the trip:

• We got to St. Louis Sunday night; the only sight still open was my favorite one, the Gateway Arch.  The arch is one of those things whose size and physical strangeness a photo really doesn’t capture, like Mt. Rushmore.  It works for me in the same way a Richard Serra sculpture works; it cuts the sky up in a way that doesn’t quite make sense.
• I thought I was doing this to be a good dad, but in fact the total eclipse was more spectacular than I’d imagined, worth it in its own right.  From the photos I imagined the whole sky going nighttime dark.  But no, it’s more like twilight. That makes it better.  A dark blue sky with a flaming hole in it.
• Underrated aspect:  the communality of it all.  An experience now rare in everyday life.  You’re in a field with thousands of other people there for the same reason as you, watching the same thing you’re watching.  Like a baseball game!  No radio call can compare with the feeling of jumping up with the crowd for a home run.  You’re just one in an array of sensors, all focused on a sphere briefly suspended in the sky.
• People thought it was going to be cloudy.  I never read so many weather blogs as I did Monday morning.  Our Hotel Tonight room was in O’Fallon, MO, right at the edge of the totality.  Our original plan was to meet Patrick LaVictoire in Hermann, west of where we were.  But the weather blogs said south, go south, as far as you can.  That was a problem, because at the end of the day we had to drive back north.  We got as far as Festus.  There were still three hours to totality and we thought it might be smart to drive further, maybe even all the way to southern Illinois.  But a guy outside the Comfort Inn with a telescope, who seemed to know what he was doing, told us not to bother, it was a crapshoot either way and we weren’t any better off there than here.  I always trust a man with a telescope.
• It became clear around Springfield we weren’t going to get home until well after midnight, so we stopped for the night in David Foster Wallace’s hometown, Normal, IL, fitting, considering we did a supposedly fun thing that turned out to be an actual fun thing which we will hardly ever have the chance to, and thus may never, do again.

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## Not to exceed 25%

Supreme Court will hear a math case!

At issue in Murphy v. Smith:  the amount of a judgment that a court can apply to covering attorney’s fees.  Here’s the relevant statute:

Whenever a monetary judgment is awarded in an action described in paragraph (1), a portion of the judgment (not to exceed 25 percent) shall be applied to satisfy the amount of attorney’s fees awarded against the defendant.

To be clear: there are two amounts of money here.  The first is the amount of attorney’s fees awarded against the defendant; the second is the portion of the judgment which the court applies towards that first amount.  This case concerns the discretion of the court to decide on the second number.

In Murphy’s case, the court decided to apply just 10% of the judgment to attorney’s fees.  Other circuit courts have licensed this practice, interpreting the law to allow the court discretion to apply any portion between 0 and 25% of the judgement to attorney’s fees.  The 7th circuit disagreed, saying that, given that the amount of attorney’s fees awarded exceeded 25% of the judgment, the court was obligated to apply the full 25% maximum.

The cert petition to the Supreme Court hammers this view, which it calls “non-literal”:

The Seventh Circuit is simply wrong in interpreting this language to mean “exactly 25 percent.” “Statutory interpretation, as we always say, begins with the text.” Ross v. Blake, 136 S. Ct. 1850, 1856 (2016). Here, the text is so clear that interpretation should end with the text as well. “Not to exceed” does not mean “exactly.”

This seems pretty clearly correct:  “not to exceed 25%” means what it means, not “exactly 25%.”  So the 7th circuit just blew it, right?

Nope!  The 7th circuit is right, the other circuits and the cert are wrong, and the Supreme Court should affirm.  At least that’s what I say.  Here’s why.

I can imagine at least three interpretations of the statuye.

1.  The court has to apply exactly 25% of the judgment to attorney’s fees.
2.  The court has to apply the smaller of the following numbers:  the total amount awarded in attorney’s fees, or 25% of the judgment.
3.  The court has full discretion to apply any nonnegative amount of the judgment to attorney’s fees.

Cert holds that 3 is correct, that the 7th circuit applied 1, and that 1 is absurdly wrong.  In fact, the 7th circuit applied 2, which is correct, and 1 and 3 are both wrong.

1 is wrong:  1 is wrong for two reasons.  One is pointed out by the cert petition:  “Not to exceed 25%” doesn’t mean “Exactly 25%.”  Another reason is that “Exactly 25%” might be more than the amount awarded in attorney’s fees, in which case it would be ridiculous to apply more money than was actually owed.

7th circuit applied 2, not 1:  The opinion reads:

In Johnson v. Daley, 339 F.3d 582, 585 (7th Cir. 2003) (en banc), we explained that § 1997e(d)(2) required that “attorneys’ compensation come[] first from the damages.” “[O]nly  if 25% of the award is inadequate to compensate counsel fully” does the defendant contribute more to the fees. Id. We continue to believe that is the most natural reading of the statutory text. We do not think the statute contemplated a discretionary decision by the district court. The statute neither uses discretionary language nor provides any guidance for such discretion.

The attorney’s compensation comes first out of the damages, but if that compensation is less than 25% of the damages, then less than 25% of the damages will be applied.  This is interpretation 2.  In the case at hand, 25% of the damages was $76,933.46 , while the attorney’s fees awarded were$108,446.54.   So, in this case, the results of applying 1 and 2 are the same; but the court’s interpretation is clearly 2, not the absurd 1.

3 is wrong:  Interpretation 3 is on first glance appealing.  Why shouldn’t “a portion of the judgment (not to exceed 25%)” mean any portion satisfying that inequality?  The reason comes later in the statute; that portion is required to “satisfy the amount of attorney’s fees awarded against the defendant.”  To “satisfy” a claim is to pay it in full, not in part.  Circuits that have adopted interpretation 3, as the 8th did in Boesing v. Spiess, are adopting a reading at least as non-literal as the one cert accuses the 7th of.

Of course, in cases like Murphy v. Smith, the two clauses are in conflict:  25% of the judgment is insufficient to satisfy the amount awarded.  In this case, one requirement must bend.  Under interpretation 2, when the two clauses are in conflict, “satisfy” is the one to give way.  The 7th circuit recognizes this, correctly describing the 25% awarded as ” toward satisfying the attorney fee the court awarded,” not “satisfying” it.

Under interpretation 3, on the other hand, the requirement to “satisfy” has no force even when it is not in conflict with the first clause.  In other words, they interpret the law as if the word “satisfy” were absent, and the clause read “shall be applied to the amount of attorney’s fees.”

Suppose the attorney’s fees awarded in Murphy had been $60,000. Under interpretation 3, the court would be free to ignore the requirement to satisfy entirely, and apply only 10% of the judgment to the attorneys, despite the fact that satisfaction was achievable within the statutory 25% limit. Even worse: imagine that the statute didn’t have the parenthetical, and said just Whenever a monetary judgment is awarded in an action described in paragraph (1), a portion of the judgment shall be applied to satisfy the amount of attorney’s fees awarded against the defendant. It would be crystal clear that the court was required to apply$60,000, the amount necessary to satisfy the award.  On interpretation 3, the further constraint imposed by the statute gives the court more discretion rather than less in a case like this one!  This can’t be right.

You could imagine switching to an interpretation 3′, in which the court is required to satisfy the amount awarded if it can do so without breaking the 25% limit, but is otherwise totally unconstrained.  Under this theory, an increase in award from $60,000 to$100,000 lessens the amount the court is required to contribute — indeed, lessens it to essentially zero.  This also can’t be right.

2 is right:  When two clauses of a statute can’t simultaneously be satisfied, the court’s job is to find some balance which satisfies each requirement to the greatest extent possible in a range of possible cases.  Interpretation 2 seems the most reasonable choice.  The Supreme Court should recognize that, contra the cert petition, this is the interpretation actually adopted by the 7th Circuit, and should go along with it.

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## Maryland flag, my Maryland flag

The Maryland flag is, in my opinion as a Marylander, the greatest state flag.

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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## Good math days

I have good math days and bad math days; we all do.  An outsider might think the good math days are the days when you have good ideas.  That’s not how it works, at least for me.  You have good ideas on the bad math days, too; but one at a time.  You have an idea, you try it, you make some progress, it doesn’t work, your mind says “too bad.”

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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## Show report: Camp Friends and Omni at the Terrace

Beautiful weather last night so I decided, why not, go to the Terrace for the free show WUD put on:  Camp Friends (Madison) and Omni (Atlanta).

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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## How many books do I read in a year?

Data:

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.)

## Lo!

“A naked man in a city street — the track of a horse in volcanic mud — the mystery of reindeer’s ears — a huge, black form, like a whale, in the sky, and it drips red drops as if attacked by celestial swordfishes — an appalling cherub appears in the sea —

Confusions.”

## When random people give money to random other people

A post on Decision Science about a problem of Uri Wilensky‘s has been making the rounds:

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

$N^2 + N - 1 \choose N-1$

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

$N^2 - 1 \choose N-1$

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

$\frac{(N^2 - 1)^N}{(N^2 + N - 1)^N} \sim (1-1/N)^N \sim 1/e$.

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

$(x_1, .. x_N) \in \mathbf{R}^N: \sum x_i = N^2$.

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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