Roasted balsamic brussels sprouts

Note to self:  this recipe is good.



Arithmetic progression joke

Why was 7.333… disgusted by 7.666….?

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The Wisconsin Supreme Court gets home rule wrong and right

The Supreme Court made a decision in the Milwaukee police officer residency requirement case I wrote about, peevishly and at length, earlier this year.  Chief Justice Michael Gableman is still claiming the home rule amendment says something it doesn’t say; whether he’s confused or cynical I can’t say.

the home rule amendment gives cities and villages the ability “to determine their local affairs and government, subject only to this constitution and to such enactments of the legislature of statewide concern as with uniformity shall affect every city or every village.”  In other words, a city or village may, under its home rule authority, create a law that deals with its local affairs, but the Legislature has the power to statutorily override the city’s or village’s law if the state statute touches upon a matter of statewide concern or if the state statute uniformly affects every city or village. See Madison
Teachers, 358 Wis. 2d 1, ¶101.

“In other words,” phooey.  The amendment says a state enactment has to be of statewide concern and uniform in its effect.  Gableman turns the “and” into an “or,” giving the state much greater leeway to bend cities to its will.  The citation, by the way, is to his own opinion in the Act 10 case, where he’s wrong for the same reason.

But here’s the good news.  Rebecca Bradley, the newest justice, wrote a blistering concurrence (scroll to paragraph 52 of the opinion) which gets the amendment right.  She agrees with the majority that the state has constitutional authority to block Milwaukee’s residency requirement.  But the majority’s means of reaching that conclusion is wrong.  Bradley explains: by the home rule amendment’s plain text and by what its drafters said at the time of its composition, it is and, not or; for a state law to override a city law, it has to involve a matter of statewide concern and apply uniformly to all muncipalities.  Here’s Daniel Hoan, mayor of Milwaukee, and one of the main authors of the home rule amendment:

We submit that this wording is not ambiguous as other constitutional Home Rule amendments may be. It does not say——subject to state laws, subject to state laws of state-wide concern, or subject to laws uniformly affecting cities, but it does say——subject only to such state laws as are therein defined, and these laws must meet two tests: First——do they involve a subject of statewide concern, and second——do they with uniformity affect every city or village?

Bradley concedes that decades of Supreme Court precedent interpret the amendment wrongly.  So screw the precedent, she writes!  OK, she doesn’t actually write that.  But words to that effect.

I know I crap on Scalia-style originalism a lot, partly because I think it’s often a put-on.  But this is the real thing.




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I got a message for you

“I got a message for you, if I could only remember.  I got a message for you, but you’re gonna have to come and get it.”  Kardyhm Kelly gave me a tape of Zopilote Machine in 1995 and I played nothing but for a month.  “Sinaloan Milk Snake Song” especially.  Nobody but the Mountain Goats ever made do-it-yourself music like this, nobody else ever made it seem so believable that the things it occurred to you to say or sing while you were playing your guitar in your bedroom at home might actually be pop songs.   The breakdown at the end of this!

“I’ve got a heavy coat, it’s filled with rocks and sand, and if I lose it I’ll be coming back one day (I got a message for you).”  I spent a lot of 1993 thinking about the chord progression in the verse of this song.  How does it sound so straight-ahead but also so weird?  Also the “la la la”s (“Sinaloan Milk Snake Song” has these too.)

“Roll me in the greenery, point me at the scenery.  Exploit me in the deanery.  I got a message for you.”

The first of these I ever heard.  Douglas Wolk used to send mixtapes to Elizabeth Wilmer at Math Olympiad training.  This was on one of them.  1987 probably. I hadn’t even started listening to WHFS yet, I had no idea who Robyn Hitchcock was.  It was on those tapes I first heard the Ramones, Marshall Crenshaw, the Mentors (OK, we were in high school, cut us some slack.)

(Update:  Douglas denies ever putting the Mentors on a mixtape, and now that I really think about it, I believe Eric Wepsic was to blame for bringing the Mentors into my life.)

Why is this line so potent?  Why is the message never explicitly presented?  It’s enough — it’s better — that the message only be alluded to, never spoken, never delivered.

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“On l-torsion in class groups of number fields” (with L. Pierce, M.M. Wood)

New paper up with Lillian Pierce and Melanie Matchett Wood!

Here’s the deal.  We know a number field K of discriminant D_K has class group of size bounded above by roughly D_K^{1/2}.  On the other hand, if we fix a prime l, the l-torsion in the class group ought to be a lot smaller.  Conjectures of Cohen-Lenstra type predict that the average size of the l-torsion in the class group of D_K, as K ranges over a “reasonable family” of algebraic number fields, should be constant.  Very seldom do we actually know anything like this; we just have sporadic special cases, like the Davenport-Heilbronn theorem, which tells us that the 3-torsion in the class group of a random quadratic field is indeed constant on average.

But even though we don’t know what’s true on average, why shouldn’t we go ahead and speculate on what’s true universally?  It’s too much to ask that Cl(K)[l] literally be bounded as K varies (at least if you believe even the most modest version of Cohen-Lenstra, which predicts that any value of dim Cl(D_K)[l] appears for a positive proportion of quadratic fields K) but people do think it’s small:

Conjecture:  |Cl(K)[l]| < D_K^ε.

Even beating the trivial bound

|Cl(K)[l]| < |Cl(K)| < D_K^{1/2 + ε}

is not easy.  Lillian was the first to do it for 3-torsion in quadratic fields.  Later, Helfgott-Venkatesh and Venkatesh and I sharpened those bounds.  I hear from Frank Thorne that he, Bhargava, Shankar, Tsimerman and Zhao have a nontrivial bound on 2-torsion for the class group of number fields of any degree.

In the new paper with Pierce and Wood, we prove nontrivial bounds for the average size of the l-torsion in the class group of K, where l is any integer, and K is a random number field of degree at most 5.  These bounds match the conditional bounds Akshay and I get on GRH.  The point, briefly, is this.  To make our argument work, Akshay and I needed GRH in order to guarantee the existence of a lot of small rational primes which split in K.  (In a few cases, like 3-torsion of quadratic fields, we used a “Scholz reflection trick” to get around this necessity.)  At the time, there was no way to guarantee small split primes unconditionally, even on average.  But thanks to the developments of the last decade, we now know a lot more about how to count number fields of small degree, even if we want to do something delicate like keep track of local conditions.  So, for instance, not only can one count quartic fields of discriminant < X, we can count fields which have specified decomposition at any specified finite set of rational primes.  This turns out to be enough — as long as you are super-careful with error terms! — to  allow us to show, unconditionally, that most number fields of discriminant < D have enough small split primes to make the bound on l-torsion go.  Hopefully, the care we took here to get counts with explicit error terms for number fields subject to local conditions will be useful for other applications too.


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Back from family vacation in Greece.  Tiny notes/memories:

  • I have a heuristic that Americans fly the national flag much more than Europeans do, but in Greece, the Greek flag is all over the place.
  • Greeks really like, or Greeks think people in hotels and restaurants really like, soft-rock covers of hits from the 80s.  Maybe both!  We heard this mix CD everywhere.

    If you don’t feel like an hour and a half of this, at least treat yourself to James Farelli’s inexplicably fascinating acoustic take on “Owner of a Lonely Heart.”

  • The city of Akrotiri in the Aegean islands, a thousand years older than classical Greece, was buried under 200 feet of ash by the massive eruption of Santorini. They’ve only just started to dig it out. There are wall frescoes whose paint is still colorful and fresh. But these wall frescoes aren’t on the walls anymore; they fell during the earthquake preceding the eruption and lie in fragments on the floors. Our guide told us that they don’t try to reconstruct these using computers; archeologists put the pieces together by hand. I was perplexed by this: why don’t they digitize the images and try to find matches? It seemed to me like exactly the sort of thing we now know how to do. But no: it turns out this is a problem CS people are already thinking about, and it’s hard. Putting together pottery turns out to be a computationally much easier problem. Why? Because pots are surfaces of revolution and so their geometry is much more constrained!
  • The 2-star Michelin molecular gastronomy restaurant Funky Gourmet, run by a member of the El Bulli disapora, is just as great as advertised. But how can you run a molecular gastronomy restaurant in Athens and not call it Grecian Formula…?

Variations on three-term arithmetic progressions

Here are three functions.  Let N be an integer, and consider:

  •  G_1(N), the size of the largest subset S of 1..N containing no 3-term arithmetic progression;
  •  G_2(N), the largest M such that there exist subsets S,T of 1..N with |S| = |T| = M such that the equation s_i + t_i = s_j + t_k has no solutions with (j,k) not equal to (i,i).  (This is what’s called  a tri-colored sum-free set.)
  • G_3(N), the largest M such that the following is true: given subsets S,T of 1..N, there always exist subsets S’ of S and T’ of T with |S’| + |T’| = M and S'+T \cup S+T' = S+T.

You can see that G_1(N) <= G_2(N) <= G_3(N).  Why?  Because if S has no 3-term arithmetic progression, we can take S = T and s_i = t_i, and get a tri-colored sum-free set.  Now suppose you have a tri-colored sum-free set (S,T) of size M; if S’ and T’ are subsets of S and T respectively, and S'+T \cup S+T' = S+T, then for every pair (s_i,t_i), you must have either s_i in S’ or t_i in T’; thus |S’| + |T’| is at least M.

When the interval 1..N is replaced by the group F_q^n, the Croot-Lev-Pach-Ellenberg-Gijswijt argument shows that G_1(F_q^n) is bounded above by the number of monomials of degree at most (q-1)n/3; call this quantity M(F_q^n).  In fact, G_3(F_q^n) is bounded above by M(F_q^n), too (see the note linked from this post) and the argument is only a  modest extension of the proof for G_1.  For all we know, G_1(F_q^n) might be much smaller, but Kleinberg has recently shown that G_2(F_2^n) (whence also G_3(F_2^n)) is equal to M(F_2^n) up to subexponential factors, and work in progress by Kleinberg and Speyer has shown this for several more q and seems likely to show that the bound is tight in general.  On the other hand, I have no idea whether to think G_1(F_q^n) is actually equal to M(F_q^n); i.e. is the bound proven by me and Dion sharp?

The behavior of G_1(N) is, of course, very much studied; we know by Behrend (recently sharpened by Elkin) that G_1(N) is at least N/exp(c sqrt(log N)).  Roth proved that G_1(N) = o(N), and the best bounds, due to Tom Sanders, show that G_1(N) is O(N(log log N)^5 / log N).  (Update:  Oops, no!  Thomas Bloom has an upper bound even a little better than Sanders, change that 5 to a 4.)

What about G_2(N) and G_3(N)?  I’m not sure how much people have thought about these problems.  But if, for instance, you could show (for example, by explicit constructions) that G_3(N) was closer to Sanders than to Behrend/Elkin, it would close off certain strategies for pushing the bound on G_1(N) downward. (Update:  Jacob Fox tells me that you can get an upper bound for G_2(N) of order N/2^{clog* N} from his graph removal paper, applied to the multicolored case.)

Do we think that G_2(N) and G_3(N) are basically equal, as is now known to be the case for F_q^n?

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Utah v. Strieff

The Supreme Court held today in Utah v. Strieff that if you stop someone illegally, then run a search on their drivers license and find they have unpaid traffic tickets, giving cause for arrest, and you then arrest them, search them, and find drugs, the drugs are admissible evidence despite arising by means of an illegal stop.  I read through the decision, following the cites and deciding whether I believed the argument.  I don’t.  But I should have saved my time and read Sotomayor’s dissent, which makes the case very clearly and in my view persuasively.

What everybody agrees on:

  • Evidence need not be excluded just because it would not have been obtained but for an illegal stop.  If the officer had stopped Strieff without reasonable cause, and in the course of their conversation, someone wandered by, pointed at Strieff, and said “that’s the guy who robbed me yesterday!” it would be OK to use the accusation as evidence even though it wouldn’t have happened had Strieff not been detained.  “But for” is necessary for exclusion, but not sufficient.
  • The criterion is, rather, supposed to be “whether, granting establishment of the primary illegality, the evidence to which instant objection is made has been come at by exploitation of that illegality or instead by means sufficiently distinguishable to be purged of the primary taint.”

The majority’s theory is that the information obtained by the offer about the arrest warrant was a “means sufficiently distinguishable.”  Sotomayor disagrees, and so do I.  Running Strieff’s name through the database wasn’t a separate interaction that just happened, by chance, to take place in the spacio-temporal neighborhood of the illegal stop:  it was an attempt to execute the purpose of the illegal stop, and has to be seen as a continuation of that stop.

What’s especially annoying is the majority’s use of cites that don’t support its case.  They say the facts in Segura v. United States are “similar” to those in Strieff.  They are not.  In fact (as the majority concedes) the decision to admit the evidence in Segura was reached under a totally different theory, because in that case, unlike this one, the evidence used at trial would have been obtained whether or not the illegal search had taken place; i.e. even the weaker “but for” standard wasn’t met.  Then they say the request for the warrant in the course of the illegal stop was a “negligibly burdensome precaution for officer safety,” citing Rodriguez v. United States.  In that case, it was remarked that it was legitimate, for the cause of traffic safety, to check for outstanding traffic warrants against a driver stopped for a traffic violation.  So far so good.  But the decision in that case goes on to say that making the driver submit to a dog sniff of his car is not permissible.  “Lacking the same close connection to roadway safety as the ordinary inquiries, a dog sniff is not fairly characterized as part of the officer’s traffic mission.”

The majority’s theory is that the officer checked Strieff for outstanding warrants because public safety required it.  Sotomayor’s theory is that the officer checked Strieff for outstanding warrants because he had no cause to search Strieff, and wanted some.  Which do you find more plausible?

What’s interesting is that the case that best supports the majority’s theory is one they don’t even directly cite: Johnson v. Louisiana.  In that case, Johnson was arrested without a warrant for a robbery, brought to the courthouse, and put in a lineup, where he was identified by a witness as perpetrator of a different robbery.  The court held that Johnson’s ID in the lineup was admissible even though it resulted from an illegal arrest, because the lineup was ordered separately by the judge after Johnson had been brought in:  this “intervening action” was held to be sufficient separation between the illegal arrest and the evidence obtained.  What I can’t tell from the decision is:  was it just by chance that the victim of the other robbery happened to be present at the lineup for the original robbery?  Or was it common practice to arrest people on a hunch and then put them in a bunch of different lineups to see if anyone IDed them as the perpetrator of a crime?  If it’s the former, I can sort of understand the Court’s reasoning.  If the latter, no way.

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Wall words

When I was a kid I had an anthology of Cyril Kornbluth stories and one detail in the introduction made a big impression on me:  Kornbluth had lots of random notes among his papers, ideas, words, and phrases, and the editor of the anthology found it kind of poignant to look at these and wonder what stories they would have been.  Especially — and this phrase has stuck with me all these years — “ghosts in a Martian department store.”

Anyway:  while cleaning up the basement, I came across a pile of Post-its, which were on my wall when I was a creative writing grad student at Johns Hopkins in 1994.  Each of these, I guess, was something I meant to use for something.  I have no memory of what any of these mean.  But here they are.  Maybe you can use them.  None of them is as good as “ghosts in a Martian department store.”

  • “every third person in the world”
  • The Admonitionist
  • assapanic
  • “Bald men, a lot of them — yeah, that might work!”
  • Rick Ziggurat
  • The First Church of Christ Plaintiff
  • Community Reaction to a Horrifying Event – Arndt
  • out of the fishy-smelling steam
  • reckless use of a highway
  • THE UNDERTAKER’S BIRTHDAY (Did the stripper shoot the corpse?)
  • The grandfatherly crook gives a motivational lecture to fifth-graders
  • “you slug in a ditch!”
  • Solomon “Duck-Duck” Goos
  • A first line:  “Here’s something you might not know.”
  • “luz” — “an unidentified bone…”
  • The hobo community, waiting /expecting (esperar) for rich couples to pick them up.
  • The apocalypse counselor “If the sun went supernova, we wouldn’t know for eight minutes.”
  • “Of course, there is some element of risk involved in building a house out of oily rags.”
  • steeped in regret
  • Chief Louis Anemone NYPD (red)
  • “This guy’s been shot.”  “No kidding — you think he’ll make it?”
  • HIDALGO – hijo de algo
  • fetus/treatise
  • fuckadelphia
  • abacinate
  • “Victory without a whimper…the only sound:  excellence.”
  • “The ghost of Knute Rockne is living and he is smiling.”
  • “That’s a big number, but it’s a heck of a lot better than infinity.”
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Creeping and boiling

Then the cannon-ball smashed through the window-sill, the opened glass panes shattering into fragments with a crash.  The ball itself rolled on until the stone wall stopped it with a heavy thud, then it burst into pieces, and a creeping gray smoke came boiling out.

I have a lot of issues with this passage.

First of all, it seems like the cannonball smashed through the window, not the windowsill.  As for the panes — if the cannonball smashed through them, doesn’t that mean they were closed, not open?  How would they shatter if not into fragments?  I object, too, to the “with a [sound effect]” construction being used in two consecutive sentences, especially given that the chosen sound effect (“crash,” “thud”) is the most obvious choice in both cases.  The second sentence has too many different actions carried out by too many different objects (the ball, the wall, then the ball again, then the smoke.)  The smoke — is it creeping or boiling?  By my lights it can’t be both.

The lines are from Naomi Novik’s Uprooted, a fantasy novel which despite this paragraph has a lot of good things about it. and which won this year’s Nebula Award.

Two idiosyncratic reactions:

  • Agnieszka’s magic is set up as being the inheritance of Baba Jaga, a kind of intuitive, sing-songy, kitcheny kind of magic, explicitly opposed to the formal, rule-governed spell-casting of Sarkan, the broody sorcery dude who kidnaps, then mentors, then eventually falls in love with her.  This works well in the story, but I’m not on board with the suggestion that formal, rule-governed manipulation is a masculine activity that needs a feminine complement in order to achieve its full power.  Math has an improvisational, intuitive aspect, to be sure; but that aspect, like the formal aspect, belongs to men and women equally.
  • Weird feature of this book:  its setting is a magical version of Poland, and Agnieszka is explicitly presented as being “rooted” in the village, the hearth, the homeland; this is, in part the source of her power.  Sarkan, by contrast, is explicitly “rootless” — without a connection of his own to the land, he has to feed on the young women of the village, one after another, cutting their connection to the village and leaving them sort of ruined, suited only for big-city life.  So my mind naturally wanders to the question of “what group of people were thought of in rural Eastern Europe as rootless cosmopolitans who hide out behind walls looking at books all day and who corrupt our women and we just have to accept it because they have access to mysterious secret powers?”  Now maybe I’m overthinking this, but I do have to point out that after I noticed this I looked up Novik on Wikipedia, and her mother is Polish and her father is Jewish.  Make of it what you will.



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