## Braid monodromy and the dual curve

Nick Salter gave a great seminar here about this paper; hmm, maybe I should blog about that paper, which is really interesting, but I wanted to make a smaller point here.  Let C be a smooth curve in P^2 of degree n. The lines in P^2 are parametrized by the dual P^2; let U be the open subscheme of the dual P^2 parametrizing those lines which are not tangent to C; in other words, U is the complement of the dual curve C*.  For each point u of U, write L_u for the corresponding line in P^2.

This gives you a fibration X -> U where the fiber over a point u in U is L_u – (L_u intersect C).  Since L_u isn’t tangent to C, this fiber is a line with n distinct points removed.  So the fibration gives you an (outer) action of pi_1(U) on the fundamental group of the fiber preserving the puncture classes; in other words, we have a homomorphism

$\pi_1(U) \rightarrow B_n$

where B_n is the n-strand braid group.

When you restrict to a line L* in U (i.e. a pencil of lines through a point in the original P^2) you get a map from a free group to B_n; this is the braid monodromy of the curve C, as defined by Moishezon.  But somehow it feels more canonical to consider the whole representation of pi_1(U).  Here’s one place I see it:  Proposition 2.4 of this survey by Libgober shows that if C is a rational nodal curve, then pi_1(U) maps isomorphically to B_n.  (OK, C isn’t smooth, so I’d have to be slightly more careful about what I mean by U.)

## Mark Metcalf

Have you ever heard of this guy?  I hadn’t.  Or thought I hadn’t.  But: he was Niedermeyer in Animal House

and the dad in Twisted Sister’s “We’re Not Gonna Take It” video

and the Master, the Big Bad of Buffy the Vampire Slayer season 1.

That’s a hell of a career! Plus: he lived in suburban Milwaukee until three years ago! And he used to go out with Glenn Close and Carrie Fisher! OK. Now I’ve heard of Mark Metcalf and so have you.

## Curry apple chicken

I didn’t have time to do a real shop and had nothing for Friday night dinner so I bought some chicken and a bag of apples and made something that came out surprisingly well; I hereby record it.

Ingredients:

2-3 lb boneless chicken breasts

5 apples, cubed

some scallions

some vegetable oil, whatever kind, doesn’t matter, I used olive

1 tbsp ground coriander

1 tbsp ground cumin

1/2 tsp turmeric

1 tsp salt

however much minced garlic you’re into

1/2-1 tsp garam masala

some crushed tomatoes but you could use actual tomatoes if it weren’t the middle of winter

Recipe:

Get oil hot.  Throw apples and scallions in.  Stir and cook 5 mins until apples soft.  Clear off some pan space and put coriander, cumin, turmeric, salt in the oil, let it cook 30 sec – 1 min, then throw in all the chicken, which by the way you cut into chunks, saute it all up until it’s cooked through.  Put the minced garlic in and let that cook for a minute.  Then put in however much tomato you need to combine with everything else in the pan and make a sauce.  (Probably less than you think, you don’t want soup.)  Turn heat down to warm and mix in garam masala.  You could just eat it like this or you could have been making some kind of starch in parallel.  I made quinoa.  CJ liked this, AB did not.

I took the spice proportions from a Madhur Jaffrey recipe but this is in no way meant as actual Indian food, obviously.  I guess I was just thinking about how when I was a kid you would totally get a “curry chicken salad” which was shredded chicken with curry powder, mayonnaise, and chunked up apple, and I sort of wanted a hot mayonnaiseless version of that.  Also, when I was in grad school learning to cook from Usenet with David Carlton, we used to make a salad with broiled chicken and curry mayonnaise and grapes.  I think it was this.  Does that sound right, David?   Yes, that recipe calls for 2 cups of mayonnaise.  It was a different time.  I feel like we would make this and then put it on top of like 2 pounds of rotini and have food for days.

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## Difference sets missing a Hamming sphere

I tend to think of the Croot-Lev-Pach method (as used, for instance, in the cap set problem) as having to do with n-tensors, where n is bigger than 2.  But you can actually also use it in the case of 2-tensors, i.e. matrices, to say things that (as far as I can see) are not totally trivial.

Write m_d for the number of squarefree monomials in x_1, .. x_n of degree at most d; that is,

$m_d = 1 + {n \choose 1} + {n \choose 2} + \ldots + {n \choose d}.$

Claim:  Let P be a polynomial of degree d in F_2[x_1, .. x_n] such that P(0) = 1.  Write S for the set of nonzero vectors x such that P(x) = 1.  Let A be a subset of F_2^n such that no two elements of A have difference lying in S.  Then |A| < 2m_{d/2}.

Proof:  Write M for the A x A matrix whose (a,b) entry is P(a-b).  By the Croot-Lev-Pach lemma, this matrix has rank at most 2m_{d/2}.  By hypothesis on A, M is the identity matrix, so its rank is |A|.

Remark: I could have said “sum” instead of “difference” since we’re in F_2 but for larger finite fields you really want difference.

The most standard context in which you look for large subsets of F_2^n with restricted difference sets is that of error correcting codes, where you ask that no two distinct elements of A have difference with Hamming weight (that is, number of 1 entries) at most k.

It would be cool if the Croot-Lev-Pach lemma gave great new bounds on error-correcting codes, but I don’t think it’s to be.  You would need to find a polynomial P which vanishes on all nonzero vectors of weight larger than k, but which doesn’t vanish at 0. Moreover, you already know that the balls of size k/2 around the points of A are disjoint, which gives you the “volume bound”

|A| < 2^n / m_{k/2}.

I think that’ll be hard to beat.

If you just take a random polynomial P, the support of P will take up about half of F_2^n; so it’s not very surprising that a set whose difference misses that support has to be small!

Here’s something fun you can do, though.  Let s_i be the i-th symmetric function on x_1, … x_n.  Then

$s_i(x) = {wt(x) \choose i}$

where wt(x) denotes Hamming weight.  Recall also that the binomial coefficient

${k \choose 2^a}$

is odd precisely when the a’th binary digit of k is 1.

Thus,

$(1-s_1(x))(1-s_2(x))(1-s_4(x))\ldots(1-s_{2^{b-1}}(x))$

is a polynomial of degree 2^b-1 which vanishes on x unless the last b digits of wt(x) are 0; that is, it vanishes unless wt(x) is a multiple of 2^b.  Thus we get:

Fact:  Let A be a subset of F_2^n such that the difference of two nonzero elements in A never has weight a multiple of 2^b.  Then

$|A| \leq 2m_{2^{b-1} - 1}$.

Note that this is pretty close to sharp!  Because if we take A to be the set of vectors of  weight at most 2^{b-1} – 1, then A clearly has the desired property, and already that’s half as big as the upper bound above.  (What’s more, you can throw in all the vectors of weight 2^{b-1} whose first coordinate is 1; no two of these sum to something of weight 2^b.  The Erdös-Ko-Rado theorem says you can do no better with those weight 2^{b-1} vectors.)

Is there an easier way to prove this?

When b=1, this just says that a set with no differences of even Hamming weight has size at most 2; that’s clear, because two vectors whose Hamming weight has the same parity differ by a vector of even weight.  Even for b=2 this isn’t totally obvious to me.  The result says that a subset of F_2^n with no differences of weight divisible by 4 has size at most 2+2n.  On the other hand, you can get 1+2n by taking 0, all weight-1 vectors, and all weight-2 vectors with first coordinate 1.  So what’s the real answer, is it 1+2n or 2+2n?

Write H(n,k) for the size of the largest subset of F_2^n having no two vectors differing by a vector of Hamming weight exactly k.  Then if 2^b is the largest power of 2 less than n, we have shown above that

$m_{2^{b-1} - 1 } \leq H(n,2^b) \leq 2m_{2^{b-1} - 1}$.

On the other hand, if k is odd, then H(n,k) = 2^{n-1}; we can just take A to be the set of all even-weight vectors!  So perhaps H(n,k) actually depends on k in some modestly interesting 2-adic way.

The sharpness argument above can be used to show that H(4m,2m) is as least

$2(1 + 4m + {4m \choose 2} + \ldots + {4m \choose m-1} + {4m-1 \choose m-1}). (*)$

I was talking to Nigel Boston about this — he did some computations which make it looks like H(4m,2m) is exactly equal to (*) for m=1,2,3.  Could that be true for general m?

(You could also ask about sets with no difference of weight a multiple of k; not sure which is the more interesting question…)

Update:  Gil Kalai points out to me that much of this is very close to and indeed in some parts a special case of the Frankl-Wilson theorem…  I will investigate further and report back!

## Vacuum cleaner on sale

I was explaining the “regular price” scam to CJ the other day. A store sells a vacuum cleaner for $79.95. One day, they put up a sign saying “SALE! Regular price,$109.95; now MARKED DOWN to $79.95.” The point is to create an imaginary past that never was, a past where vacuum cleaners cost$109.95, a difficult past from which the store has generously granted you respite.
This is what Trump’s team is doing. They’re trying to create an imaginary past in which the last 5 years of life in America was characterized by ubiquitous street crime, unchecked terrorism, and mass unemployment. So that life in America in 2017 and 2018 will seem comparatively placid, safe, and prosperous. Look how much I saved you on this goddamn vacuum cleaner. You’re welcome.

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## Holocaust Remembrance Day

I was always skeptical of the U.S. Holocaust Memorial Museum.  I understand why there’s a Holocaust museum in Israel, and I understand why there’s one in Berlin, and I understand why there are memorials in the places where the slaughter was carried out.  But why in Washington D.C.?

Then I went there, I got it.  The reason there’s a Holocaust Museum in America is because it’s a Holocaust Museum that’s about America.  The museum tells the story, as all Holocaust museums must, of the German slaughter of the Jews, and of gay people, disabled people, Roma and Slavs besides.  But the focus is on what happened here in America, and what didn’t happen.  The point of the museum is to ask a simple question:  what is the responsibility of a rich, safe, comfortable country to the human beings being hunted down and killed outside its borders?

OK, I’ll admit it, that’s not a simple question.  But it’s a question worth thinking about today.

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Couldn’t find my phone yesterday morning.  I definitely remembered having it in the car on the way home from the kids’ swim lesson, so I knew I hadn’t left it.  “Find my iPhone” told me the phone was on the blacktop of the elementary school, about 1000 feet from my house.  What?  Why?  Then a few minutes later the location updated to the driveway of a bank, closer to my house but in the other direction.  So I went over to the bank and looked around in the driveway, even peering into the garbage shed and seeing if my phone was in their dumpster.

But why did I do that?  It was terrible reason.  There was no chain of events leaving my phone at the bank, or at the school, which wasn’t incredibly a prior unlikely.  I should have reasoned:  “The insistence of Find my iPhone that my phone is at the bank drastically increases the probability my phone is at the bank, but that probability started out so tiny that it remains tiny, and the highest-expected-utility use of my time is to keep looking around my house and my car until I find it.”

Anyway, it was in the basement.

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## In which South Dakota can’t actually do that

Voters in South Dakota approved a sweeping new government ethics law by referendum in November.  The South Dakota state legislature just overruled them.

Can they do that?

Yes and no.  A law passed by referendum is, in the end, a law; and laws can be repealed by the legislature.  A lot of states have protections against this practice, which is called “legislative tampering.”  South Dakota is one of 12 states that doesn’t.  So if you think the people can band together and pass a law by ballot measure there, you’re only sort of right; if the people’s will goes against the will of the legislative majority, as in this case, right out the window goes the popular vote.

But the legislature did more.  The bill, HB 1069, finishes off with the following language:

Section 35. Whereas, this Act is necessary for the support of the state government and its existing public institutions, an emergency is hereby declared to exist, and this Act shall be in
full force and effect from and after its passage and approval.

The declaration of an “emergency” does two things:  it makes the bill active immediately upon passage, thus preventing the ethics commission from coming into existence even temporarily, and it prevents the people of South Dakota from launching a new referendum to veto the repeal and restore the ethics law.

I don’t think they can do this.

Under the South Dakota Constitution, some kinds of laws are subject to veto by popular referendum and some are not.  Those laws protected from referendum are categorized as follows:

However, the people expressly reserve to themselves the right to propose measures, which shall be submitted to a vote of the electors of the state, and also the right to require that any laws which the Legislature may have enacted shall be submitted to a vote of the electors of the state before going into effect, except such laws as may be necessary for the immediate preservation of the public peace, health or safety, support of the state government and its existing public institutions.

The legislature doesn’t have the power to protect a law from referendum just by declaring an emergency.  It has to fall under one of the two protected categories delineated in Section 1 above.  This was laid out in Lindstrom v. Goetz (1951):

Only those laws which are not subject to the referendum, according to § 1 are subject to the emergency clause authorized by § 22. State ex rel. Richards v. Whisman, 36 S.D. 260, 154 N.W. 707, L.R.A. 1917B, 1. The same rule applies to municipalities. City of Colome v. Von Seggern Bros. Ludden, 56 S.D. 390, 228 N.W. 800. Whether a law or ordinance is subject to the referendum is a judicial question. If it is found that the law is not subject to the referendum, the legislative declaration of an emergency is conclusive. If it be found that the law is subject to the referendum, the declaration of an emergency is void, for then no emergency could exist.

The Legislature, in HB 1069,  invoked the second category of protection.  That’s what I don’t think they can do.  The interpretation of what counts as “necessary for … support of state government” in South Dakota has traditionally been pretty broad, encompassing laws designed to enhance or even redistribute state revenue.  But it’s not unlimited.  Check this out, from “Restrictions on Initiative and Referendum Powers in South Dakota,” Lowe, Chip J., 28 S.D. L. Rev. 53 (1982-1983), p.61:

I don’t think you can say, with a straight face, that eliminating the independent ethics commission is an appropriation bill or a taxing measure.  So their claim dies here:  they can overrule the referendum, but they can’t prevent the public from overruling them right back.

## Booklist 2016 — the year of translation

This year my reading project was for the majority of the books I read to be translated from a language other than English.  Here’s the list:

• 31 Dec 2016:  Troubling Love, by Elena Ferrante (Ann Goldstein, trans.)
• 27 Dec 2016:  The Civil Servant’s Notebook, by Wang Xiaofang (Eric Abrahamsen, trans.)
• 16 Dec 2016:  Nirmala, by Premchand (David Rubin, trans.)
• 16 Dec 2016:  A Long Walk to Water, by Linda Sue Park
• 1 Dec 2016:  Nabokov’s Favorite Word is Mauve, by Ben Blatt
• 24 Nov 2016: HHhH, by Laurent Binet (Sam Taylor, trans.)
• 21 Nov 2016:  Secondhand Time, by Svetlana Alexievich (Bela Shayevich, trans.)
• 20 Nov 2016:  Twenty-Four Hours in the Life of a Woman, by Stefan Zweig (Anthea Bell, trans.)
• 6 Nov 2016:  Houseboy, by Ferdinand Oyono (John Reed, trans.)
• 3 Nov 2016:  The Good Life Elsewhere, by Vladimir Lorchenkov (Ross Ufberg, trans.)
• 12 Oct 2016:  Tales of the Hasidim:  The Early Masters, by Martin Buber (Olga Marx, trans.)
• 1 Oct 2016:  Hit Makers, by Derek Thompson
• 25 Sep 2016:  The Fireman, by Joe Hill
• 19 Sep 2016:  Ghosts, by Raina Telgemeier
• 3 Sep 2016:  The Queue, by Basma Abdel Aziz (Elizabeth Jaquette, trans.)
• 11 Aug 2016:  City of Mirrors, by Justin Cronin
• 26 Jul 2016:  Why I Killed My Best Friend, by Amanda Michalopoulou (Karen Emmerich, trans.)
• 19 Jul 2016:  1Q84, by Haruki Murakami (Philip Gabriel and Jay Rubin, trans.)
• 10 Jul 2016:  The Story of My Teeth, by Valeria Luiselli (Christina MacSweeney, trans.)
• 1 Jul 2016:  So You Don’t Get Lost In The Neighborhood, by Patrick Modiano (Euan Cameron, trans.)
• 13 May 2016:  Weapons of Math Destruction, by Cathy O’Neil
• 2 May 2016:  Sh*tty Mom for All Seasons, by Erin Clune
• 20 Apr 2016:  There’s Nothing I Can Do When I Think of You Late at Night, by Cao Naiqian (John Balcom, trans.)
• 1 Apr 2016:  The Story of the Lost Child, by Elena Ferrante (Ann Goldstein, trans.)
• 25 Feb 2016:  Those Who Leave and Those Who Stay, by Elena Ferrante (Ann Goldstein, trans.)
• 10 Feb 2016:  Voices from Chernobyl, by Svetlana Alexievich (Keith Gessen, trans.)
• 1 Feb 2016:  The Story of a New Name, by Elena Ferrante (Ann Goldstein, trans.)
• 9 Jan 2016:  Amy and Laura, by Marilyn Sachs
• 7 Jan 2016:  My Brilliant Friend, by Elena Ferrante (Ann Goldstein, trans.)

Note that I’m behind on these posts:  I covered the 2013 booklist about a year ago,  but still have to do 2015 (the year of reading mostly women) and 2014.  I’ll get to it.

20 translated books, 9 books in English.  One thing to note is that I read few books this year; I think reading in translation is just a little slower for me.

The languages:

• 5 Italian (all Ferrante)
• 3 French (two from France, one from Cameroon)
• 3 Russian (but no Russian authors!  Lorchenkov is Moldovan, Alexievich is Belarussian.)
• 2 Chinese
• 2 German
• 1 Japanese, 1 Arabic, 1 Greek, 1 Hindi, 1 Spanish.

Overall thoughts:  My plan, I guess, was to expand my horizons.  Did I?  I’m not sure I found these books to be as different from my usual reading as I expected.  Maybe because when American and British writers translate foreign books they somehow press them into the mold of the American and British novel I’m so at ease with?  Or because the novel is fundamentally a cosmopolitan form that works roughly the same way in different national traditions?

The one exception was There’s Nothing I Can Do When I Think of You Late at Night, a kind of Chinese Winesburg,Ohio:  very short, linked stories all set in a remote and desperately impoverished village.  It’s sort of incantatory, phrases repeated several times, in a way that really feels alien to the prose fiction tradition I know.  Niqian wasn’t trained as a writer; apparently he was a detective who started writing as a bet.  Here’s a review with some excerpts.

Best of the year:  No way to choose between Ferrante and Alexievich.  They are too different.  Also the same, of course, in that they always come back to women and the men from whom they expect little and get even less.  And the men from whom they expect something bad and get something even worse.

The books are oral history, interviews collected and transcribed into something like an epic.  Here’s a young woman in Belarus, released from prison after being arrested in a demonstration, telling her story in Secondhand Time:

Do I still like the village?  People here live the same way year in and year out.  They dig for potatoes in their vegetable patches, crawl around on their knees.  Make moonshine.  You won’t find a sngle sober man after dark, they all drink every single day.  They vote for Lukashenko and mourn the Soviet Union.  The undefeatable Soviet Army.  On the bus, one of our neighbors sat down next to me.  He was drunk.  He talked about politics:  “I would beat every moron democrat’s face in myself if I could.  They let you off easy.  I swear to God!  All of them ought to be shot.  America is behind all this, they’re paying for it … Hillary Clinton … but we’re a strong people.  We lived through perestroika, and we’ll make it through another revolution.  One wise man told me that the kikes are the ones behind it.”  The whole bus supported him.  “Things wouldn’t be any worse than they are now.  All you see on TV is bombings and shootings everywhere.”

The same woman, on her time in jail:

I learned that happiness can come from something as small as a bit of sugar or a piece of soap.  In a cell intended for five people — thirty-two square meters — there were seventeen of us.  You had to learn how to fit your entire life into two square meters.  It was especially hard at night, there was no air to breathe, it was stifling.  We wouldn’t get to sleep for a long time.  We stayed up talking.  The first few days, we discussed politics, but after that, we only ever talked about love.

Other Notes:  1Q84 was my first Murakami.  A fascinating example of a book that in many ways I view as  objectively poorly written but which I found captivating, even though it was 1000 pages long.  So maybe this, like Cao, is another book doing something with prose which I’m not used to and which I can’t completely understand.  Twenty-Four Hours in the Life of a Woman was compelling melodrama.  Tales of the Hasidim helped me remember that my idea of what “Jewish culture” means (intellectual, verbal, rule-governed, repressed)  is only one small part of our tradition, and not necessarily the biggest one.  The Lorchenkov was blackly funny.  The Aziz and the Michalopoulou were dull, though this could have been the translator’s fault.  The Civil Servant’s Notebook is a multivocal roman a clef (really multivocal; some of the chapters are narrated by desk furniture) about municipal corruption in China; it was apparently a huge bestseller there and has touched off an entire popular genre of “officialdom literature.”  Maybe we should have that here!

Worst of the year:  Easy, City of Mirrors.  I just dumped a huge ball of words on this terrible book so I went ahead and broke it out as a separate post so as not to dominate my nice year of translations.

## City of Mirrors

Remember how much I liked the first book in this series?  It wasn’t perfect, but I admired the idea of depicting the destruction of a world that’s already kind of ruined though the people in the world don’t fully realize it.  (See also:  Station Eleven.) I was going to write a long post about how lousy this book was but didn’t get around to it and now I’ve mercifully forgotten most of the worst parts.  Still, I did save a lot of highlights of terrible sentences to my Kindle so here are some.

“such was the bittersweet beauty of life”

“Here, tacked to the neutral plaster walls, are the pennants of sports teams and the conundrumous M.C. Escher etching of hands drawing each other and, opposite the sagging single bed, the era-appropriate poster of the erect-nippled Sports Illustrated swimsuit model, beneath whose lubricious limbs and come-hither gaze and barely concealed pudenda the boy has furiously masturbated night after adolescent night.”

“I’d known that Lucessi had a younger sister; he had failed to mention that she was a bona fide Mediterranean goddess, quite possibly the most beautiful girl I’d ever laid eyes on — regally tall, with lustrous black hair, a complexion so creamy I wanted to drink it, and a habit of traipsing into a room wearing nothing more than a slip…. striding through the house in tall riding boots and clanking spurs and tight breeches, a costume no less powerful than the slip in its ability to send the blood dumping to my loins.”

“Though I knew I had done well, I was still astonished to see my first-semester report with its barricade of A’s”

“Between these carnal escapades — Carmen and I would often race back to her room between classes for an hour of furious copulation — and my voluminous classwork and, of course, my hours at the library — time well spent replenishing myself for our next encounter — I saw less and less of Lucessi.”

“On a Saturday afternoon, escorted by my father, I entered this sacred masculine space.  The details were intoxicating.  The odors of tonic, leather, talc.  The combs lounging in their disinfecting aquamarine bath.  The hiss and crackle of AM radio, broadcasting manly contests upon green fields.  My father beside me.  I waited on a chair of cracked red vinyl. Men were being barbered, lathered, whisked.”

“Caleb had peeked at her journals a few times over the years, unable to resist this small crime; like her letters, her entries were wonderfully written.  While they sometimes expressed doubts or concern over various matters, generally they communicated an optimistic view of life.”

The combination of pretentiousness (“manly contests upon green fields”), cliche (“hiss and crackle”), thesaurus-wrangling (“barricade”,”traipsing”), and general vagueness (“various matters”) is really something special.  It is not just bad but BAD in the sense of Paul Fussell.

Oh yeah and also he’s obsessed with the word “possessed” where he means to say “had” or just express the idea in a defter way.

“His flesh, a sickly yellow, possessed a damp, translucent appearance, like the inner layers of an onion.”

“His thoughts possessed a lazy, unmoored quality.”

“His limbs possessed a thin-boned delicacy”

So much more is wrong!  The way characters are often referred to as “the man” or “the woman,” the general concern with manliness throughout, but manliness of a very bent kind:  a weird you-know-how-it-is sympathy towards characters when they get so frustrated that they just have to hit / strangle / kill / sexually assault a woman?  Which culminates in the big action set piece at the end — so of course the big giant ship they have to escape on has been tiresomely personified as a woman,  Michael’s actual one true love for 400 pages — then the big moment comes and the ship can’t consummate, all seems lost, until Michael solves the problem by calling the ship a bitch and hitting it with wrench, at which point it immediately settles down and behaves.  Then the end is an epilogue set 1000 years in the future where a tenured professor (yes, there’s still tenure in 1000 years) gets lucky with a young journalist who’s captivated by his hidden depth and middle-aged loneliness.

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