Author Archives: JSE

American Revolution

I was in Philadelphia a couple of weeks ago with AB and we went to the brand-new Museum of the American Revolution.  It’s a great work of public history.  Every American, and everybody else who cares about America, should see it.

The museum scrapes away the layer of inevitability and myth around our founding.  Its Revolution is something that might easily not have succeeded.  Or that might have succeeded but with different aims.  There were deep contemporary disagreements about what kind of nation we should be.  The museum puts you face to face with them.

E Pluribus Unum was an aspiration, not a fact.  There was a lot of pluribus.  The gentility in Massachusetts and the Oneida and frontierspeople in Maryland and the French and the enslaved Africans and their American slavemasters were different people with different interests and each had their own revolution in mind.

Somehow it came together.  George Washington gets his due.  The museum presents him as a real person, not just a face on the money.  A person who knew that the decisions he made, in a hurry and under duress, would reverberate through the lifespan of the new country.  We were lucky to have him.  And yes, I choked up, seeing his tent, fragile and beaten-up and confined to a climate-controlled chamber, but somehow still here and standing.

The Haggadah tells us that every generation of Jews has to read the story of Exodus as if we, ourselves, personally, were among those brought out from Egypt.  The museum reminded me of that commandment.  It demands that we find the General Washington in ourselves.  In each generation we have to tell the story of the American Revolution as if we, ourselves, personally, are fighting for our freedom, and are responsible for what America will be.

Because we are!  We are still in the course of human events.  The American Revolution isn’t over.  It won’t ever be over.  It’s right that we call it a “revolution” and not an “overthrow” or a “liberation.”  We’re still revolving, still turning this place over, we’re still plural, we’re still arguing.  We still have the chance, and so we still have the obligation, to make the lives of our children more free than our own.

Happy Independence Day.

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Heights on stacks and heights on vector bundles over stacks

I’ve been giving a bunch of talks about work with Matt Satriano and David Zureick-Brown on the problem of defining the “height” of a rational point on a stack.  The abstract usually looks something like this:

Here are two popular questions in number theory:

1.  How many degree-d number fields are there with discriminant at most X?
2.  How many rational points are there on a cubic surface with height at most X?

Our expectations about the first question are governed by Malle’s conjecture; about the second, by the Batyrev-Manin conjecture.  The forms of the conjectures are very similar, predicting in both cases an asymptotic of the form c X^a (log X)^b, and this is no coincidence: I will explain how to think of both questions in a common framework, that of counting points of bounded height on an algebraic stack.  A serious obstacle is that there is no definition of the height of a rational point on a stack.  I will propose a definition and try to convince you it’s the right one.  If there’s time, I’ll also argue that when we talk about heights with respect to a line bundle we have always secretly meant “vector bundle,” or should have.

(joint work with Matt Satriano and David Zureick-Brown)

Frank Calegari asked a good question after I talked about this at Mazur’s birthday conference.  And other people have asked me the same question!  So I thought I’d write about it here on the blog.

An actual (somewhat tangential) math question about your talk: when it comes (going back to the original problem) of extensions with Galois group G, there is (as you well know) a natural cover \mathbf{A}^n/G \rightarrow \cdot/G, and the source has a nice smooth unirational open subscheme which is much less stacky object and could possibly still be used to count G-extensions (or rather, to count G-polynomials). How does this picture interact (if at all) with your talk or the Malle conjecture more generally?

Here’s an answer.  Classically, how do we count degree-n extensions of Q?  We count monic degree-n polynomials with bounded coefficients; that is, we count integral points of bounded height on A^n / S_n, which is isomorphic to A^n, the space of monic degree-n polynomials.

Now A^n / S_n is the total space of a vector bundle over the stack B(S_n).  So you might say that what we’re doing is using “points on the total space of a vector bundle E/X as a proxy for points on X.”  And when you put it that way, you see that it’s what people who work on rational points do all the time!  What do we do when we count rational points on P^1?  We count pairs of coprime integers in a box; in other words, we count integral points on A^2 – 0, which is the total space (sans zero section) of a line bundle on P^1.  More generally, in many cases where people can prove the Batyrev-Manin conjecture for a variety X, it’s precisely by means of passing to a “universal torsor” — the total space of a vector bundle (or an torus bundle sitting in a vector bundle) over X.  Sometimes you can use this technique to get actual asymptotics for rational points on X; other times you just get bounds; if you can prove that, for any x in X(Q), there is a point on the fiber E_x whose height is at most F(height(x)) for some reasonable function F, you can parlay upper bounds for points on E into upper bounds for points on X.  In the classical case, this is the part where we argue that (by Minkowski) a number field with discriminant D contains an algebraic integer whose characteristic polynomial has coefficients bounded in terms of D.

So coming back to the original question:  how do you know which vector bundle on BG is a good one to think about?  Actually, this is far from clear!  The very first thing I ever wrote about counting number fields, my first paper with Akshay, gave new upper bounds for the number of degree-n extensions, by counting points on

(\mathbf{A}^n)^m / S_n

where S_n acts diagonally.  In other words, we used a different vector bundle on B(S_n) than the “standard” one, and showed that by optimizing m (and being careful about stripping out loci playing the role of accumulating subvarieties) we could get better upper bounds than the ones coming from counting polynomials.

So apparently I’ve been counting points on vector bundles on stacks all along…!

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Two Hebrew words

  1. As part of my 2018 plan to read mostly books older than me, I’m reading Bleak House.  Found this, said of an urchin:

“He’s as obstinate a young gonoph as I know.”

This is of course via the Hebrew ganav (thief) via the Yiddish gonif.  Had no idea it had 19th century London demimonde currency.  Dickens is generally said to have been the first writer to put it in print in English, though Judith Flanders found a somewhat obscure reference a decade older.

“Gonoph” is overtaken by “gonif” as preferred English spelling sometime in the 1940s:

2. During L’cha Dodi last week I was struck by the word “Hitna’ari” (“shake yourself off!”).  That’s a word I know “Na’ar,” as in “Na’ar hayiti, gam zakanti,” (“I have been a youth and I have been old.”)  But does it mean “young” or does it mean “shake yourself off?”  Well, kind of both.  It seems there are two words, the na’ar of youth and the na’ar of shaking off.

Or is there only one word?  People have tried to connect the two senses:

I don’t know.  Hebrew speakers should feel free to weigh in on either of these words!

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Rebecca Dallet and the gerrymandered Assembly map

The fate of the current Wisconsin Assembly district map, precision-engineered to maintain a Republican majority in the face of anything short of a major Democratic wave election, is in the hands of the Supreme Court, which could announce a decision in Gill v. Whitford any day.

One theory of gerrymandering is that the practice isn’t much of a problem, because the power of a gerrymandered map “decays” with time — a map that suits a party in 2010 may, due to shifting demographics, be reasonably fair a few years later.

How’s the Wisconsin gerrymander doing in 2018?  We just had a statewide election in which Rebecca Dallet, the more liberal candidate, beat her conservative rival by 12 points, an unusually large margin for a Wisconsin statewide race.

The invaluable J. Miles Coleman broke the race down by Assembly district:

Dallet won in 58% of seats while getting 56% of the vote.  That sounds fair, but in fact a candidate who wins by 12 points is typically going to win in more seats than that.  (That’s why the courts are right to say proportional representation isn’t a reasonable expectation!)

Here’s the breakdown by Assembly district, shown a little bigger:

Dallet won by 2 points or less in 8 of the Assembly districts.  So, as a rough estimate, if she’d gotten 2% of the vote less, and won 54-46 instead of 56-44, you might guess she’d have won 49 out of 99 seats.  That’s consistent with the analysis of Herschlag, Ravier, and Mattingly conducted last year, which estimates that under current maps Democrats would need an 8-12 point statewide lead in order to win half the Assembly seats. (Figure 5 in the linked paper.)

I don’t think the gerrymander is decaying very much.  I think it’s robust enough to make GOP legislative control very likely through 2020, at which point it can be updated to last another ten years, and so on and so on.  This isn’t the same kind of softcore gerrymandering the Supreme Court allowed to stand in 1986, and I hope the 2018 Supreme Court decides to do something about it.

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Missing LeBron

When I was a postdoc in Princeton I subscribed to the Trenton Times, because I felt it was important to be in touch with what was going on in my local community and not just follow national news.  The only story I remember was one that said “hey, a basketball team from Akron is coming to play against a top prep-school team in Trenton, and they’ve got this kid LeBron James they say is incredible, you should come check it out.”  And I really did think about it, but I was a postdoc, I was trying to write papers, I was busy, too busy to drive into Trento for a high-school basketball game.

So I guess what I’m trying to say is, yes, subscribe to your local paper because local journalism badly needs financial support, and maybe actually take seriously the local events it alerts you to.


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Did Tim Burns voters come through for Rebecca Dallet?

Two liberal candidates, Rebecca Dallet and Tim Burns, combined for 54% of the vote in this February’s primary for the Wisconsin Supreme Court.  Dallet and the conservative candidate Michael Screnock, who got 46% in the primary, moved on to the general election in April.

There was some worry among liberal political types that voters who went for Burns, the vocally left candidate, would sit out the general rather than show up for the more conventionally liberal Dallet.  Did that happen?  Here’s something cool:  Wisconsin offers full statewide ward-level election results, which helps us figure that out!

First of all, here’s a ward-by-ward picture of the primary:

Each circle is a ward and its position in the triangle shows the proportion of votes going to Screnock (top vertex), Burns (left vertex), and Dallet (right vertex.)  The size of the circle is the total number of votes in that ward.  You can see that there’s no visible clustering, and that Dallet did much better than Burns.

So what happened in the general?

Well, first of all, Dallet won, and won big:  56-44.  But that doesn’t mean Burns voters showed up.  We can’t really know!  But the ward-by-ward data at least helps us make some guesses.

Quick and dirty:  you can do a linear regression on Dallet’s share of the general in terms of Burns’s and Dallet’s share of the primary vote.  I stripped out wards with fewer than 100 general-election votes, which still left 1827 wards.  You get

Dallet general ~ 0.724*Burns primary + 0.892* Dallet primary + 0.112

with a pretty decent fit

Screen Shot 2018-05-03 at 3 May 7.27.AM

The Burns coefficient is a little bit lower but I don’t see strong evidence that a lot of Burns voters skipped the general election.

Here’s a test I like a little bit more.  There are 79 wards where Burns and Dallet together got between 54 and 56% of the vote in the primary.  Among these wards, Burns’s voteshare ranged from 6.5% (Milwaukee ward 211) to 35% (Town of Moscow wards 1-2, a bit on the nose, don’t you think?)  If Burns voters were skipping the general election, you might expect Dallet to do worse in April in those wards where Burns did better in February.  Here’s the scatter.  If there’s a downward trend here, it’s not very strong.

Screen Shot 2018-05-02 at 2 May 7.59.PM

My conclusion:  liberals gonna liberal.

Update:  I got the last scatter wrong when I originally posted this; if you remember the post being a little different, you’re right!

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Brewers 6, Marlins 5 / Bucks 104, Celtics 102 / Orioles 6, Tigers 0

I’ve lived in Madison for 13 years and this is the first time I’ve noticed anybody caring about the Milwaukee Bucks.  It’s definitely the first time I’ve cared about the Milwaukee Bucks.  But now the Bucks have a legitimate superstar in Giannis Antetokoumnpo  and a likeable cast of supporting characters like 19-year-old former refugee and skinny blockmaster Thon Maker.  The kids had a rare unscheduled day on Sunday and the Bucks were in the playoffs against the Celtics and there were nosebleed tickets on Stubhub for $40 apiece so why not?

You may know that I kind of hate driving so if I’m gonna drive all the way to Milwaukee it’s got to be for more than a Bucks game.  When I thought about what the kids would really want to do it was pretty clear — see the Brewers, stay over, then see the Bucks.  So that’s what we did!

Notes on the Brewers:

  • I got lost in the impossible off-ramp spaghetti surrounding Miller Park and we ended up not getting into the ballpark until the second inning.  The Brewers were already down 4-0.  4-0!  To the sad Miami Marlins, the team Derek Jeter is using as a tax dodge, the team so bad Marlins Man cancelled his season tickets!
  • But as soon as we sat down, Travis Shaw muscled a huge home run to left center.  Didn’t even look like he got all of it, he kind of sliced it.  But Travis Shaw is a big strong man.
  • Brewers just keep creeping back.  Crowd stays in the game, at no point do you really feel the Brewers are out of it.  Three straight Brewers hit what look like go-ahead home runs but each dies at the wall.  (Ryan Braun at least gets a sacrifice fly out of it.)   In the 8th, Derek Dietrich loses an Eric Sogard fly ball in the, I dunno, the lights?  The roof?  He plays for the Marlins and he just doesn’t care?  Anyway the ball plunked down right next to him, Shaw hustles in from second to tie it, Eric Thames, who starts the play on first, tries to get in behind with the go-ahead run but is tagged out at or rather substantially before the plate because Eric Thames made a bad decision.
  • Josh Hader looks like he should be playing bass in Styx.
  • Then comes the bottom of the 9th and the play you might have read about.  Still tied 5-5.  Jesus Aguilar, who’s already warmed up twice in the on-deck circle, finally gets his chance to pinch-hit against Junichi Tazawa.  Gets behind 0-2.  And then just starts fouling, fouling, fouling.  Takes a few pitches here and there.  Full count.  Foul, foul, foul.  And on the 13th pitch, Aguilar launches it to center field.  I thought it was gonna be one more death on the warning track.  But nope; ball gets out, game over, fireworks.  I felt like my kids got to see true baseball.

On to Milwaukee.  Bucks play the Celtics at noon, in what, if they lose, could be the last ever game played at Bradley Center.  (This is a bit of a sore point for UW folks, who absorbed as a budget cut the $250m state contribution to the arena’s cost.)  We have breakfast at the hotel and chat with a nice older couple in Packers/Celtics gear — what?  — who turn out to be Boston forward Al Horford’s aunt and uncle from Green Bay.

This is only the third NBA game I’ve been to, CJ’s second, AB’s first.  We wander around inside the arena for a bit.  Two separate groups of Bucks cheerleaders come up to AB and applaud her curly hair.  I think people are especially struck by it when they see us together, because I don’t have curly hair, except here’s a little-known fact:  I do have curly hair!  I just keep it short so it doesn’t curl.  In 1995 or so it looked like this:

Anyway.  The atmosphere, as I have promised AB, is more intense than baseball.  Bucks build up a 19-point lead and seem poised to coast but the Celtics come back, and back, and back, and finally go ahead with 52 seconds left.  Jaylen Brown plainly capable of taking over a game.  Aron Baynes has a very dumb-looking haircut.  Milwaukee’s Thon Maker is ridiculously skinny and has very long arms.  He’s just 21, a former refugee from South Sudan.  We saw his first game as a Buck, an exhibition against the Mavericks at Kohl Center.  Those long skinny arms can block a shot.

Game tied at 102, 5 seconds left, Malcom Brogdon (called “The President” — why?) misses a layup, and there, rising like a Greek column above the scene, is the Greek arm of Giannis Antetokounmpo — the tip-in is good, Celtics miss the desperation last shot, Bucks win 104-102, crowd goes berserk.


I was going to blog about this last week but got busy so let’s throw in more sports.  Bucks eventually lose this series in 7, home team winning every game a la Twins-Braves 1991.  The next Friday, I’m giving a talk at Maryland, and the Orioles are playing that night.  It’s been five years since I’ve seen OPACY.  I brought CJ along this time, too.  The Orioles are not in a good way; they’ve won 6 and lost 19, though 3 of those 6 were against New York at least.  Attendance at the game, on a beautiful Friday night, was just over 14,000.  The last baseball game I went to that felt this empty and mellow was the AAA Tucson Toros, several months before they moved to El Paso and became the Chihuahuas.  Chris Tillman, tonight’s starter, was the Orioles’ ace five years ago.  Now he’s coming off a 1-7 season and has an ERA over 9.

So who would have thought he’d toss seven shutout innings and take a no-hitter into the fifth?  Never looked overpowering but kept missing bats.  His first win in almost a year.  Manny Machado, surely now in his last year as an Oriole, strokes a home run to dead center to get things started.  It’s a beautiful thing.  It doesn’t even look like he’s working hard.  It’s like he’s just saying “Out there. Out there is where this ball should be.”  Pedro Alvarez homers twice, in exactly the opposite manner, smashing the ball with eye-popping force.  Jace Peterson, who the Orioles picked up off the Yankees’ scrap heap, steals third on the shift when the Tigers third baseman forgets to pay attention to him.  He did the same thing against the Rays the night before.  I am already starting to love him the way I love Carlos Gomez.  Maybe now the Orioles are going to go back to being a bad team that makes good use of players nobody else wants, like Melvin Mora and Rodrigo López.

Besides me and CJ, this guy was at the game:

Never get tired of that flag.


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The Lovasz number of the plane is about 3.48287

As seen in this comment on Polymath and explicated further in Fernando de Oliveira Filho’s thesis, section 4.4.

I actually spent much of today thinking about this so let me try to explain it in a down-to-earth way, because it involved me thinking about Bessel functions for the first time ever, surely a life event worthy of recording.

So here’s what we’re going to do.  As I mentioned last week, you can express this problem as follows:  suppose you have a map h: R^2 -> V, for some normed vector space V, which is a unit-distance embedding; that is, if |x-x’|_{R^2} = 1, then |h(x)-h(x’)|_V = 1.  (We don’t ask that h is an isometry, only that it preserves the distance-1 set.)

Then let t be the radius of the smallest hypersphere in V containing h(R^2).

Then any graph embeddable in R^2 with all edges of length 1 is sent to a unit-distance graph in V contained in the hyperplane of radius t; this turns out to be equivalent to saying the Lovasz number of G (ok, really I mean the Lovasz number of the complement of G) is at most 1/(1-2t).  So we want to show that t is bounded below 1, is the point.  Or rather:  we can find a V and a map from R^2 to V to make this the case.

So here’s one!  Let V be the space of L^2 functions on R^2 with the usual inner product.  Choose a square-integrable function F on R^2 — in fact let’s normalize to make F^2 integrate to 1 — and for each a in R^2 we let h(a) be the function F(x-a).

We want the distance between F(x-a) and F(x-b) to be the same for every pair of points at distance 1 from each other; the easiest way to arrange that is to insist that F(x) be a radially symmetric function F(x) = f(|x|); then it’s easy to see that the distance between F(x-a) and F(x-b) in V is a function G(a-b) which depends only on |a-b|.  We write

g(r) = \int_{\mathbf{R}^2} F(x)F(x-r) dx

so that the squared distance between F(x) and F(x-r) is

\int F(x)^2 dx - 2 \int F(x)F(x-r) dx + \int F(x-r)^2 dx = 2(1-g(r)).

In particular, if two points in R^2 are at distance 1, the squared distance between their images in V is 2(1-g(1)).  Note also that g(0) is the square integral of F, which is 1.

What kind of hypersphere encloses all the points F(x-a) in V?  We can just go ahead and take the “center” of our hypersphere to be 0; since |F| = 1, every point in h(R^2) lies in (indeed, lies on) the sphere of radius 1 around the origin.

Hey but remember:  we want to study a unit-distance embedding of R^2 in V.  Right now, h sends unit distances to the distance 2(1-g(1)), whatever that is.  We can fix that by scaling h by the square root of that number.  So now h sends unit distances to unit distances, and its image is enclosed in a hypersphere of radius


The more negative g(1) is, the smaller this sphere is, which means the more we can “fold” R^2 into a small space.  Remember, the relationship between hypersphere number and Lovasz theta is

2t + 1/\theta = 1

and plugging in the above bound for the hypersphere number, we find that the Lovasz theta number of R^2, and thus the Lovasz theta number of any unit-distance graph in R^2, is at most


So the only question is — what is g(1)?

Well, that depends on what g is.

Which depends on what F is.

Which depends on what f is.

And of course we get to choose what f is, in order to make g(1) as negative as possible.

How do we do this?  Well, here’s the trick.  The function G is not arbitrary; if it were, we could make g(1) whatever we wanted.  It’s not hard to see that G is what’s called a positive definite function on R^2.  And moreover, if G is positive definite, there exists some f giving rise to it.  (Roughly speaking, this is the fact that a positive definite symmetric matrix has a square root.)  So we ask:  if G is a positive definite (radially symmetric) function on R^2, and g(0) = 1, how small can g(1) be?

And now there’s an old theorem of (Wisconsin’s own!) Isaac Schoenberg which helpfully classifies the positive definite functions on R^2; they are precisely the functions G(x) = g(|x|) where g is a mixture of scalings of the Bessel function $J_0$:

g(r) = \int_0^\infty J_0(ur) A(u)

for some everywhere nonnegative A(u).  (Actually it’s more correct to say that A is a distribution and we are integrating J_0(ur) against a non-decreasing measure.)

So g(1) can be no smaller than the minimum value of J_0 on [0,infty], and in fact can be exactly that small if you let A become narrowly supported around the minimum argument.  This is basically just taking g to be a rescaled version of J_0 which achieves its minimum at 1.  That minimum value is about -0.4, and so the Lovasz theta for any unit-distance subgraph on the plane is bounded above by a number that’s about 1 + 1/0.4 = 3.5.

To sum up:  I give you a set of points in the plane, I connect every pair that’s at distance 1, and I ask how you can embed that graph in a small hypersphere keeping all the distances 1.  And you say:  “Oh, I know what to do, just assign to each point a the radially symmetrized Bessel function J_0(|x-a|) on R^2, the embedding of your graph in the finite-dimensional space of functions spanned by those Bessel translates will do the trick!”

That is cool!

Remark: Oliveira’s thesis does this for Euclidean space of every dimension (it gets more complicated.)  And I think (using analysis I haven’t really tried to understand) he doesn’t just give an upper bound for the Lovasz number of the plane as I do in this post, he really computes that number on the nose.

Update:  DeCorte, Oliveira, and Vallentin just posted a relevant paper on the arXiv this morning!

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The first pancake is always strangely shaped

Alena Pirutka gave a great algebraic geometry seminar here last week, about (among many other things!) families of smooth projective varieties containing both rational and non-rational members.   We were talking about how you have to give a talk several times before it really starts to be well-put together, and she told me there’s a Russian proverb on the subject:  “The first pancake is always strangely shaped.”  I am totally going to go around saying this from now on.


Ocular regression

A phrase I learned from Aaron Clauset’s great colloquium on the non-ubiquity of scale-free networks.  “Ocular regression” is the practice of squinting at the data until it looks linear.

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