Eighth Grade

Saw this with CJ.  Good movie.  If you’re wondering, can you see this with your adolescent, definitely yes.  If you’re wondering, will my adolescent have a deep conversation with me afterwards about the challenges of growing up, well, that’s not really CJ’s style but good luck with it!

My favorite thing about Eighth Grade is the way it captures the adolescent challenge seeing other human beings as actual people, like oneself, with their own interior lives.  Other people, for Kayla, are still mostly instruments, things to do something with, or things from which to get a response.  Or maybe she’s just at the moment of learning that other people are not just that?  Very good the way she records Olivia’s name in her phone as “Olivia High School” — other people are roles, they fit in slots — the crush, the shadow, the rival.  Olivia, older, engages with Kayla’s real self in a way that Kayla isn’t yet ready to reciprocate.

But why did she have to say at the end that high school was going to be cool, except math?  Come on, teen movie, be better.


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Before the Golden Age, and memories of memories

When I was a kid, one of my favorite books was Before the Golden Age, a thousand-page anthology Isaac Asimov edited of his favorite stories from the pulp era of science fiction, the early 1930s, when Asimov was a teenager.  I was reading those stories at about the same age Asimov was when he read them.  Asimov put this anthology together in 1974, and remarks in his afterwords on his surprise at how well he remembered these stories.  I, reading them in my own adulthood, am surprised by the same thing.  The armored fighting suits with all the different-colored rays!  1930s science fiction was really into “rays.”

On the other hand, reading these stories again now, and thinking about whether I’d want to lend this book to CJ, I’m stopped short by, well, how super-racist a lot of these stories are?  I hadn’t remembered this at all.  Like, you write a story (“Awlo of Ulm”) about a guy who makes himself smaller than an atom and discovers an entirely new subnuclear universe, and the wildest thing you can imagine finding there is… that the black-skinned subnuclear people are cannibalistic savages, and the yellow-skinned, slant-eyed ones are hyperrational, technically advanced, and cruel, and the white-skinned ones are sort of feudal and hapless but really standup guys when you get to know them?

Anyway, then I read the story, and then I read Asimov’s 1974 afterwords, when he writes about how he was stopped short, reading the stories again then, by how super-racist a lot of the stories were, and that he hadn’t remembered that at all.

So not only did I forget the stories had a lot of racism, I also forgot about Asimov forgetting about the stories having a lot of racism!

1930s SF was really worried about (but also, I think, kind of attracted to) the idea that humans, by relying on machines for aid, would become less and less physically capable, transforming first into big-headed weaklings and finally into animate brains, maybe with tiny eyes or beaks or tentacles attached.  This image comes up in at least three of the stories I’ve read so far (but is most vividly portrayed in “The Man Who Evolved.”)

Of course, you can ask:  was this actually a dominant concern of 1930s SF, or was it a dominant concern of nerdy teen Isaac Asimov?  What I know about the pulps is what I know from this anthology, so my memory of it is my memory of his memory of it.

When I was a kid, by the way, I sent Isaac Asimov a fan letter.  I was really into his collections of popular science essays, which I read again and again.  I told him “I’ll bet I’m your only seven-year-old fan.”  He sent back a postcard that said “I’ll bet you are not my only seven-year-old fan.”  Damn, Asimov, you burned me good.

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Unitarians, legislative districting, and fairness

I gave a talk about gerrymandering at the Prairie Unitarian Universalist society.  As usual, I showed how you can pretty easily district a state with a 60-40 partisan split to give the popular majority party 60% of the seats, 40% of the seats, or 100% of the seats.  After I do that, I ask the audience which map they consider the most fair and which they consider the least fair.  Usually, people rate the proportional representation map the fairest, and the map where the popular minority controls the legislature the least fair.

But not this time!  The Unitarians, almost to a one, thought the districting where the popular majority controlled all the seats was the least fair.  I take from this that the Unitarian ethos rates “the minority rules over the majority” as a lesser evil than “the minority is given no voice at all.”




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Wisconsin municipal vexillology update

Madison is changing its flag!  The old one

has a Zuni sun symbol in the middle of it, which people correctly feel is a sort of random and annoying and unrelated-to-Madison vic of somebody else’s religious symbol.  On the other hand, on pure design grounds it’s kind of a great flag!  Simple, but you see the lakes, the isthmus, the Capitol.  The new flag elegantly keeps all that while skimming off the cultural appropriation:


Meanwhile, in Milwaukee, pressure is mounting to adopt the People’s Flag. Milwaukee’s existing flag is an ungepotchkit mess, routinely ranked among the nation’s worst city banners.  I mean:

I think my favorite part of this mess is that there are two miniflags inside this flag, and the one that’s not the U.S. flag nobody even remembers what it is!

Anyway, this is the proposed new flag, currently the subject of hot civic dissent:

I think this is great.  Daring color choices, you get your lake, you get your big flat lake, you get your optimistic sense of sunrise.  Make the right choice, Cream City!

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Look how much I saved you on this goddamn vacuum cleaner

I wrote this on Facebook about a year and a half ago.

Thought of it today when I saw this tweet from Donald Trump Jr.


A type of unproductivity

There is a certain very special type of unproductivity that I have experienced only in math.  You are working on something and you feel almost certain your strategy is not going to work.  In fact, it is more likely than not that your strategy is not even a new strategy, but a variant on something you’ve already tried unsuccessfully — or maybe not even actually a variant, but just a rephrasing you’ve fooled yourself into thinking is a variant.  So you are not sure whether you are actually working on something at all.  In fact, you doubt it.

And yet you keep going!  Because what if it works?


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Dinner experiment

Ground lamb from Double Ewe Farm (Arena, WI) bought at Conscious Carnivore, stir-fried with scallions/mushrooms/cabbage/garlic/soy sauce/sesame oil and served on top of shiso leaves from Crossroads Community Farm (Cross Plains, WI) with Hot Mama’s habanero sauce from Belize.

I would include a picture of it but it actually didn’t look very pretty.  It tasted great, though!


Dane County Fair

Dane County fair.  Food consumed:  some Chocolate Shoppe ice cream, lemonade, a banh mi, Thai iced tea, chicken yassa from Keur Fatou.  We saw two young cows auctioned for $400 and $425.  We rode the Blizzard and the Ferris wheel.  (The trip is now just two revolutions long — sorry, but that’s a ripoff, for five bucks I should get at least four top-offs.)  This year’s circus acts were frisbee-catching dogs and two motorcyclists in the Globe of Death.  (Good, but not as good as the year the Flying Wallendas themselves came to town.)  We didn’t stay for the Christian rock act but heard them soundcheck a very faithful “Can’t Stop The Feeling!”

First place hay:


First place oats:

The T-shirt concession failed to stake out a clear ideological location.

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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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