Cities, Villages, Towns, and Scott Walker

Speaker of the Wisconsin Assembly Robin Vos, still smarting from Scott Walker’s loss in his re-election bid, said “Evers win was due to Dane County and the City of Milwaukee.”  It’s typical GOP politics here to split off Madison and Milwaukee like this, as if liberalism in Wisconsin is a pair of dark blue inkstains on an otherwise conservative shirt.

Not so.  There are liberals all over your shirt, Mr. Vos.

You can find tons of interesting data about Wisconsin elections in Excel spreadsheets at the Wisconsin Elections Commission page.  This already gives you the ability to do some quick and dirty analysis of where Evers’ victory was won.  In Wisconsin, every municipality is either a City, a Village, or a Town, in roughly decreasing order of urbanization.  So it’s easy to separate out Wisconsin into three parts, the Cities, the Villages, and the Towns.  This is what you get:

CITIES:  Walker 542148 (40%), Evers 808145 (60%)

VILLAGES: Walker 257858 (55%), Evers 208596 (45%)

TOWNS:  Walker 495074 (62%), Evers 307566 (38%).

That’s a pretty clear story.  Evers won in the cities, Walker won by a bit in the villages and by a lot in the most rural segment of the state, the towns.

But wait — Madison and Milwaukee are cities!  Is that all we’re seeing in this data, a distinction between Madison and Milwaukee on the one hand and real Wisconsin, Republican Wisconsin, Robin Vos’s Wisconsin, on the other?  Nope.  Take out the cities of Milwaukee and Madison from the city total and Evers still gets 521265 votes to Walker’s 477447, drawing 52% of the vote to Walker’s 28%.  There are decent-sized cities all over the state, and Evers won almost all of them.  Evers won Green Bay, he won Sheboygan, he won Appleton, he won Wausau.  Evers won Chippewa Falls and Viroqua and Oshkosh and Neenah and Fort Atkinson and Rhinelander and Beloit.  He won all over the place, wherever Wisconsinites congregate in any fair number.

Craig Gilbert has a much deeper dive into this data in the Journal-Sentinel.  The shift away from Scott Walker wasn’t just in the biggest cities; it was pretty uniform over localities with population 10,000 or more.  Update: An even deeper dive by John Johnson at Marquette, which brings in data from presidential elections too.

There’s a general feeling that the urban-rural split is new, a manifestation of Trumpian anti-city feeling.  Let’s look back at the 2010 election between Scott Walker and Tom Barrett, an election Walker won by 7 points.  2010, when Donald Trump was just Jeff Probst in a tie.  But the urban-rural split is still there:

CITIES: Walker 505213 (46%), Barrett 603905 (54%)

VILLAGES:  Walker 207243 (59%), Barrett 143297 (41%)

TOWNS:  Walker 416485 (62%) , Barrett 257101 (38%)

Here’s the thing, though.  You can see that Walker actually didn’t do any worse in the towns in 2018 than he did in 2010.  But his support dropped off a lot in the villages and the cities.  And if you take Madison and Milwaukee out of the 2010 totals, Walker won the remainder of Wisconsin’s cities 53-47, which is actually a bit ahead of his overall 2010 statewide margin.

I don’t think Donald Trump has made Wisconsin politics very different.  I think it’s still a state that calls its own tune, and a state where either a Democrat or a Republican can win big — if they have something to say that makes sense all across the state, as Walker and Ron Johnson used to, as Evers and Tammy Baldwin do now.

 

 

 

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The great qualities with which dullness takes lead in the world

He firmly believed that everything he did was right, that he ought on all occasions to have his own way — and like the sting of a wasp or serpent his hatred rushed out armed and poisonous against anything like opposition. He was proud of his hatred as of everything else. Always to be right, always to trample forward, and never to doubt, are not these the great qualities with which dullness takes lead in the world?

(William Makepeace Thackeray, from Vanity Fair)

 

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Why is a Markoff number a third of a trace?

I fell down a rabbit hole this week and found myself thinking about Markoff numbers again.  I blogged about this before when Sarnak lectured here about them.  But I understood one minor point this week that I hadn’t understood then.  Or maybe I understood it then but I forgot.  Which is why I’m blogging this now, so I don’t forget again, or for the first time, as the case may be.

Remember from the last post:  a Markoff number is (1/3)Tr(A), where A is an element of SL_2(Z) obtained by a certain construction.  But why is this an integer?  Isn’t it a weird condition on a matrix to ask that its trace be a multiple of 3?  Where is this congruence coming from?

OK, here’s the idea.  The Markoff story has to do with triples of matrices (A,B,C) in SL_2(Z) with ABC = identity and which generate H, the commutator subgroup of SL_2(Z).  I claim that A, B, and C all have to have trace a multiple of 3!  Why?  Well, this is of course just a statement about triples (A,B,C) of matrices in SL_2(F_3).  But they actually can’t be arbitrary in SL_2(F_3); they lie in the commutator.  SL_2(F_3) is a double cover of A_4 so it has a map to Z/3Z, which is in fact the full abelianization; so the commutator subgroup has order 8 and in fact you can check it’s a quaternion group.  What’s more, if A is central, then A,B, and C = A^{-1}B^{-1} generate a group which is cyclic mod its center, so they can’t generate all of H.  We conclude that A,B, and C are all non-central elements of the quaternion group.  Thus they have exact order 4, and so their eigenvalues are +-i, so their trace is 0.

In other words:  any minimal generating set for the commutator subgroup of SL_2(Z) consists of two matrices whose traces are both multiples of 3.

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It’s right by the airport

I went to California last week to talk math and machine learning with Ben Recht (have you read his awesome blogstravaganza about reinforcement learning and control?) My first time on the brand-new Madison – San Francisco direct flight (the long-time wish of Silicon Isthmus finally realized!) That flight only goes once a day, which means I landed at SFO at 6:15, in the middle of rush hour, which meant getting to Berkeley by car was going to take almost an hour and a half.  So maybe it made more sense to have dinner near SFO and then go to the East Bay.  But where can you have dinner near SFO?

Well, here’s what I learned.  When I was at MSRI for the Galois Groups and Fundamental Groups semester in 1999, there was an amazing Chinese restaurant in Albany, CA called China Village.  I learned about it from my favorite website at the time, Chowhound.com.  China Village is still there and apparently still great, but the original chef, Zongyi Liu, left long ago.  Chowhound, too, is still there, but a thin shadow of its old self.  When I checked Chowhound this week, though, I learned something fantastic — Liu is back and cooking in Millbrae!  At Royal Feast, a 10-minute drive from SFO.  So what started as a plan to dodge traffic turned into the best Chinese meal I’ve eaten in forever.  Now I’m thinking I’ll probably stop there every time I fly to San Francisco!  And it’s right by the Millbrae BART station, so if you’re going into the city, it’s as convenient as being at the airport.

So that got me thinking:  what are good things to know about that are right near the airport in other cities?  The neighborhood around the airport is often kind of unpromising, so it’s good to have some prior knowledge of places worth stopping.  And I actually have a pretty decent list!

LAX:  This is easy — you can go to the beach!  Dockweiler State Beach is maybe 5 minutes from the airport.  It’s a state park, not developed, so there’s no boardwalk, no snack stand, and, when I went there, no people.  You just walk down to the ocean and look at the waves and every thirty seconds or so a jumbo jet blasts by overhead on its way to Asia because did I mention 5 minutes from the airport?  You’re right under the takeoff path.  And it’s great.  A sensory experience like no other beach there is.  I just stood there for an hour thinking about math.

BOSTON:  There is lots of great pizza in Boston, of course, but Santarpio’s in East Boston might be the very best I’ve had, and it’s only 7 minutes from Logan airport.  Stop there and get takeout on your way unless you want to bring yet another $13 cup of Legal Seafood chowder on your flight.

MILWAUKEE:  I have already blogged about the unexpectedly excellent Jalapeño Loco, literally across the street from the airport.  Best chile en nogada in the great state of Wisconsin.

SEATTLE:  The Museum of Flight isn’t quite as close to Sea-Tac as some of these other attractions are to their airports — 12 minutes away per Google Maps.  But it’s very worth seeing, especially if you happen to be landing in Seattle with an aircraft-mad 11-year-old in tow.

MADISON:  “The best barbecue in Madison, Wisconsin” is not going to impress my friends south of the Mason-Dixon line, or even my friends south of the Beloit-Rockford line, but Smoky Jon’s, just north of the airport on Packers Avenue (not named for the football team, but for the actual packers who worked at the Oscar Mayer plant that stood on this road until 2017) is the real thing, good enough for out of town visitors and definitely better than what’s on offer at MSN.

CHICAGO:  No, O’Hare is terrible in this way as in every other way.  I once got stuck there for the night and tried to find something exciting in the area to do or eat.  I didn’t succeed.

You guys travel a lot — you must have some good ones!  Put them in the comments.

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Ringo Starr rebukes the Stoics

I’ve been reading Marcus Aurelius and he keeps returning to the theme that one must live “according to one’s nature” in order to live a good life.  He really believes in nature.  In fact, he reasons as follows:  nature wouldn’t cause bad things to happen to the virtuous as well as the wicked, and we see that both the virtuous and the wicked often die young, so early death must not be a bad thing.

Apparently this focus on doing what is according to one’s nature is a standard feature of Stoic philosophy.  It makes me think of this song, one of the few times the Beatles let Ringo sing.  It’s not even a Beatles original; it’s a cover of a Buck Owens hit from a couple of years previously.  Released as a B-side to “Yesterday” and then on the Help! LP.

Ringo has a different view on the virtues of acting according to one’s nature:

They’re gonna put me in the movies
They’re gonna make a big star out of me
We’ll make a film about a man that’s sad and lonely
And all I gotta do is act naturally
Well, I’ll bet you I’m a-gonna be a big star
Might win an Oscar you can’t never tell
The movie’s gonna make me a big star,
‘Cause I can play the part so well
Well, I hope you come and see me in the movie
Then I’ll know that you will plainly see
The biggest fool that’s ever hit the big time
And all I gotta do is act naturally

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On not staying in your lane

This week I’ve been thinking about some problems outside my usual zone of expertise — namely, questions about the mapping class group and the Johnson kernel.  This has been my week:

  • Three days of trying to prove a cohomology class is nonzero;
  • Then over Thanksgiving I worked out an argument that it was zero and was confused about that for a couple of days because I feel quite deeply that it shouldn’t be zero;
  • This morning I was able to get myself kind of philosophical peace with the class being zero and was working out which refined version of the class might not be zero;
  • This afternoon I was able to find the mistake in my argument that the class was zero so now I hope it’s not zero again.
  • But I still don’t know.

There’s a certain frustration, knowing that I’ve spend a week trying to compute something which some decently large number of mathematicians could probably sit down and just do, because they know their way around this landscape.  But on the other hand, I would never want to give up the part of math research that involves learning new things as if I were a grad student.  It is not the most efficient way, in the short term, to figure out whether this class is zero or not, but I think it probably helps me do math better in a global sense that I spend some of my weeks stumbling around unfamiliar rooms in the dark.  Of course I might just be rationalizing something I enjoy doing.  Even if it’s frustrating.  Man, I hope that class isn’t zero.

Wisconsin post-election post

A few thoughts.

  • Crazy that, once again, a statewide election in Wisconsin is decided when a county clerk, late into the night, reveals a stash of not-yet-counted ballots.
  • Evers winning and Baldwin cruising while national media darling Randy Bryce got soundly beaten by boring former UW Regent Bryan Steil in a not-that-Republican district is further evidence for the ham sandwich theory of Wisconsin politics.  Bryce ran about a point behind Tony Evers in Racine, his own home county!
  • A lot of people asking “How can there be so many Baldwin-Walker voters?  It makes no sense!”  I think it makes sense.  In a state not experiencing any visible crisis, incumbency is an advantage.  The last Marquette poll had “state on the right track / state going the wrong direction” at 55/40.  Roughly speaking, if incumbency is a 5% boost and the state’s “mood” was +5% Democratic, you’d get a Democratic incumbent winning by 10 and a Republican incumbent locked in a tie, which is pretty much what happened.  There are plenty of people whose votes aren’t strongly ideological.
  • With all the focus on the governor’s race, hardly anyone was watching the state legislative elections.  They didn’t go well for Democrats, who lost Caleb Frostman’s Senate seat and may lose seats in the Assembly too, while Democrats elsewhere were making pretty big gains in state legislatures.  Some of that, probably most of it, is our ridiculously gerrymandered Assembly map.  I’ll write more about that when I know more of the final numbers in these races.  But there’s another thing going on, too — I think Wisconsin Democrats have figured out that turnout efforts in Dane and Milwaukee are the most efficient way to turn effort into votes.  Turnout for this election was high statewide but in Dane County it was crazy; Tony Evers got more votes in the county than Hillary Clinton did in a presidential year!  And it’s not cause Dane County doesn’t love Hillary Clinton.  But that focus doesn’t shift the Assembly or Senate map.
  • In 2014, the total votes for Attorney General were 95% of the governor votes.  This time it was 99%.  I read that as:  more voters voting party-line, fewer voting only for governor and leaving the AG line blank because all they know is the party.  But maybe I’m reading too much into it.

 

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Wisconsin pre-election post

I see no reason to doubt the polls that show a very close race between Tony Evers and Scott Walker for governor, but a healthy lead for incumbent Senator Tammy Baldwin over her challenger, Leah Vukmir.

In the current hyperpartisan environment, why are these two races so different?  For one thing, Baldwin and Walker are incumbents, and people in Wisconsin seem to mostly feel things are OK here (55% said the state was “on the right track” in the latest Marquette poll.)

But there’s something else.  I think in Wisconsin we like our politicians, well, not too salty.  Scott Walker is a bland guy.  He famously eats two ham sandwiches in a paper bag for lunch every day, and the thing is, I don’t think that’s an affectation, Walker really is a guy who doesn’t mind eating the same thing for lunch every day.  Tony Evers is bland, too, and he fought off seven spicier opponents in a wild primary, winning just about every county in the state.  When Tammy Baldwin announced for Senate people thought there was no way a movement liberal and out lesbian from Madison could win a statewide race.  She won it easily.  Because she is is a movement liberal out lesbian from Madison who in every way comes off as the super-nice mom at the PTA meeting who you go ahead and let make all the decisions because it kind of seems like she’s got this.

Leah Vukmir, by contrast, is a cookie-cutter Fox News Republican who wants to bring a meaner, harder-edged style to Wisconsin politics.  I don’t think it’s gonna work.  I think Wisconsinites, both Democrats and Republicans, prefer the ham sandwich in the paper bag.

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The greatest Red Sox / Dodger

After game 2 it was already clear this was an NLCS so great it had to go seven, and it did.  But game seven wasn’t a great game seven.  After six hard-fought games, the Brewers never really mounted a threat, and went down 5-1.  Keenest pain of all was that I got what I’d been waiting for the whole series; a chance for my beloved Jonathan Schoop to be the hero.  He came in to pinch hit for starter Joulys Chacin in the bottom of the second, with two on and two out and the Brewers down by 1.  Schoop grounded out.  He was 0 for the postseason in 6 plate appearances.

So here we have it, a Red Sox / Dodgers series, and so it’s time for my annual post about what player had the best combined career for both teams.  (Last year:  Jimmy Wynn was the greatest Astro/Dodger.)

The greatest Red Sox / Dodger?  A player I’d never heard of, even though he was just a little before my time:  Reggie Smith.  Played in one World Series for the Red Sox (1967) and three for the Dodgers (1977,1978,1981).  Went to the All-Star Game with both teams.  Hit 300 home runs, cannon of an arm in the outfield, got 0.7% of the vote the one and only time he was up for the Hall of Fame.  Well, here’s his all time distinction; with 34.2 WAR for the Red Sox and 19.4 for the Dodgers, he’s the greatest Red Sox / Dodger of all time.

Surprisingly, given how old these teams are, the top Red Sox / Dodgers of all time are mostly recent players.  Derek Lowe is the top pitcher (19.4 WAR for Boston, 13.3 for LA.)  Adrian Gonzalez, Manny Ramirez, and Adrian Beltre are also worthy of mention.   The only old-time player who was a contender was Dutch Leonard, who actually pitched for Boston in the last Red Sox – Dodgers World Series in 1916, notching a complete game win.  But that guy never actually pitched for the Dodgers!  My search got confused because it turns out there were two Dutch Leonards, the second of whom was a Dodger to start his career.  Doesn’t count!

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Naser Talebizadeh Sardari, Hecke eigenvalues, and Chabauty in the deformation space

Naser Sardari is finishing a postdoc at Wisconsin this year and just gave a beautiful talk about his new paper.  Now Naser thinks of this as a paper about automorphic forms — and it is — but I want to argue that it is also a paper which develops an unexpected new form of the Chabauty method!  As I will now explain.  Tell me if you buy it.

First of all, what does Naser prove?  As the title might suggest, it’s a statement about the multiplicity of Hecke eigenvalues a_p; in this post, we’re just going to talk about the eigenvalue zero.  The Hecke operator T_p acts on the space of weight-k modular forms on Gamma_0(N); how many zero eigenvectors can it have, as k goes to infinity with N,p fixed?  If you believe conjectures of Maeda type, you might expect that the Hecke algebra acts irreducibly on the space S_k(Gamma_0(N)); of course this doesn’t rule out that one particular Hecke operator might have some zeroes, but it should make it seem pretty unlikely.

And indeed, Naser proves that the number of zero eigenvectors is bounded independently of k, and even gives an explicit upper bound. (When the desired value of a_p is nonzero, T_p has finite slope and we can reduce to a problem about modular forms in a single p-adic family; in that context, a uniform bound is easy, and one can even show that the number of such forms of weight <k grows very very very very slowly with k, where each "very" is a log; this is worked out on Frank Calegari’s blog.. On the other hand, as Naser points out below in comments, if you ask about the “Hecke angle” a_p/p^{(k-1)/2}, we don’t know how to get any really good bound in the nonzero case. I think the conjecture is that you always expect finite multiplicity in either setting even if you range over all k.)

What I find most striking is the method of proof and its similarity to the Chabauty method!  Let me explain.  The basic idea of Naser’s paper is to set this up in the language of deformation theory, with the goal of bounding the number of weight-k p-adic Galois representations rho which could be the representations attached to weight-k forms with T_p = 0.

We can pin down the possible reductions mod p of such a form to a finite number of possibilities, and this number is independent of k, so let’s fix a residual representation rhobar once and for all.

The argument takes place in R_loc, the ring of deformations of rhobar|G_{Q_p}.  And when I say “the ring of deformations” I mean “the ring of deformations subject to whatever conditions are important,” I’m just drawing a cartoon here.  Anyway, R_loc is some big p-adic power series ring; or we can think of the p-adic affine space Spec R_loc, whose Z_p-points we can think of as the space of deformations of rhobar to p-adic local representations.  This turns out to be 5-dimensional in Naser’s case.

Inside Spec R_loc, we have the space of local representations which extend to global ones; let’s call this locus Spec R_glob.  This is still a p-adic manifold but it’s cut out by global arithmetic conditions and its dimension will be given by some computation in Galois cohomology over Q; it turns out to be 3.

But also inside Spec R_loc, we have a submanifold Z cut out by the condition that a_p is not just 0 mod p, it is 0 on the nose, and that the determinant is the kth power of cyclotomic for the particular k-th power you have in mind.  This manifold, which is 2-dimensional, is something you could define without ever knowing there was such a thing as Q; it’s just some closed locus in the deformation space of rhobar|Gal(Q_p).

But the restriction of rho to Gal(Q_p) is a point psi of R_loc which has to lie in both these two spaces, the local one which expresses the condition “psi looks like the representation of Gal(Q_P) attached to a weight-k modular form with a_p = 0” and the global one which expresses the condition “psi is the restriction to Gal(Q_p) of representation of Gal(Q) unramified away from some specified set of primes.”  So psi lies in the intersection of the 3-dimensional locus and the 2-dimensional locus in 5-space, and the miracle is that you can prove this intersection is transverse, which means it consists of a finite set of points, and what’s more, it is a set of points whose cardinality you can explicitly bound!

If this sounds familiar, it’s because it’s just like Chabauty.  There, you have a curve C and its Jacobian J.  The analogue of R_loc is J(Q_p), or rather let’s say a neighborhood of the identity in J(Q_p) which looks like affine space Q_p^g.

The analogue of R_glob is (the p-adic closure of) J(Q), which is a proper subspace of dimension r, where r is the rank of J(Q), something you can compute or at least bound by Galois cohomology over Q.  (Of course it can’t be a proper subspace of dimension r if r >= g, which is why Chabauty doesn’t work in that case!)

The analogue of Z is C(Q_p); this is something defined purely p-adically, a locus you could talk about even if you had no idea your C/Q_p were secretly the local manifestation of a curve over Q.

And any rational point of C(Q), considered as a point in J(Q_p), has to lie in both C(Q_p) and J(Q), whose dimensions 1 and at most g-1, and once again the key technical tool is that this intersection can be shown to be transverse, whence finite, so C(Q) is finite and you have Mordell’s conjecture in the case r < g.  And, as Coleman observed decades after Chabauty, this method even allows you to get an explicit bound on the number of points of C(Q), though not an effective way to compute them.

I think this is a very cool confluence indeed!  In the last ten years we've seen a huge amount of work refining Chabauty; Matt Baker discusses some of it on his blog, and then there’s the whole nonabelian Chabauty direction launched by Minhyong Kim and pushed forward by Jen Balakrishnan and Netan Dogra and many others.  Are there other situations in which we can get meaningful results from “deformation-theoretic Chabauty,” and are the new technical advances in Chabauty methods relevant in this context?

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