I’ve been thinking about his talk all weekend and I’m just going to write down a bit about what I learned. In a liquid crystal, the molecules are like little rods; they have an orientation and nearby molecules want to have nearby orientations. In a *nematic crystal*, that’s all that’s going on — the state of the crystal in some region B is given by a line field on B. A *smectic crystal* has a little more to it — here, the rods are aligned into layers

(image via this handy guide to liquid crystal phases)

separated by — OK, I’m not totally clear on whether they’re separated by a sheet of *something else* or whether that’s just empty space. Doesn’t matter. The point is, this allows you to tell a really interesting topological story. Let’s focus on a smectic crystal in a simply connected planar region B. At every point of B, you have, locally, a structure that looks like a family of parallel lines in the plane, each pair of lines separated by a unit distance. (The unit is the length of the molecule, I think.)

Alternatively, you can think of such a “local smectic structure” as a line in the plane, where we consider two lines equivalent if they are parallel and the distance between them is an integer. What’s the moduli space M — the “ground state manifold” — of such structures? Well, the line family has a direction, so you get a map from M to S^1. The lines in a given direction are parametrized by a line, and the equivalence relation mods out by the action of a lattice, so the fiber of M -> S^1 is a circle; in fact, it’s not hard to see that this surface M is a Klein bottle.

Of course this map might be pretty simple. If B is the whole plane, you can just choose a family of parallel lines on B, which corresponds to the constant map. Or you can cover the plane with concentric circles; the common center doesn’t have a smectic structure, and is a *defect*, but you can map B = R^2 – 0 to M. Homotopically, this just gives you a path in M, i.e. an element of pi_1(M), which is a semidirect product of Z by Z, with presentation

The concentric circle smectic corresponds the map which sends the generator of pi_1(B) to F.

So already this gives you a nice topological invariant of a plane smectic with k defects; you get a map from pi_1(B), which is a free group on k generators, to pi_1(M). Note also that there’s a natural notion of equivalence on these maps; you can “stir” the smectic, which is to say, you can apply a diffeomorphism of the punctured surface, which acts by precomposition on pi_1(B). The action of (the connected components of) Diff(B) on Hom(pi_1(B), pi_1(M)) is my favorite thing; the Hurwitz action of a mapping class group on the space of covers of a Riemann surface! In particular I think the goal expressed in Chen et al’s paper of “extending our work to the topology of such patterns on surfaces of nontrivial topology (rather than just the plane)” will certainly involve this story. I think in this case the Hurwitz orbits are pretty big; i.e. if what you know is the local appearance of the defects (i.e. the image in pi_1(M) of the conjugacy class in pi_1(B) corresponding to the puncture) you should *almost* be able to reconstruct the homotopy type of the map (up to stirring.) If I understood Randy correctly, those conjugacy classes are precisely what you can actually measure in an experiment.

There’s more, though — a lot more! You can’t just choose a map from B to M and make a smectic out of it. The layers won’t line up! There’s a differential criterion. This isn’t quite the way they express it, but I think it amounts to the following: the tangent bundle of M has a natural line bundle L sitting inside it, consisting of those directions of motion that move a line parallel to itself. I think you want to consider only those maps from B to M such that the induced map on tangent bundles TB -> TM takes image in L. More concretely, in coordinates, I think this means the following: if you think of the local smectic structure at p as the preimage of Z under some real-valued function f in the neighborhood of p, then f should satisfy

This restricts your maps a lot, and it accounts for all kinds of remarkable behavior. For one thing, it forbids certain conjugacy classes in pi_1(M) from appearing as local monodromy; i.e. the set of possible defect types is strictly smaller than the set of conjugacy classes in pi_1(M). Moreover, it forbids certain kinds of defects from colliding and coalescing — for algebraic geometers, this naturally makes you feel like there’s a question about *boundaries* of Hurwitz spaces floating around.

Best of all, the differential equation forces the appearance of families of parallel ellipses, involute spirals, and other plane curves of an 18th century flavor. The cyclides of Dupin put in an appearance. Not just in the abstract — in actual liquid crystals! There are pictures! This is great stuff.

**Update:** Wait a minute — I forgot to say anything about fingerprints! Maybe because I don’t have anything to say at the moment. Except that the lines of a fingerprint are formally a lot like the lines of a smectic crystal, the defects can be analyzed in roughly the same way, etc. Whether the diffeomorphism type of a fingerprint is an interesting forensic invariant I don’t rightly know. I’ll bet whoever made my iPhone home button knows, though.

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No! In fact, we show there are O(n^{3/2}). (Megyesi and Szabo had previously given an upper bound of the form n^{2-delta} in the case where the curves are all conics.)

Is n^{3/2} best possible? Good question. The best known lower bound is given by a configuration of n circles with about n^{4/3} mutual tangencies.

Here’s the main idea. If a curve C starts life in A^2, you can lift it to a curve C’ in A^3 by sending each point (x,y) to (x,y,z) where z is the slope of C at (x,y); of course, if multiple branches of the curve go through (x,y), you are going to have multiple points in C’ over (x,y). So C’ is isomorphic to C at the smooth points of C, but something’s happening at the singularities of C; basically, you’ve blown up! And when you blow up a tacnode, you get a regular node — the two branches of C through (x,y) have the same slope there, so they remain in contact even in C’.

Now you have a bunch of bounded degree curves in A^3 which have an unexpectedly large amount of intersection; at this point you’re right in the mainstream of incidence geometry, where incidences between points and curves in 3-space are exactly the kind of thing people are now pretty good at bounding. And bound them we do.

Interesting to let one’s mind wander over this stuff. Say you have n curves of bounded degree. So yes, there are roughly n^2 intersection points — generically, these will be distinct nodes, but you can ask how non-generic can the intersection be? You have a partition of const*n^2 coming from the multiplicity of intersection points, and you can ask what that partition is allowed to look like. For instance, how much of the “mass” can come from points where the multiplicity of intersection is at least r? Things like that.

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- The space of quadrilaterals in R^2, up to the action of affine linear transformations, is basically just R^2, right? Because you can move three vertices to (0,0), (0,1), (1,0) and then you’re basically out of linear transformations. And the property “can be decomposed into n triangles of equal area” is invariant under those transformations. OK, so — for which choices of the “fourth vertex” do you get a quadrilateral that has a decomposition into an odd number of equal-area triangles? (I think once you’re not a parallelogram you lose the easy decomposition into 2 equal area triangles, so I suppose generically maybe there’s NO equal-area decomposition?) When do you have a decomposition into triangles whose area has odd denominator?
- What if you replace the square with the torus R^2 / Z^2; for which n can you decompose the torus into equal-area triangles? What about a Riemann surface with constant negative curvature? (Now a “triangle” is understood to be a geodesic triangle.) If I have this right, there are plenty of examples of such surfaces with equal-area triangulations — for instance, Voight gives lots of examples of Shimura curves corresponding to cocompact arithmetic subgroups which are finite index in triangle groups; I think that lets you decompose the Riemann surface into a union of fundamental domains each of which are geodesic triangles of the same area.

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I share this dream with you mostly because I think Bugbiter is actually a legitimately good band name.

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**Update:** In the 34th percentile at ESPN after one day of play — thanks, Yale!

**Update: **Down to the 5th percentile and only Duke and UVa are left out of my final 8 picks. Not gonna be the math bracket’s finest year.

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There are a bunch of cool pictures on the SIAGA flyer: UC-Berkeley grad student Anna Seigal is explaining them on her blog, one by one. The first entry depicts the algebraic variety you encounter when you try to find a point minimizing the summed distances to k other points in the plane. Handsome!

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You might think this would be clear. On November 4, 1924, voters in Wisconsin overwhelmingly approved the Home Rule Amendment, which added to the state Constitution:

Cities and villages organized pursuant to state law may determine their local affairs and government, subject only to this constitution and to such enactments of the legislature of statewide concern as with uniformity shall affect every city or every village. The method of such determination shall be prescribed by the legislature.

It turns out it hasn’t been so simple, in practice, to figure out what those 51 words mean. In a recent high-profile case, the Wisconsin Supreme Court upheld Act 10, Governor Walker’s signature legislation; among other things, the law forbade Milwaukee from contributing to its employees’ pension funds. The plaintiffs argued that this provision violated home rule. Michael Gableman, writing for the court majority, said it was fine.

This raises questions. First of all: if a state law needs to affect every city uniformly in order to supersede local government, how can it be OK to specifically target Milwaukee’s pension fund? Here the exact wording of 62.623 is critical. The law doesn’t mention Milwaukee: it applies to “any employee retirement system of a 1st class city.” The “uniformity” requirement in the Home Rule amendment has generally been understood very liberally, allowing laws which affect cities in different size classes differently as long as the application *within* each class is uniform.

To construe the amendment as meaning that every act of the Legislature relating to cities is subject to a charter ordinance unless the act of the Legislature affected with uniformity every city from the smallest to the greatest, practically destroys legislative control over municipal affairs, assuming that laws could be drawn which would meet the requirements of the amendment so construed.

That’s from *Van Gilder v. City of Madison* (1936), one of the first Wisconsin Supreme Court cases to wrestle with the limits of home rule. I will have more to say about Chief Justice Marvin Rosenberry’s decision in that case, some of it pretty salty. But for now let’s stick to the point at hand. The law can be argued to pass the “uniformity” test because it applies equally to all cities of the first class. There is only one city of the first class in Wisconsin, and there has only ever been one city of the first class in Wisconsin, and it’s Milwaukee.

That’s the argument the Walker administration made in defense of the law. But the court’s upholding the law rejects that defense, and the uniformity clause as a whole, as irrelevant the question before it.

In sum, our home rule case law instructs us that, when reviewing a legislative enactment under the home rule amendment, we apply a two-step analysis. First, as a threshold matter, the court determines whether the statute concerns a matter of primarily statewide or primarily local concern. If the statute concerns a matter of primarily statewide interest, the home rule amendment is not implicated and our analysis ends. If, however, the statute concerns a matter of primarily local affairs, the reviewing court then examines whether the statute satisfies the uniformity requirement. If the statute does not, it violates the home rule amendment.

Thus:

no merit exists in the plaintiffs’ contention that the legislative enactment at issue in a home rule challenge must be a matter of statewide concern and uniformly applied statewide to withstand constitutional scrutiny.

Now this is weird, right? Because what’s described and rejected as “the plaintiff’s contention” is what the constitution says. Gableman replaces the Constitution’s **and **with an **or**: in his analysis, a state law supersedes local powers if it’s *either* of statewide concern or applied uniformly to all cities.

Is this an act of wanton judicial activism? Well, not exactly. The phrase “as home rule case law instructs us” is important here. The opinion marshals a long line of precedents showing that the Home Rule amendment has typically been read as an or, not an and. It goes all the way back to Rosenberry’s opinion in *Van Gilder v. City of Madison*; and the reason there’s such a long list is that all those other cases rely on *Van Gilder, *which has become the foundation of Wisconsin’s theory of home rule.

Which brings us to the main point. I’m not a legal scholar, but what the hell, this is blogging, I get to have an opinion, and here’s mine: *Van Gilder v. City of Madison***was wrongly decided and has been screwing up home rule jurisprudence for 80 years.**

Rosenberry’s first go at explaining home rule goes like this:

The home–rule amendment certainly confers upon cities plenary powers to deal with local affairs and government subject to the limitations contained in the amendment itself and other provisions of the Constitution. The powers of municipalities are subject to the limitation that the municipality cannot by its charter deal with matters which

are of state–wide concern and its power to enact an organic law dealing with local affairs and government is subject to such acts of the Legislature relating thereto as are of state–wide concern and affect with uniformity all cities.

The “and” between statewide concern and uniformity is clear here. But Rosenberry also says that municipalities simply have no power to address matters of statewide concern: its powers, he says, are restricted to “local affairs and government” *as distinct* from matters of statewide concern. So what cases are the second clause (“its power to enact an organic law….”) referring to? Only those matters which are *not* of statewide concern, but which are affected by state laws which *are* of statewide concern. Rosenberry gives no examples of such a situation, nor can I really imagine one, so I don’t think that’s really the conclusion he means to draw. Later in the opinion, he settles more clearly on the policy adopted by Gableman in *Madison Teachers Inc. v. Walker:*

when the Legislature deals with local affairs as distinguished from

matters which are primarily of state–wide concern, it can only do so effectually by an act which affects with uniformity every city. It is true that this leaves a rather narrow field in which the home–rule amendment operates freed from legislative restriction, but there is no middle ground.

and

the limitation contained in the section upon the power of the Legislature is a limitation upon its power to deal with the local affairs and government of a city or village. Care must be taken to distinguish between the power of the Legislature to deal with local affairs and its power to deal with matters primarily of state–wide concern. When the Legislature deals with local affairs and government of a city, if its act is not to be subordinate to a charter ordinance, the act must be one which affects with uniformity every city. If in dealing with the local affairs of a city the Legislature classifies cities so that the act does not apply with uniformity to every city, that act is subordinate to a charter ordinance relating to the same matter. A charter ordinance of a city is not subject to an act of the Legislature dealing with local affairs unless the act affects with uniformity every city. State ex rel. Sleeman v. Baxter, supra. When the Legislature deals with matters which are primarily matters of state–wide concern, it may deal with them free from any restriction contained in the home rule amendment.

Now the ground has shifted. In Rosenberry’s reading, when the home rule amendment refers to “local affairs and government” it specifically intends to exclude any “matters of statewide concern.” I can accept this as a reading of those four words, but not as a reading of the whole sentence. If Roseberry is correct, then the phrase “of statewide concern” is *never active* in the amendment: a local affair is, by definition, not a matter of statewide concern. I think when your interpretation of a constitutional passage means that part of the text never applies, you need to think twice about your interpretation.

What’s more, Rosenberry holds that the state has the power to override local officials on *purely* local matters, of no statewide concern whatsoever, as long as it does so uniformly. If that is so, what does he think the words “of statewide concern” are doing in the Home Rule amendment at all?

To me, the amendment has a pretty plain meaning. Something like a residency requirement for city employees or a fiscal decision about a city pension plan is plainly a local affair. It may also be a matter of statewide concern. The state legislature can enact a law overriding local legislation if the matter is of statewide concern **and** the law in question applies uniformly to all cities. I think Rosenberry just plain got this wrong in *Van Gilder* and it’s been wrong ever since.

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Nina Konstantinovnva, a literature teacher:

I teach Russian literature to kids who are not like the kids I taught ten years ago. They are constantly seeing someone or something get buried, get placed underground. Houses and trees, everything gets buried. If they stand in line for fifteen, twenty minutes, some of them start fainting, their noses bleed. You can’t surprise them with anything and you can’t make them happy. They’re always tired and sleepy. Their faces are pale and gray. They don’t play and they don’t fool around. If they fight or accidentally break a window, the teachers are pleased. We don’t yell at them, because they’re not like kids. And they’re growing so slowly. You ask them to repeat something during a lesson, and the child can’t, it gets to the point where you simply ask him to repeat a sentence, and he can’t. You want to ask him, “Where are you? Where?”

Major Oleg Pavlov, a helicopter pilot:

Every April 26 we get together, the guys who were there. We remember how it was. You were a soldier, at war, you were necessary. We forget the bad parts and remember that. We remember that they couldn’t have made it without us. Our system, it’s a military system, essentially, and it works great in emergencies. You’re finally free there, and necessary. Freedom! And in those times the Russian shows how great he is. How unique. We’ll never be Dutch or German. And we’ll never have proper asphalt or manicured lawns. But there’ll always be plenty of heroes.

Translated by Keith Gessen.

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