## Smectic crystals, fingerprints, Klein bottles

Amazing colloquium this week by Randall Kamien, who talked about this paper with Chen and Alexander, this one with Liarte, Bierbaum, Mosna, and Sethna, and other stuff besides.

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

$\langle S,F: FSF^{-1} = S^{-1} \rangle$

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

$(df/dx)^2 + (df/dy)^2 = 1.$

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.

• Algebraists eat corn row by row, analysts eat corn circle by circle.  Yep, I eat down the rows like a typewriter.  Why?  Because it is the right way.
• This short paper by Johan de Jong and Wei Ho addresses an interesting question I’d never thought about; does a Brauer-Severi variety over a field K contain a genus-1 curve defined over K?  They show the answer is yes in dimensions up to 4 (3 and 4 being the new cases.)  In dimension 1, this just asks about covers of Brauer-Severi curves by genus 1 curves; I remember this kind of situation coming up in Ekin Ozman’s thesis, where certain twists of modular curves end up being covers of Brauer-Severi curves.  Which Brauer-Severi varieties are split by twisted modular curves?
• Always nice to see a coherent description of the p-adic numbers in the popular press; and George Musser delivers the goods in Scientific American, in the context of recent work in cosmology by Harlow, Shenker, Stanford, and Susskind.  Two quibbles:  first, if I understood Susskind’s talk on this stuff correctly, the point is to model things by an infinite regular tree.  The fact that when the out-degree is a prime power this happens to look like the Bruhat-Tits tree is in some sense tangential, though very useful for getting an intuitive picture of what’s going on — as long as your intuition is already p-adic, of course!  Second quibble is that Musser seems to suggest at the end that p-adic distances can’t get arbitrarily small:

On top of that, distance is always finite. There are no p-adic infinitesimals, or infinitely small distances, such as the dx and dy you see in high-school calculus. In the argot, p-adics are “non-Archimedean.” Mathematicians had to cook up a whole new type of calculus for them.

Prior to the multiverse study, non-Archimedeanness was the main reason physicists had taken the trouble to decipher those mathematics textbooks. Theorists think that the natural world, too, has no infinitely small distances; there is some minimal possible distance, the Planck scale, below which gravity is so intense that it renders the entire notion of space meaningless. Grappling with this granularity has always vexed theorists. Real numbers can be subdivided all the way down to geometric points of zero size, so they are ill-suited to describing a granular space; attempting to use them for this purpose tends to spoil the symmetries on which modern physics is based.

## Higgs

Since someone asked me today: yes, the Stanley Higgs who appears in my novel was named after the Higgs boson. I thought it would add a very slight tinge of cosmic mystery to the character. Not any more, I guess.

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## Put the second law of thermodynamics down and slowly step away, New York Times

Yet given her professional background, Dr. Oakley couldn’t help doubting altruism’s exalted reputation. “I’m not looking at altruism as a sacred thing from on high,” she said. “I’m looking at it as an engineer.”

And by the first rule of engineering, she said, “there is no such thing as a free lunch; there are always trade-offs.” If you increase order in one place, you must decrease it somewhere else.

Moreover, the laws of thermodynamics dictate that the transfer of energy will itself exact a tax, which means that the overall disorder churned up by the transaction will be slightly greater than the new orderliness created. None of which is to argue against good deeds, Dr. Oakley said, but rather to adopt a bit of an engineer’s mind-set, and be prepared for energy losses and your own limitations.

Stop hurting physics!

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## How the Hippies Saved Physics: David Kaiser visits UW April 1

David Kaiser from MIT will give Physics colloquium next Friday, April 1, about his appealing new book How the Hippies Saved Physics.  Did you ever see a dog-eared copy of The Dancing Wu Li Masters at a used book store?  It’s about those guys, and how their thoughts turned out to be not totally irrelevant to academic physics after all.  He’s also talking in History of Science at noon, about the Cold War explosion, credentialization, and finally contraction of the physics profession, a story told in his book in progress American Physics and the Cold War Bubble. Both will be interesting, I bet.

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## Bill Foster, D-Physics

Bill Foster,  U.S. Representative from the 14th district of Illinois and a former Fermilab physicist, gave the physics colloquium here last week.  He, like Cass Sunstein, feels that American politics is getting more polarized, and blames the large number of “safe districts,” carefully drawn to ensure long tenure for incumbents of whatever party.  The idea is that a Democrat in a Democratic district doesn’t have to seriously defend against a general-election opponent, but only to guard the left flank against a more radical primary opponent.  Thus, almost everyone has an incentive to move away from the center.  Good theory, though not true according to my favorite quantitative poli-sci team.

Foster talked a bit about growing up in the 1950s in a fiercely Democratic family in what was a much more Republican Madison.  His parents met on Capitol Hill working for Democratic Senators, and his sister‘s middle name is “Adlai.”  I’ve heard people say — and I have no idea whether this statement has empirical support — that women in math are disproportionately likely to have a parent who’s an academic mathematician; presumably the presence of an actual mathematician in the house does something to counteract cultural stereotypes about math.  I wonder whether the same is true for scientists in politics?  Rush Holt, like Foster a physicist turned Congressman, is the son of a U.S. Senator and a West Virginia Secretary of State.  The web tells me nothing about the family history of Jerry McNerney, the only member of Congress with a Ph.D. in math.

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## Do you like Richard Feynman?

Twice in the last few weeks I’ve had heated discussions with friends about Richard Feynman — more precisely, about the character of Feynman as he presents himself in his popular memoirs Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?

I read these books as a kid and found the experience profoundly off-putting — like being trapped in a room for hours with a guy who keeps saying, “You know, that reminds me of yet another occasion on which small-minded types were first startled, then chastened, by my unconventional yet plainly superior approach!” The second book, in particular, might better have been titled What Do You Care What Other People Think? Other People Are Stupid! It was very popular with the toxic nerds who liked to refer to science fiction non-fans as “mundanes.”

And yet: I have recently found that lots of gentle and thoughtful grownups find the protagonist of these books charming, and even admirable. So again, I ask: am I the weird one here? Am I selectively remembering these books as meaner than they were? Are normal people fond of Feynman?

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## The bipartisan physics caucus grows by one

Bill Foster, a particle physicist at Fermilab, won a special election in Illinois last week to become the newest member of the House of Representatives. He’s the third physicist in Congress! The other two are Rush Holt (D-NJ), a former professor at Swarthmore and assistant director of the Princeton Plasma Physics Lab, and Vern Ehlers (R-MI), who has a Ph.D. in nuclear physics from Berkeley and used to chair the physics department at Calvin College in Grand Rapids.

Why three physicists and no mathematicians? Maybe because physicists are much more likely to have experience overseeing large pots of money — and by extension, to have experience asking people for large pots of money, a critical skill for aspirants to public office.

There is one Congressman with a Ph.D. in math, first-termer Jerry McNerney (D-CA). But in his professional life he’s an engineer. And algebraic topologist Daniel Biss will be a Democratic candidate this fall for State Representative in Illinois. Are there any other mathematicians in state legislatures?

If you’re running for office and want to cash in on this physics trend, you might start with Chad Orzel’s enjoyable post explaining how to talk like a physicist. Some of his suggestions are indeed things I say all the time, without thinking of them as markers of membership in the math tribe: for instance, referring to difficult things as “non-trivial,” or bad things as “sub-optimal.”