I’ve supervised a lot of student projects in my nine years at MIT, but my inner nerdy teenager has never been as *personally* delighted by a project as it is right now. Today, I’m proud to announce that Adam Yedidia, a PhD student at MIT (but an MEng student when he did most of this work), has explicitly constructed a one-tape, two-symbol Turing machine with 7,918 states, whose behavior (when run on a blank tape) can never be proven from the usual axioms of set theory, under reasonable consistency hypotheses. Adam has also constructed a 4,888-state Turing machine that halts iff there’s a counterexample to Goldbach’s Conjecture, and a 5,372-state machine that halts iff there’s a counterexample to the Riemann hypothesis. In all three cases, this is the first time we’ve had an explicit upper bound on how many states you need in a Turing machine before you can see the behavior in question.

Here’s our research paper, on which Adam generously included me as a coauthor, even though he did the heavy lifting. Also, here’s a github repository where you can download all the code Adam used to generate these Turing machines, and even use it to build your own small Turing machines that encode interesting mathematical statements. Finally, here’s a YouTube video where Adam walks you through how to use his tools.

A more precise statement of our main result is this: we give a 7,918-state Turing machine, called Z (and actually explicitly listed in our paper!), such that:

- Z runs forever, assuming the consistency of a large-cardinal theory called SRP (Stationary Ramsey Property), but
- Z can’t be
*proved*to run forever in ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice, the usual foundation for mathematics), assuming that ZFC is consistent.

A bit of background: it follows, as an immediate consequence of Gödel’s Incompleteness Theorem, that there’s *some* computer program, of *some* length, that eludes the power of ordinary mathematics to prove what it does, when it’s run with an unlimited amount of memory. So for example, such a program could simply enumerate all the possible consequences of the ZFC axioms, one after another, and halt if it ever found a contradiction (e.g., a proof of 1+1=3). Assuming ZFC is consistent, this program must run forever. But again assuming ZFC is consistent, ZFC can’t *prove* that the program runs forever, since if it did, then it would prove its own consistency, thereby violating the Second Incompleteness Theorem!

Alas, this argument still leaves us in the dark about *where*, in space of computer programs, the “Gödelian gremlin” rears its undecidable head. A program that searches for an inconsistency in ZFC is a fairly complicated animal: it needs to encode not only the ZFC axiom schema, but also the language and inference rules of first-order logic. Such a program might be thousands of lines long if written in a standard programming language like C, or millions of instructions if compiled down to a bare-bones machine code. You’d certainly never run across such a program by chance—not even if you had a computer the size of the observable universe, trying one random program after another for billions of years in a “primordial soup”!

So the question stands—a question that strikes me as *obviously* important, even though as far as I know, only one or two people ever asked the question before us; see here for example. Namely: do the axioms of set theory suffice to analyze the behavior of every computer program that’s at most, let’s say, 50 machine instructions long? Or are there super-short programs that *already* exhibit “Gödelian behavior”?

Theoretical computer scientists might object that this is “merely a question of constants.” Well yes, OK, but the origin of life in our universe—a not entirely unrelated puzzle—is also “merely a question of constants”! In more detail, we know that it’s *possible* with our laws of physics to build a self-replicating machine: say, DNA or RNA and their associated paraphernalia. We also know that tiny molecules like H_{2}O and CO_{2} are not self-replicating. But we don’t know *how small* the smallest self-replicating molecule can be—and that’s an issue that influences whether we should expect to find ourselves alone in the universe or find it teeming with life.

Some people might also object that what we’re asking about has already been studied, in the half-century quest to design the smallest universal Turing machine (the subject of Stephen Wolfram’s $25,000 prize in 2007, to which I responded with my own $25.00 prize). But I see that as fundamentally different, for the following reason. A universal Turing machine—that is, a machine that simulates any other machine that’s described to it on its input tape—has the privilege of offloading almost all of its complexity onto the description format for the input machine. So indeed, that’s exactly what all known tiny universal machines do! But a program that checks (say) Goldbach’s Conjecture, or the Riemann Hypothesis, or the consistency of set theory, on an initially blank tape, has no such liberty. For such machines, the number of states really *does* seem like an intrinsic measure of complexity, because the complexity can’t be shoehorned anywhere else.

One can also phrase what we’re asking in terms of the infamous Busy Beaver function. Recall that BB(n), or the n^{th} Busy Beaver number, is defined to be the maximum number of steps that any n-state Turing machine takes when run on an initially blank tape, assuming that the machine eventually halts. The Busy Beaver function was the centerpiece of my 1998 essay Who Can Name the Bigger Number?, which *might* still attract more readers than anything else I’ve written since. As I stressed there, if you’re in a biggest-number-naming contest, and you write “BB(10000),” you’ll *destroy* any opponent—however otherwise mathematically literate they are—who’s innocent of computability theory. For BB(n) grows faster than any computable sequence of integers: indeed, if it didn’t, then one could use that fact to solve the halting problem, contradicting Turing’s theorem.

But the BB function has a second amazing property: namely, it’s a perfectly well-defined integer function, and yet once you fix the axioms of mathematics, only finitely many values of the function can ever be *proved*, even in principle. To see why, consider again a Turing machine M that halts if and only if there’s a contradiction in ZF set theory. Clearly such a machine could be built, with some finite number of states k. But then ZF set theory can’t possibly determine the value of BB(k) (or BB(k+1), BB(k+2), etc.), unless ZF is inconsistent! For to do so, ZF would need to prove that M ran forever, and therefore prove its own consistency, and therefore be inconsistent by Gödel’s Theorem.

OK, but we can now ask a quantitative question: *how many* values of the BB function is it possible for us to know? Where exactly is the precipice at which this function “departs the realm of mortals and enters the realm of God”: is it closer to n=10 or to n=10,000,000? In practice, *four* values of BB have been determined so far:

- BB(1)=1
- BB(2)=6
- BB(3)=21 (Lin and Rado 1965)
- BB(4)=107 (Brady 1975)

We also know two lower bounds: BB(5) ≥ 47,176,870 (Marxen and Buntrock 1990), and BB(6) ≥ 7.4 × 10^{36,534}. See Heiner Marxen’s page for more.

Some Busy Beaver enthusiasts have opined that even BB(6) will never be known exactly. On the other hand, the abstract argument from before tells us only that, if we confine ourselves to (say) ZF set theory, then there’s *some* k—possibly in the tens of millions or higher—such that the values of BB(k), BB(k+1), BB(k+2), and so on can never be proven. So again: is the number of knowable values of the BB function more like 10, or more like a million?

This is the question that Adam and I (but mostly Adam) have finally addressed.

It’s hopeless to design a Turing machine by hand for all but the simplest tasks, so as a first step, Adam created a new programming language, called Laconic, specifically for writing programs that compile down to small Turing machines. Laconic programs actually compile to an intermediary language called TMD (Turing Machine Descriptor), and from there to Turing machines.

Even then, we estimate that a direct attempt to write a Laconic program that searched for a contradiction in ZFC would lead to a Turing machine with millions of states. There were three ideas needed to get the state count down to something reasonable.

The first was to take advantage of the work of Harvey Friedman, who’s one of the one or two people I mentioned earlier who’s written about these problems before. In particular, Friedman has been laboring since the 1960s to find “natural” arithmetical statements that are provably independent of ZFC or other strong set theories. (See this *AMS Notices* piece by Martin Davis for a discussion of Friedman’s progress as of 2006.) Not only does Friedman’s quest continue, but some of his most important progress has come only within the last year. His statements—typically involving objects called “order-invariant graphs”—strike me as alien, and as far removed from anything number theorists or combinatorialists have thus far had independent reasons to think about (but is that just a sign of their limited perspectives?). Be that as it may, Friedman’s statements *still* seem a lot easier to encode as short computer programs than the full apparatus of first-order logic and set theory! So that’s what we started with; our work wouldn’t have been possible without Friedman (who we consulted by email throughout the project).

The second idea was something we called “on-tape processing.” Basically, instead of compiling directly from Laconic down to Turing machine, Adam wrote an *interpreter* in Turing machine (which took about 4000 states—a single, fixed cost), and then had the final Turing machine first write a higher-level program onto its tape and then interpret that program. Instead of the compilation process producing a huge multiplicative overhead in the number of Turing machine states (and a repetitive machine), this approach gives us only an additive overhead. We found that this one idea decreased the number of states by roughly an order of magnitude.

The third idea was first suggested in 2002 by Ben-Amram and Petersen (and refined for us by Luke Schaeffer); we call it “introspective encoding.” When we write the program to be interpreted onto the Turing machine tape, the naïve approach would use one Turing machine state per bit. But that’s clearly wasteful, since in an n-state Turing machine, every state contains ~log(n) bits of information (because of the other states it needs to point to). A better approach tries to exploit as many of those bits as it can; doing that gave us up to a factor-of-5 additional savings in the number of states.

For Goldbach’s Conjecture and the Riemann Hypothesis, we paid the same 4000-state overhead for the interpreter, but then the program to be interpreted was simpler, giving a smaller overall machine. Incidentally, it’s not intuitively obvious that the Riemann Hypothesis is equivalent to the statement that some particular computer program runs forever, but it is—that follows, for example, from work by Lagarias (which we used).

To preempt the inevitable question in the comments section: yes, we *did* run these Turing machines for a while, and no, none of them had halted after a day or so. But before you interpret that as evidence in favor of Goldbach, Riemann, and the consistency of ZFC, you should probably know that a Turing machine to test whether *all perfect squares are less than 5*, produced using Laconic, needed to run for more than an hour before it found the first counterexample (namely, 3^{2}=9) and halted. Laconic Turing machines are optimized only for the number of states, not for speed, to put it mildly.

Of course, three orders of magnitude still remain between the largest value of n (namely, 4) for which BB(n) is known to be knowable in ZFC-based mathematics, and the smallest value of n (namely, 7,918) for which BB(n) is known to be unknowable. I’m optimistic that further improvements are possible to the machine Z—whether that means simplifications to Friedman’s statement, a redesigned interpreter (possibly using lambda calculus?), or a “multi-stage rocket model” where a bare-bones interpreter would be used to unpack a second, richer interpreter which would be used to unpack a third, etc., until you got to the actual program you cared about. But I’d be *shocked* if anyone in my lifetime determined the value of BB(10), for example, or proved the value independent of set theory. Even after the Singularity happens, I imagine that our robot overlords would find the determination of BB(10) quite a challenge.

In an early *Shtetl-Optimized* post, I described theoretical computer science as “quantitative epistemology.” Constructing small Turing machines whose behavior eludes set theory is not conventional theoretical computer science by any stretch of the imagination: it’s closer in practice to programming languages or computer architecture, or even the recreational practice known as code-golfing. On the other hand, I’ve never been involved with any project that was so clearly, explicitly about pinning down the quantitative boundary between the knowable and the unknowable.

Comments on our paper are welcome.

**Addendum:** Some people might wonder “why Turing machines,” as opposed to a more reasonable programming language like C or Python. Well, first of all, we needed a language that could address an unlimited amount of memory. Also, the BB function is traditionally defined in terms of Turing machines. But the most important issue is that we wanted there to be *no suspicion whatsoever* that our choice of programming language was artificially helping to make our machine small. And hopefully everyone can agree that one-tape, two-symbol Turing machines aren’t designed for *anyone’s* convenience!

I am happy to announce that this year we will run the 5th international conference on *Experimental Search for Quantum Gravity* here in Frankfurt, Germany. The meeting will take place Sep 19-23, 2016.

We have a (quite preliminary) website up here. Application is now open and will run through June 1st. If you're a student or young postdoc with an interest in the phenomenology of quantum gravity, this conference might be a good starting point and I encourage you to apply. We cannot afford handing out travel grants, but we will waive the conference fee for young participants (young in terms of PhD age, not biological age).

The location of the meeting will be at my new workplace, the Frankfurt Institute for Advanced Studies, FIAS for short. When it comes to technical support, they seem considerably better organized (not to mention staffed) than my previous institution. At this stage I am thus tentatively hopeful that this year we'll both record and livestream the talks. So stay tuned, there's more to come.

We have a (quite preliminary) website up here. Application is now open and will run through June 1st. If you're a student or young postdoc with an interest in the phenomenology of quantum gravity, this conference might be a good starting point and I encourage you to apply. We cannot afford handing out travel grants, but we will waive the conference fee for young participants (young in terms of PhD age, not biological age).

The location of the meeting will be at my new workplace, the Frankfurt Institute for Advanced Studies, FIAS for short. When it comes to technical support, they seem considerably better organized (not to mention staffed) than my previous institution. At this stage I am thus tentatively hopeful that this year we'll both record and livestream the talks. So stay tuned, there's more to come.

If you’re not British, or you live under a stone somewhere, then you may not have heard about one of the most extraordinary sporting stories ever. Leicester City, a football (in the British sense) team that last year only just escaped relegation from the top division, has just won the league. At the start of […]

If you’re not British, or you live under a stone somewhere, then you may not have heard about one of the most extraordinary sporting stories ever. Leicester City, a football (in the British sense) team that last year only just escaped relegation from the top division, has just won the league. At the start of the season you could have bet on this happening at odds of 5000-1. Just 12 people availed themselves of this opportunity.

Ten pounds bet then would have net me 50000 pounds now, so a natural question arises: should I be kicking myself (the appropriate reaction given the sport) for not placing such a bet? In one sense the answer is obviously yes, as I’d have made a lot of money if I had. But I’m not in the habit of placing bets, and had no idea that these odds were being offered anyway, so I’m not too cut up about it.

Nevertheless, it’s still interesting to think about the question hypothetically: if I *had* been the betting type and had known about these odds, should I have gone for them? Or would regretting not doing so be as silly as regretting not choosing and betting on the particular set of numbers that just happened to win the national lottery last week?

Here’s a possible argument that the 5000-1 odds at the beginning of the season were about right, or at least not too low, and an attempted explanation of why hardly anybody bet on Leicester. If you’ve watched football for any length of time, you know that the league is dominated by the big clubs, with their vast resources to spend on top players and managers. Just occasionally a middle-ranking club has a surprisingly good season and ends up somewhere near the top. But a bottom-ranking club that hasn’t just been lavished with money doesn’t become a top club overnight, and since to win the league you have to do consistently well over an entire season, it just ain’t gonna happen that a club like Leicester will win.

And here are a few criticisms of the above argument.

1. The argument that we know how things work from following the game for years or even decades is convincing if all you want to prove is that it is very unlikely that a team like Leicester will win. But here we want to prove that the odds are not just low, but one-in-five-thousand low. What if the probability of it happening in any given season were 100 to 1? We haven’t had many more than 100 seasons ever, so we might well never have observed what we observed this season.

2. The argument that consistency is required over a whole season is a very strong one if the conclusion to be established is that a mediocre team will almost never win. Indeed, for a mediocre team to beat a very good team some significantly good luck is required. And the chances of that kind of luck happening enough times during a season for the team to win the league are given by the tail of a binomial distribution, so they are tiny.

However, in practice it is not at all true that results of different matches are independent. Once Leicester had won a few matches against far bigger and richer clubs, a simple Bayesian calculation would have shown that it was far more likely that Leicester had somehow become a much better team since last season than that it had won those matches by a series of independent flukes. I think the bookmakers probably made a big mistake by offering odds of 1000-1 three months into the season, at which point Leicester were top. Of course we all expected them to fall off, but were we 99.9% sure of that? Surely not. (I think if I’d known about those odds, I probably would have bet £20 or so. Oh well.)

3. Although it was very unlikely that Leicester would suddenly become far better, there were changes, such as a new manager and some unheralded new players who turned out to be incredibly good. How unlikely is it that a player who has caught someone’s eye will be much better than anybody expected? Pretty unlikely but not impossible, I’d have thought: it’s quite common for players to blossom when they move to a new club.

4. The fact that Leicester had a remarkable escape from relegation at the end of last season, winning seven of their last nine matches, was already fairly strong evidence that something had changed (see point 2 above). Had they accumulated their meagre points total in a more uniform manner, it would have reduced the odds of their winning this season.

The first criticism above is not itself beyond criticism, since we have more data to go on than just the English league. If nothing like the Leicester story had happened in any league anywhere in the world since the beginning of the game, then the evidence would be more convincing. But from what I’ve read in the papers the story isn’t *completely* unprecedented: that is, pretty big surprises do just occasionally happen. Though against that, the way that money has come into the game has made the big clubs more dominant in recent years, which would seem to reduce Leicester’s chances.

I’m not going to come to any firm conclusion here, but my instinct is that 5,000-1 was a very good bet to take at the beginning of the season, even without hindsight, and that 1000-1 three months later was an amazing chance. I’m ignoring here the well-known question of whether it is sensible to take unlikely bets just because your expected gain is positive. I’m just wondering whether the expected gain *was* positive. Your back-of-envelope calculations on the subject are welcome …

*[I have been out on vacation; hence the lack of posts recently.]*

Out of the blue came an email from So Hattori (NYUAD), who has found many single transits (long-period planet candidates) in the *Kepler* data. This is awesome! Foreman-Mackey and I discussed the goals for Hattori's project, and its relationships with Foreman-Mackey's current project (which has found a Saturn analog and has some occurrence rate results).

In the morning, I spent time with Leslie Greengard discussing various matters related to the description (on a computer, say) of continuous (and infinitely differentiable) surfaces. There are some outstanding problems, which seem like simple math problems but are unsolved. This has nothing to do with anything I am working on, but I could get hooked. Of course my position is that the determination of a surface given control points ought to be cast as an inference problem!

The remainder of the day was spent planning and outlining my argument in my “Inference of Variance” project: How the question of inferring the variance of a process (that generated some points or data, say) is related to the problem of cosmological parameter estimation, and how we can help the latter with work on the former.

New paper up on the arXiv, with Jozsef Solymosi and Josh Zahl. Suppose you have n plane curves of bounded degree. There ought to be about n^2 intersections between them. But there are intersections and there are intersections! Generically, an intersection between two curves is a node. But maybe the curves are mutually tangent at […]

New paper up on the arXiv, with Jozsef Solymosi and Josh Zahl. Suppose you have n plane curves of bounded degree. There ought to be about n^2 intersections between them. But there are intersections and there are intersections! Generically, an intersection between two curves is a node. But maybe the curves are mutually tangent at a point — that’s a more intense kind of singularity called a *tacnode*. You might think, well, OK, a tacnode is just some singularity of bounded multiplicity, so maybe there could still be a constant multiple of n^2 mutual tangencies.

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.

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 […]

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

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.

As mentioned in passing a little while ago, we spent last week on a Disney cruise in the Caribbean, with the kids and my parents. We had sort of wondered for a while what those trips are like, and since the first reaction of most parents I’ve mentioned it to has been “Oh, we’ve thought about that– let us know how it is,” I figure it’s worth a blog post to say a bit about the trip, which the kids enjoyed just a little bit:

The ship we were on was the Disney Magic, which is the smallest of the Disney ships, and had something like 2500 guests on board (the others hold closer to 4000, I think). It’s got three pools (a wading pool, a 4′ deep kid pool with a giant video screen above it, and a 4′ deep adults-only pool with classy jazz guitar music to drown out the noise from the other two), and two big waterslides. There were also a couple of “kids club” areas in the lower decks, with various activities and counselors to keep kids occupied while their parents relax elsewhere. This being a cruise ship, there were also a million dining options, of which we only used a few– the buffet restaurant at the back of the ship for breakfast and lunch, and three formal dining rooms for dinner. SteelyKid was a big fan of the self-serve soft ice cream machines near the pools, and there were also multiple snack-bar sorts of things in those areas.

The formal dinner arrangement has you rotate through the three dining rooms, but you have the same service team in each place. Our kids are insanely picky eaters, and rejected most of the official menu options, but the servers bent over backwards to accommodate them, bringing a bunch of stuff that wasn’t on the menu. The kids were surprisingly cheerful about dinner, and even sitting around after they had finished waiting for us to eat. It helped that they had some sort of entertainment almost every night, and our waiter helped divert SteelyKid with math-y puzzles (mostly of the “here’s a shape made out of a dozen crayons, move two to make a different shape” variety).

We didn’t make all that much use of the floating day care options, though about once a day we’d drop the kids there for a little while so the adults could shower or otherwise relax. They were pretty cheerful about playing down there, but also perfectly happy to leave when we came to get them. We picked the itinerary to minimize at-sea days in favor of stops at interesting places, and when we were in port we did stuff all together, because why would you take your family to an interesting place and then not spend time with them?

When we were on board, the kids were very happy with the pools– The Pip would’ve been content to splash around in the “circle pool” all day, and SteelyKid was happy to float around in the deeper pool watching Pixar movies on the giant video screen. We also went to a number of the shows and activities– SteelyKid has gotten interested in magic (she’s constantly asking to watch Penn and Teller videos), so we went to both the Magic Dave show in the evening and the “class” that he ran for kids the next day. We also saw a hypnotist, a stage musical version of *Tangled*, and a couple of comedy juggling things. These were all pitched really well– mostly under an hour of running time, and kind of broad humor-wise but not too eye-rollingly so. The kids also went to see *Zootopia* and we carefully kept them from finding out about the live-action *Jungle Book* and *The Force Awakens*, which were also playing but are probably too scary for The Pip.

In port, we did a bunch of ocean-oriented activities, because we were in the Caribbean for God’s sake. In Key West, we took a glass-bottom boat ride (a bit windy, so the visibility wasn’t fantastic and there were some minor motion sickness issues). In Grand Cayman we visited the Cayman Turtle Farm, where SteelyKid was cranky and overheated but cheered up enormously when she got to pick up some green sea turtles (as the name suggests, these are raised in captivity…); The Pip was delighted by the big freshwater pool. In Cozumel, we went to the Dolphin Discovery where SteelyKid and I did a “Push, Pull, and Swim” activity with the trained dolphins– the photo above is from the “Pull” portion, where you grab the pectoral fins of a dolphin who swims upside down towing you back to the dock. The “Push” involves a boogie-board with dolphins pushing on your feet. SteelyKid was absolutely over the moon about this, and I was very impressed with how well the whole thing was run (I did a “swim with dolphins” thing years and years ago up in the Florida Keys, which was much seedier). The Pip wasn’t old enough to swim with dolphins (and isn’t quite comfortable enough in the water yet to really enjoy it), but cheerfully passed an hour or two fighting imaginary crimes in the freshwater pool at the park.

The last stop was “Castaway Cay,” which is Disney’s corporate island in the Bahamas, seen in the “featured image” up top and here for those reading via RSS or too lazy to scroll up and back down:

There’s a nice sandy beach with three inlets; the first is for boat rentals, the second swimming and snorkeling, and the third has a water slide platform that we never did get to. SteelyKid wanted to try snorkeling, so she and I rented gear (my parents brought their own gear) and got in the water. The first attempt was only moderately cool– we saw a big red snapper and a stingray (she climbed onto my back while we were swimming above the ray, and peeked at it around my shoulder). After lunch, she wanted to go again, which is when we discovered all the stuff they sank in the lagoon for people to look at (character statues, fake boats, and lots of big pots and urns), and ended up spending more than an hour in the water, going from one end of the beach (just behind the buildings in the front right of the picture) to the other (by the buildings in the back middle) by way of the lifeguard stand just left of the center. Grandpa and I gave SteelyKid occasional breaks by letting her hang on our shoulders, and we saw a huge array of fish- not much coral, because the lagoon is artificial and of recent origin– but there were snappers and angelfish and blue tang and parrotfish and at the end of the swim two good-sized green turtles. Again, she was over the moon excited about the whole thing, so there will definitely be more snorkeling trips in our future.

(That kid has stamina like you wouldn’t believe– after all that swimming, I could hardly walk, but she was running and splashing and then she and The Pip went down the waterslide on the ship ten times while Kate and I were packing up the room…)

There were some suboptimal aspects, of course– the wifi on the ship was expensive and metered in a way that seemed to us to be massively overcounting the bandwidth Kate and I used. The on-board phones were pretty bad– old-school texting with a phone keyboard, but they didn’t always work– and the chat function of the Disney cruise app was awful– messages were routinely delayed, and after a couple of days it developed a bug where every time we would reconnect to the wi-fi, it would blort out a couple dozen old messages from the second day of the trip. And there were some issues with the sheer number of people on board– the pools got very crowded during the at-sea days, at a level that was frankly pretty scary when The Pip decided he wanted to “glide” from one adult to another in the deeper pool on one at-sea day.

And, of course, that’s the really big issue for anybody thinking about this sort of thing: just how much of other people do you have to tolerate? As I said, we were on the relatively small ship of the bunch, but it’s still a BIG mob of people, a large fraction of them with kids.

While there were occasional displays of, let’s call it “baffling parenting strategies”– mostly involving overstimulated and undersupervised children in the pools and on the waterslide– it was actually pretty reasonable. There’s some amount of forced jollity pushed at you, but the cheesiest bits were easy enough to avoid (it helps that the “Character Appearances” don’t hold much attraction for our kids– they were mostly happy to look and wave from a distance, and didn’t force us to wait on lines to pose for pictures with people in character suits). You can’t completely get away from crowds, but it wasn’t notably worse than most other activities you might choose to do with kids the age of ours. And Disney as an organization is very, very good at managing large crowds of children, with most of their programming well matched to the attention spans of kids about the age of SteelyKid and the Pip.

In fact, it was probably less stressful than a lot of other things we might’ve done with the kids, precisely because dealing with kids is What Disney Does– they’re such terribly picky eaters that it’s really hard to take them out, but they have a good variety of stuff that kids like on board, and as noted above, they were awesome about accommodating our oddball requests. And all the staff on the ship were fantastically cheerful and patient with kids– one of the guys busing tables at breakfast distracted a slightly grumpy Pip by doing magic tricks, which he totally didn’t have to do, but we appreciated enormously.

This is, of course, not remotely cheap, and we were able to do it mostly because my parents are way too good to us, and bought us the tickets as a gift. Despite SteelyKid’s expressed desire to do this all again next year (if not even more frequently), it’s not going to be a regular thing. But it was an excellent experience overall, so if you vacation with kids and have the cash, I’d recommend it.

And now, I need to try to get back to doing actual, you know, work. And also find a way to reconcile myself to being back in Niskayuna where it’s 50 degrees and raining…

As an undergraduate student at RWTH Aachen University, I asked Prof. Barbara Terhal to supervise my bachelor thesis. She told me about qCraft and asked whether I could implement PR-boxes in Minecraft. PR-boxes are named after their inventors Sandu Popescu and … Continue reading

As an undergraduate student at RWTH Aachen University, I asked Prof. Barbara Terhal to supervise my bachelor thesis. She told me about qCraft and asked whether I could implement PR-boxes in Minecraft. PR-boxes are named after their inventors Sandu Popescu and Daniel Rohrlich and have a rather simple behavior. Two parties, let’s call them Alice and Bob, find themselves at two different locations. They each have a box in which they can provide an input bit. And as soon as one of them has done this, he/she can obtain an output bit. The outcomes of the boxes are correlated and satisfy the following condition: If both input bits are 1, the output bits will be different, each 0 or 1 with probability 1/2. If at least one of the input bits is 0, the output bits will be the same, 0 or 1 with probability 1/2. Thus, input bits x and y, and output bits a and b of the PR-box satisfy x AND y = a⊕b, where ⊕ denotes addition modulo two. Neither Alice nor Bob can learn anything about the other one’s input from his/her input and output. This means that Alice and Bob cannot use the PR-boxes to signal to each other.

The motivation for PR-boxes arose from the Clauser-Horne-Shimony-Holt (CHSH) inequality. This Bell-like inequality bounds the correlation that can exist between two remote, non-signaling, classical systems described by local hidden variable theories. Experiments have now convincingly shown that quantum entanglement cannot be explained by local hidden variable theories. Furthermore, the CHSH inequality provides a method to distinguish quantum systems from super-quantum correlations. The correlation between the outputs of the PR-box goes beyond any quantum entanglement. If Alice and Bob were to share an entangled state they could only realize the correlation of the PR-box with probability at most cos²(π/8). PR-boxes are therefore, as far as we know, not physically realizable.

But PR-boxes would have impressive consequences. One of the most remarkable was shown by Wim van Dam in his Oxford PhD thesis in 1999. He proved that two parties can use these PR-boxes to compute any Boolean function f(x,y) of Alice´s input bit string x and Bob´s input bit string y, with only one bit of communication. This is fascinating due to the non-signaling condition fulfilled by PR-boxes. For instance, Alice and Bob could compare their two bit strings x and y of arbitrary length and compute whether or not they are the same. Using classical or quantum systems, one can show that there are lower bounds for the number of bits that need to be communicated between Alice and Bob, which grow with the length of the input bit strings. If Alice and Bob share PR-boxes, they only need sufficiently many PR-boxes (unfortunately, for arbitrary Boolean functions this number grows exponentially) and either Alice or Bob only has to send one bit to the other party. Another application is one-out-of-two oblivious transfer. In this scenario, Alice provides two bits and Bob can choose which of them he wants to know. Ideally, Alice does not learn which bit Bob has chosen and Bob does not learn anything about the other bit. One can use a PR-box to obtain this ideal behavior.

An exciting question for theorists is: why does nature allow for quantum correlations and entanglement but not for super-quantum correlations such as the PR-box? Is there a general physical principle at play? Research on PR-boxes could unveil such principle and explain why PR-boxes are not physically realizable but quantum entanglement is.

But now in the Minecraft world PR-boxes are physically realized! I have built a modification that includes these non-local boxes as an extension of the qCraft modification. Each PR-box is divided into two blocks in order to give the two parties the possibility of spatially partitioning the inputs and outputs. The inputs and outputs are provided by using the in-built Redstone system. This works pretty much like building electrical circuits. The normal PR-boxes function similar as measurements on quantum mechanical states. An input is provided and the corresponding random output is obtained by energizing a block (like measuring the quantum state). This can only be done once. Afterwards, the output is maintained throughout the game. To avoid laborious redistribution and replacement after each usage, I have introduced a timed version of the PR-box in Minecraft. To get a better idea of what this all looks like, visit this demo video.

PR-boxes are interesting in particular in multiplayer scenarios since there are two parties needed to use them appropriately. For example, these new elements could be used to create multiplayer dungeons where the players have to communicate using only a small number of bits or provide a combined password to deactivate a trap. The timed PR-box may be used as a component of a Minecraft computer to simplify circuits using the compatibility with clocks.

I hope that you will try this modification and show how they can enhance gameplay in Minecraft! This mod as well as my thesis can be downloaded here. For me it was much fun to go from the first ideas how to realize PR-boxes in Minecraft to this final implementation. Just as qCraft, this is a playful way of exploring theoretical physics.

**[ Warning: This movie review contains spoilers, as well as a continued fraction expansion.]**

These days, it takes an extraordinary occasion for me and Dana to arrange the complicated, rocket-launch-like babysitting logistics involved in *going out for a night at the movies*. One such an occasion was an opening-weekend screening of *The Man Who Knew Infinity—*the new movie about Srinivasa Ramanujan and his relationship with G. H. Hardy—followed by a Q&A with Matthew Brown (who wrote and directed the film), Robert Kanigel (who wrote the biography on which the film was based), and Fields Medalist Manjul Bhargava (who consulted on the film).

I read Kanigel’s *The Man Who Knew Infinity* in the early nineties; it was a major influence on my life. There were equations in that book to stop a nerdy 13-year-old’s pulse, like

$$1+9\left( \frac{1}{4}\right) ^{4}+17\left( \frac{1\cdot5}{4\cdot8}\right)

^{4}+25\left( \frac{1\cdot5\cdot9}{4\cdot8\cdot12}\right) ^{4}+\cdots

=\frac{2^{3/2}}{\pi^{1/2}\Gamma\left( 3/4\right) ^{2}}$$

$$\frac{1}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\frac{e^{-6\pi}}{1+\cdots}}%

}}=\left( \sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}\right)

\sqrt[5]{e^{2\pi}}$$

A thousand pages of exposition about Ramanujan’s mysterious self-taught mathematical style, the effect his work had on Hardy and Littlewood, his impact on the later development of analysis, etc., could never replace the experience of just *staring* at these things! Popularizers are constantly trying to “explain” mathematical beauty by comparing it to art, music, or poetry, but I can best understand art, music, and poetry if I assume other people experience them like the above identities. Across all the years and cultures and continents, can’t you feel Ramanujan himself leaping off your screen, still trying to make you see this bizarre aspect of the architecture of reality that the goddess Namagiri showed him in a dream?

Reading Kanigel’s book, I was also entranced by the culture of early-twentieth-century Cambridge mathematics: the Tripos, Wranglers, High Table. I asked, why was I *here* and not *there*? And even though I was (and remain) at most 1729^{-1729} of a Ramanujan, I could strongly identify with his story, because I knew that I, too, was about to embark on the journey from total scientific nobody to someone who the experts might at least take seriously enough to try to prove him wrong.

Anyway, a couple years after reading Kanigel’s biography, I went to the wonderful Canada/USA MathCamp, and there met Richard K. Guy, who’d actually *known* Hardy. I couldn’t have been more impressed had Guy visited Platonic heaven and met π and e there. To put it mildly, no one in my high school had known G. H. Hardy.

I often fantasized—this was the nineties—about writing the screenplay myself for a Ramanujan movie, so that millions of moviegoers could experience the story as I did. Incidentally, I also fantasized about writing screenplays for Alan Turing and John Nash movies. I do have a few mathematical biopic ideas that *haven’t* yet been taken, and for which any potential buyers should get in touch with me:

*Radical: The Story of Évariste Galois**Give Me a Place to Stand: Archimedes’ Final Days**Mathématicienne: Sophie Germain In Her Prime**The Prime Power of Ludwig Sylow*(OK, this last one would be more of a limited-market release)

But enough digressions; how was the Ramanujan movie?

Just as Ramanujan himself wasn’t an infallible oracle (many of his claims, e.g. his formula for the prime counting function, turned out to be wrong), so *The Man Who Knew Infinity* isn’t a perfect movie. Even so, there’s no question that this is one of the best and truest movies ever made about mathematics and mathematicians, if not the best and truest. If you’re the kind of person who reads this blog, go see it now. Don’t wait! As they stressed at the Q&A, the number of tickets sold in the first couple weeks is what determines whether or not the movie will see a wider release.

More than *A Beautiful Mind* or *Good Will Hunting* or *The Imitation Game*, or the play *Proof*, or the TV series *NUMB3RS*, the Ramanujan movie seems to me to respect math as a thing-in-itself, rather than just a tool or symbol for something else that interests the director much more. The background to the opening credits—and what better choice could there be?—is just page after page from Ramanujan’s notebooks. Later in the film, there’s a correct explanation of what the partition function P(n) is, and of one of Ramanujan’s and Hardy’s central achievements, which was to give an asymptotic formula for P(n), namely $$ P(n) \approx \frac{e^{π \sqrt{2n/3}}}{4\sqrt{3}n}, $$ and to prove the formula’s correctness.

The film also makes crystal-clear that pure mathematicians do what they do not because of applications to physics or anything else, but simply because they feel compelled to: for the devout Ramanujan, math was literally about writing down “the thoughts of God,” while for the atheist Hardy, math was a religion-substitute. Notably, the movie explores the tension between Ramanujan’s untrained intuition and Hardy’s demands for rigor in a way that does them both justice, resisting the Hollywood urge to make intuition 100% victorious and rigor just a stodgy punching bag to be defeated.

For my taste, the movie could’ve gone even further in the direction of “letting the math speak”: for example, it could’ve explained just one of Ramanujan’s infinite series. Audiences might even have *liked* some more T&A (theorems and asymptotic bounds). During the Q&A that I attended, I was impressed to see moviegoers repeatedly pressing a somewhat-coy Manjul Bhargava to *explain Ramanujan’s actual mathematics* (e.g., what exactly were the discoveries in his first letter to Hardy? what was in Ramanujan’s Lost Notebook that turned out to be so important?). Then again, this was Cambridge, MA, so the possibility should at least be entertained that what I witnessed was unrepresentative of American ticket-buyers.

From what I’ve read, the movie is also true to South Indian dress, music, religion, and culture. Yes, the Indian characters speak to each other in English rather than Tamil, but Brown explained that as a necessary compromise (not only for the audience’s sake, but also because Dev Patel and the other Indian actors didn’t speak Tamil).

Some reviews have mentioned issues with casting and characterization. For example, Hardy is portrayed by Jeremy Irons, who’s superb but also decades older than Hardy was at the time he knew Ramanujan. Meanwhile Ramanujan’s wife, Janaki, is played by a fully-grown Devika Bhise; the real Janaki was nine (!) when she married Ramanujan, and fourteen when Ramanujan left for England. J. E. Littlewood is played as almost a comic-relief buffoon, so much so that it feels incongruous when, near the end of the film, Irons-as-Hardy utters the following real-life line:

I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.”

Finally, a young, mustachioed Bertrand Russell is a recurring character. Russell and Hardy really *were* friends and fellow WWI pacifists, but Hardy seeking out Bertie’s advice about each Ramanujan-related development seems like almost certainly just an irresistible plot device.

But none of that matters. What bothered me more were the dramatizations of the prejudice Ramanujan endured in England. Ramanujan is shown getting knocked to the ground, punched, and kicked by British soldiers barking anti-Indian slurs at him; he then shows up for his next meeting with Hardy covered in bruises, which Hardy (being aloof) neglects to ask about. Ramanujan is also depicted getting shoved, screamed at, and told never to return by a math professor who he humiliates during a lecture. I understand why Brown made these cinematic choices: there’s no question that Ramanujan experienced prejudice and snobbery in Cambridge, and that he often felt lonely and unwelcome there. And it’s surely *easier* to show Ramanujan literally getting beaten up by racist bigots, than to depict his alienation from Cambridge society as the subtler matter that it most likely was. To me, though, that’s precisely why the latter choice would’ve been even more impressive, had the film managed to pull it off.

Similarly, during World War I, the film shows not only Trinity College converted into a military hospital, and many promising students marched off to their deaths (all true), but also a shell exploding on campus near Ramanujan, after which Ramanujan gazes in horror at the bleeding dead bodies. Like, isn’t the truth here dramatic enough?

One other thing: the movie leaves you with the impression that Ramanujan died of tuberculosis. More recent analysis concluded that it was probably hepatic amoebiasis that he brought with him from India—something that could’ve been cured with the medicine of the time, had anyone correctly diagnosed it. (Incidentally, the film completely omits Ramanujan’s final year, back in India, when he suffered a relapse of his illness and slowly withered away, yet with Janaki by his side, continued to do world-class research and exchanged letters with Hardy until the very last days. Everyone I read commented that this was “the right dramatic choice,” but … I dunno, I would’ve shown it!)

But enough! I fear that to harp on these defects is to hold the film to impossibly-high, Platonic standards, rather than standards that engage with the reality of Hollywood. An anecdote that Brown related at the end of the Q&A session brought this point home for me. Apparently, Brown struggled for an entire decade to attract funding for a film about a turn-of-the-century South Indian mathematician visiting Trinity College, Cambridge, whose work had no commercial or military value whatsoever. At one point, Brown was actually told that he could get the movie funded, *if he’d agree to make Ramanujan fall in love with a white nurse*, so that a British starlet who would sell tickets could be cast as his love interest. One can only imagine what a battle it must have been to get a correct explanation of the partition function onto the screen.

In the end, though, nothing made me appreciate *The Man Who Knew Infinity* more than reading negative reviews of it, like this one by Olly Richards:

Watching someone balancing algorithms or messing about with multivariate polynomials just isn’t conducive to urgently shovelling popcorn into your face. Difficult to dislike, given its unwavering affection for its subject, *The Man Who Knew Infinity* is nevertheless hamstrung by the dryness of its subject … Sturdy performances and lovely scenery abound, but it’s still largely just men doing sums; important sums as it turns out, but that isn’t conveyed to the audience until the coda [which mentions black holes] tells us of the major scientific advances they aided.

On behalf of mathematics, on behalf of my childhood self, I’m grateful that Brown fought this fight, and that he won as much as he did. Whether you walk, run, board a steamship, or take taxi #1729, go see this film.

**Addendum:** See also this review by Peter Woit, and this in *Notices of the AMS* by Ramanujan expert George Andrews.

In a short research day, I had a very useful conversation about robot fiber positioners for multi-object spectrographs with Peter Mao (Caltech), who is working on the *Prime Focus Spectrograph* for Subaru. He gave me a sense of the cost scale, the human effort scale, and the technical precision of such systems. This is critical information for the Letter of Intent that I and various others are putting in for the use of the *SDSS* hardware after the end of the current survey, *SDSS-IV*. We would like to go really big with the two *APOGEE* spectrographs, but if we want to do really large numbers of stars (think: millions or even tens of millions) we need to have robots place the fibers.

The wildflower patch continues to produce surprises. You never know exactly what's going to come up, and in what quantities. I've been fascinated by this particular flower, for example, which seems to be constructed out of several smaller flowers! What a wonder, and of course, there's just one example of its parent plant in the entire patch, so once it is gone, it's gone.

-cvj Click to continue reading this post

The post Wild Thing appeared first on Asymptotia.

Let $(M, \otimes)$ be a monoidal category and let $C$ be a left module category over $M$, with action map also denoted by $\otimes$. If $m \in M$ is a monoid and $c \in C$ is an object, then we can talk about an **action** of $m$ on $c$: it’s just a map

$\alpha : m \otimes c \to c$

satisfying the usual associativity and unit axioms. (The fact that all we need is an action of $M$ on $C$ to define an action of $m$ on $c$ is a cute instance of the microcosm principle.)

This is a very general definition of monoid acting on an object which includes, as special cases (at least if enough colimits exist),

- actions of monoids in $\text{Set}$ on objects in ordinary categories,
- actions of monoids in $\text{Vect}$ (that is, algebras) on objects in $\text{Vect}$-enriched categories,
- actions of monads (letting $M = \text{End}(C)$), and
- actions of operads (letting $C$ be a symmetric monoidal category and $M$ be the monoidal category of symmetric sequences under the composition product)

This definition can be used, among other things, to straightforwardly motivate the definition of a monad (as I did here): actions of a monoidal category $M$ on a category $C$ correspond to monoidal functors $M \to \text{End}(C)$, so every action in the above sense is equivalent to an action of a monad, namely the image of the monoid $m$ under such a monoidal functor. In other words, monads on $C$ are the universal monoids which act on objects $c \in C$ in the above sense.

Corresponding to this notion of action is a notion of endomorphism object. Say that the **relative endomorphism object** $\text{End}_M(c)$, if it exists, is the universal monoid in $M$ acting on $c$: that is, it’s a monoid acting on $c$, and the action of any other monoid on $c$ uniquely factors through it.

This is again a very general definition which includes, as special cases (again if enough colimits exist),

- the endomorphism monoid in $\text{Set}$ of an object in an ordinary category,
- the endomorphism algebra of an object in a $\text{Vect}$-enriched category,
- the endomorphism monad of an object in an ordinary category, and
- the endomorphism operad of an object in a symmetric monoidal category.

If the action of $M$ on $C$ has a compatible enrichment $[-, -] : C^{op} \times C \to M$ in the sense that we have natural isomorphisms

$\text{Hom}_C(m \otimes c_1, c_2) \cong \text{Hom}_M(m, [c_1, c_2])$

then $\text{End}_M(c)$ is just the endomorphism monoid $[c, c]$, and in fact the above discussion could have been done in the context of enrichments only, but in the examples I have in mind the actions are easier to notice than the enrichments. (Has anyone ever told you that symmetric monoidal categories are canonically enriched over symmetric sequences? Nobody told me, anyway.)

Here’s another example where the action is easier to notice than the enrichment. If $D, C$ are two categories, then the monoidal category $\text{End}(C) = [C, C]$ has a natural left action on the category $[D, C]$ of functors $D \to C$. If $G : D \to C$ is a functor, then the relative endomorphism object $\text{End}_{\text{End}(C)}(G)$, if it exists, turns out to be the codensity monad of $G$!

This actually follows from the construction of an enrichment: the category $[D, C]$ of functors $D \to C$ is (if enough limits exist) enriched over $\text{End}(C)$ in a way compatible with the natural left action. This enrichment takes the following form (by a straightforward verification of universal properties): if $G_1, G_2 \in [D, C]$ are two functors $D \to C$, then their hom object

$[G_1, G_2] = \text{Ran}_{G_1}(G_2) \in \text{End}(C)$

is, if it exists, the right Kan extension of $G_2$ along $G_1$. When $G_1 = G_2$ this recovers the definition of the codensity monad of a functor $G : D \to C$ as the right Kan extension of $G$ along itself, and neatly explains why it’s a monad: it’s an endomorphism object.

**Question:** Has anyone seen this definition of relative endomorphisms before?

It seems pretty natural, but I tried guessing what it would be called on the nLab and failed. It also seems that “relative endomorphisms” is used to mean something else in operad theory.

I'd like to ask my readers that own Quantum Design PPMS or MPMS instruments for help regarding a technical glitch. My aging PPMS superconducting magnet power supply (the kind QD calls the H-plate version) has developed a problem. For high fields (say above 7 T) the power supply fails to properly put the magnet in persistent mode and throws up an error in the control software. After talking with QD, it seems like options are limited. They no longer service this model of power supply, and therefore one option would be to buy a new one. However, I have a sense that other people have dealt with this issue before, and I would feel dumb buying a new supply if the answer was that this is a known issue involving a $ 0.30 diode or something. Without a schematic it's difficult to do diagnostics ourselves. Has anyone out there seen this issue and knows how to correct it?

If you fall into a black hole, you’ll die. That much is pretty sure. But what happens before that?

Leaving aside lots of hot gas and swirling particles, you have good chances to survive crossing the horizon of a supermassive black hole, like that in the center of our galaxy. You would, however, probably be torn apart before crossing the horizon of a solar-mass black hole.

It takes you a finite time to reach the horizon of a black hole. For an outside observer however, you seem to be moving slower and slower and will never quite reach the black hole, due to the (technically infinitely large) gravitational redshift. If you take into account that black holes evaporate, it doesn’t quite take forever, and your friends will eventually see you vanishing. It might just take a few hundred billion years.

In an article that recently appeared on “Quick And Dirty Tips” (featured by SciAm), Everyday Einstein Sabrina Stierwalt explains:

I suspect this confusion was caused by the idea of black hole complementarity. Which is indeed a highly contest area of current physics research. According to black hole complementarity the information that falls into a black hole both goes in and comes out. This is in contradiction with quantum mechanics which forbids making exact copies of a state. The idea of black hole complementarity is that nobody can ever make a measurement to document the forbidden copying and hence, it isn’t a real inconsistency. Making such measurements is typically impossible because the infalling observer only has a limited amount of time before hitting the singularity.

Black hole complementarity is actually a pretty philosophical idea.

Now, the black hole firewall issue points out that black hole complementarity is inconsistent. Even if you can’t measure that a copy has been made, pushing the infalling information in the outgoing radiation changes the vacuum state in the horizon vicinity to a state which is no longer empty: that’s the firewall.

Be that as it may, even in black hole complementarity the infalling observer still falls in, and crosses the horizon at a finite time.

The real question that drives much current research is how the information comes out of the black hole before it has completely evaporated. It’s a topic which has been discussed for more than 40 years now, and there is little sign that theorists will agree on a solution. And why would they? Leaving aside fluid analogies, there is no experimental evidence for what happens with black hole information, and there is hence no reason for theorists to converge on any one option.

The theory assessment in this research area is purely non-empirical, to use an expression by philosopher Richard Dawid. It’s why I think if we ever want to see progress on the foundations of physics we have to think very carefully about the non-empirical criteria that we use.

Anyway, the lesson here is: Everyday Einstein’s*Quick and Dirty Tips* is not a recommended travel guide for black holes.

The gravitational pull of a black hole depends on its mass. At a fixed distance from the center, it isn’t any stronger or weaker than that of a star with the same mass. The difference is that, since a black hole doesn’t have a surface, the gravitational pull can continue to increase as you approach the center.

The gravitational pull itself isn’t the problem, the problem is the change in the pull, the tidal force. It will stretch any extended object in a process with technical name “spaghettification.” That’s what will eventually kill you. Whether this happens before or after you cross the horizon depends, again, on the mass of the black hole. The larger the mass, the smaller the space-time curvature at the horizon, and the smaller the tidal force.Leaving aside lots of hot gas and swirling particles, you have good chances to survive crossing the horizon of a supermassive black hole, like that in the center of our galaxy. You would, however, probably be torn apart before crossing the horizon of a solar-mass black hole.

It takes you a finite time to reach the horizon of a black hole. For an outside observer however, you seem to be moving slower and slower and will never quite reach the black hole, due to the (technically infinitely large) gravitational redshift. If you take into account that black holes evaporate, it doesn’t quite take forever, and your friends will eventually see you vanishing. It might just take a few hundred billion years.

In an article that recently appeared on “Quick And Dirty Tips” (featured by SciAm), Everyday Einstein Sabrina Stierwalt explains:

“As you approach a black hole, you do not notice a change in time as you experience it, but from an outsider’s perspective, time appears to slow down and eventually crawl to a stop for you [...] So who is right? This discrepancy, and whose reality is ultimately correct, is a highly contested area of current physics research.”No, it isn’t. The two observers have different descriptions of the process of falling into a black hole because they both use different time coordinates. There is no contradiction between the conclusions they draw. The outside observer’s story is an infinitely stretched version of the infalling observer’s story, covering only the part before horizon crossing. Nobody contests this.

I suspect this confusion was caused by the idea of black hole complementarity. Which is indeed a highly contest area of current physics research. According to black hole complementarity the information that falls into a black hole both goes in and comes out. This is in contradiction with quantum mechanics which forbids making exact copies of a state. The idea of black hole complementarity is that nobody can ever make a measurement to document the forbidden copying and hence, it isn’t a real inconsistency. Making such measurements is typically impossible because the infalling observer only has a limited amount of time before hitting the singularity.

Black hole complementarity is actually a pretty philosophical idea.

Now, the black hole firewall issue points out that black hole complementarity is inconsistent. Even if you can’t measure that a copy has been made, pushing the infalling information in the outgoing radiation changes the vacuum state in the horizon vicinity to a state which is no longer empty: that’s the firewall.

Be that as it may, even in black hole complementarity the infalling observer still falls in, and crosses the horizon at a finite time.

The real question that drives much current research is how the information comes out of the black hole before it has completely evaporated. It’s a topic which has been discussed for more than 40 years now, and there is little sign that theorists will agree on a solution. And why would they? Leaving aside fluid analogies, there is no experimental evidence for what happens with black hole information, and there is hence no reason for theorists to converge on any one option.

The theory assessment in this research area is purely non-empirical, to use an expression by philosopher Richard Dawid. It’s why I think if we ever want to see progress on the foundations of physics we have to think very carefully about the non-empirical criteria that we use.

Anyway, the lesson here is: Everyday Einstein’s

It is far from clear to me that Springer’s copyright agreement (quoted below) is compatible with posting your paper on the arXiv under the CC-BY license. Happily, a Springer representative has just confirmed for me that this is allowed: Let me first say that the cc-by-0 license is no problem at all as it allows […]

It is far from clear to me that Springer’s copyright agreement (quoted below) is compatible with posting your paper on the arXiv under the CC-BY license. Happily, a Springer representative has just confirmed for me that this is allowed:

Let me first say that the cc-by-0 license is no problem at all as it allows for other publications without restrictions. Second, our copyright statement of course only talks about the version published in one of our journals, with our copyright line (or the copyright line of a partner society if applicable, or the author’s copyright if Open Access is chosen) on it.

At least if you are publishing in a Springer journal, and more generally, I would strongly encourage you to post your papers to the arXiv under the more permissive CC-BY-0 license, rather than the minimal license the arXiv requires.

As a question to any legally-minded readers: does copyright law genuinely distinguish between “the version published in one of our journals, with our copyright line”, and the “author-formatted post-peer-review” version which is substantially identical, barring the journals formatting and copyright line?

Today Boris Leistedt and Michael Troxel (Manchester) came to Simons to hack on a proposal to change the latter years of *DES* observing strategy. Their argument is that a small amount of *u*-band imaging (currently *DES* does none) could have a huge impact on photometric redshifts (particularly bias), which, in turn, could have a huge impact on the accuracy of the convergence mapping and large-scale structure constraints. They spent the day doing complete end-to-end simulations of observing, photometry, data analysis, and parameter estimation. I shouldn't really blog this, because it isn't my research, but it is very impressive!/p>

On the side, Leistedt and I checked in on our project to build a generative model of galaxy photometry, in which the full family of possible spectral energy distributions would be latent variables. Leistedt had a great breakthrough: If the SEDs are drawn from a Gaussian process, then the observables are also drawn from a Gaussian process, because projection onto redshifted bandpasses is a linear operation! He has code that implements this and some toy problems that seem to work, so I am cautiously optimistic.

At group meeting today, Michael Troxel (Manchester) showed up, as he and Boris Leistedt are working on proposals to modify *DES* observing strategy. Troxel showed us his convergence maps from the first look or engineering data from *DES*. Incredible! They look high in signal-to-noise and correlate very well with measured large-scale structure along the line of sight. It appears that this kind of mass density (or convergence) mapping is really mature.

After that, Dun Wang summarized a talk from last week by Jason Wright (PSU); this led to a conversation about the alien megastructures and Tabby's star. We discussed projects we might do on this. I asked what we would need to observe in order to be *really* convinced that this is alien technology. Since this question is very hard to answer, it is not clear that the “alien megastructure” explanation is really a scientific explanation at all. Oh so many good April Fools' projects.

30 years ago, on 26 April 1986, the biggest nuclear accident happened at the Chernobyl nuclear power station. The picture above is of my 8th grade class (I am in the front row) on a trip from Leningrad to Kiev. We wanted to make sure that we’d spend May 1st (Labor Day in the Soviet […]

30 years ago, on 26 April 1986, the biggest nuclear accident happened at the Chernobyl nuclear power station.

The picture above is of my 8th grade class (I am in the front row) on a trip from Leningrad to Kiev. We wanted to make sure that we’d spend May 1st (Labor Day in the Soviet Union) in Kiev! We took that picture in Gomel, which is about 80 miles away from Chernobyl, where our train made a regular stop. We were instructed to bury some pieces of clothing and shoes after coming back to Leningrad due to excess of radioactive dust on them…

A couple of weeks ago I visited the Texas A&M Physics and Engineering Festival. It was a busy trip — I gave a physics colloquium and a philosophy colloquium as well as a public talk — but the highlight for … Continue reading

A couple of weeks ago I visited the Texas A&M Physics and Engineering Festival. It was a busy trip — I gave a physics colloquium and a philosophy colloquium as well as a public talk — but the highlight for me was an hourlong chat with the Davidson Young Scholars, who had traveled from across the country to attend the festival.

The Davidson Young Scholars program is an innovative effort to help nurture kids who are at the very top of intellectual achievement in their age group. Every age and ability group poses special challenges to educators, and deserves attention and curricula that are adjusted for their individual needs. That includes the most high-achieving ones, who easily become bored and distracted when plopped down in an average classroom. Many of them end up being home-schooled, simply because school systems aren’t equipped to handle them. So the DYS program offers special services, including most importantly a chance to meet other students like themselves, and occasionally go out into the world and get the kind of stimulation that is otherwise hard to find.

These kids were awesome. I chatted just very briefly, telling them a little about what I do and what it means to be a theoretical physicist, and then we had a free-flowing discussion. At some point I mentioned “wormholes” and it was all over. These folks *love* wormholes and time travel, and many of them had theories of their own, which they were eager to come to the board and explain to all of us. It was a rollicking, stimulating, delightful experience.

You can see from the board that I ended up talking about Einstein’s equation. Not that I was going to go through all of the mathematical details or provide a careful derivation, but I figured that was something they wouldn’t typically be exposed to by either their schoolwork or popular science, and it would be fun to give them a glimpse of what lies ahead if they study physics. Everyone’s life is improved by a bit of exposure to Einstein’s equation.

The kids are all right. If we old people don’t ruin them, the world will be in good hands.

The so-called III-V semiconductors, compounds that combine a group III element (Al, Ga, In) and a group V element (N, As, P, Sb), are mainstays of (opto)electronic devices and condensed matter physics. They have never taken over for Si in logic and memory like some thought they might, for a number of materials science and economic reasons. (To paraphrase an old line, "GaAs is the material of the future [for logic] and always will be.") However, they are tremendously useful, in part because they are (now) fortuitously easy to grow - many of the compounds prefer the diamond-like "zinc blende" structure, and it is possible to prepare atomically sharp, flat, abrupt interfaces between materials with quite different semiconducting properties (very different band gaps and energetic alignments relative to each other). Fundamentally, though, the palette is limited - these materials are very conventional semiconductors, without exhibiting other potentially exciting properties or competing phases like ferroelectricity, magnetism, superconductivity, etc.

Enter oxides. Various complex oxides can exhibit all of these properties, and that has led to a concerted effort to develop materials growth techniques to create high quality oxide thin films, with an eye toward creating the same kind of atomically sharp heterointerfaces as in III-Vs. A foundational paper is this one by Ohtomo and Hwang, where they used pulsed laser deposition to produce a heterojunction between LaAlO_{3}, an insulating transparent oxide, and SrTiO_{3}, another insulating transparent oxide (though one known to be *almost* a ferroelectric). Despite the fact that both of those parent constituents are band insulators, the interface between the two was found to play host to a two-dimensional gas of electrons with remarkable properties. The wikipedia article linked above is pretty good, so you should read it if you're interested.

When you think about it, this is really remarkable. You take an insulator, and another insulator, and yet the interface between them acts like a metal. Where did the charge carriers come from? (It's complicated - charge transfer from LAO to STO, but the free surface of the LAO and its chemical termination is hugely important.) What is happening right at that interface? (It's complicated. There can be some lattice distortion from the growth process. There can be oxygen vacancies and other kinds of defects. Below about 105 K the STO substrate distorts "ferroelastically", further complicating matters.) Do the charge carriers live more on one side of the interface than the other, as in III-V interfaces, where the (conduction) band offset between the two layers can act like a potential barrier, and the same charge transfer that spills electrons onto one side leads to a self-consistent electrostatic potential that holds the charge layer right against that interface? (Yes.)

Even just looking at the LAO/STO system, there is a ton of exciting work being performed. Directly relevant to the meeting I just attended, Jeremy Levy's group at Pitt has been at the forefront of creating nanoscale electronic structures at the LAO/STO interface and examining their properties. It turns out (one of these fortunate things!) that you can use a conductive atomic force microscope tip to do (reversible) electrochemistry at the free LAO surface, and basically *draw* conductive structures with nm resolution at the buried LAO/STO interface right below. This is a very powerful technique, and it's enabled the study of the basic science of electronic transport at this interface at the nanoscale.

Beyond LAO/STO, over the same period there has been great progress in complex oxide materials growth by groups at a number of universities and at national labs. I will refrain from trying to list them since I don't know them all and don't want to offend with the sin of inadvertent omission. It is now possible to prepare a dizzying array of material types (ferromagnetic insulators like GdTiO_{3}; antiferromagnetic insulators like SmTiO_{3}; Mott insulators like LaTiO_{3}; nickelates; superconducting cuprates; etc.) and complicated multilayers and superlattices of these systems. It's far too early to say where this is all going, but historically the ability to grow new material systems of high quality with excellent precision tends to pay big dividends in the long term, even if they're not the benefits originally envisioned.

If you tell me you’re going to take a compact smooth 2-dimensional manifold and subdivide it into polygons, I know what you mean. You mean something like this picture by Norton Starr:

or this picture by Greg Egan:

(Click on the images for details.) But what’s the usual term for this concept, and the precise definition? I’m writing a paper that uses this concept, and I don’t want to spend my time proving basic stuff. I want to just refer to something.

I’m worried that CW decompositions of surfaces might include some things I don’t like, though I’m not sure.

Maybe I want PLCW decompositions, which seem to come in at least two versions: the old version discussed in C. Hog-Angeloni, W. Metzler, and A. Sieradski’s book *Two-dimensional Homotopy and Combinatorial Group Theory*, and a new version due to Kirillov. But I don’t even know if these two versions are the same!

One big question is whether one wants polygons to include ‘bigons’ or even ‘unigons’. For what I’m doing right now, I don’t much care. But I want to know what I’m including!

Another question is whether we can glue one edge of a polygon to another edge of that same polygon.

Surely there’s some standard, widely used notion here…

Adrian Price-Whelan and I met at Columbia to discuss various things. We looked at the new paper of Reid *et al* on distances; which appeared on twitter this month as an argument that the Milky Way has spiral structure. Although this paper is not really dishonest—it explains what it did (though with a few lacunae)—it is misleading and wrong in various ways. The most important: It is misleading because it is being used as evidence for spiral structure (its figure 5 is being tweeted around!). But it also shows (in its figure 6) that even if there was no evidence *at all* for spiral structure in the data, their analysis would find a spiral pattern in the posterior pdf and distance estimators! It is wrong because it (claims to) multiply together posteriors (rather than likelihoods). That is, it violates the rules of probability that I tried to set out clearly here. I try not to use the word "wrong" when talking about other people's work; I don't mean to be harsh! The team on this paper includes some of the best observational astrophysicists in the world. I just mean that if you want to do probabilistic data analysis, you should obey the rules, and clearly state what you can and cannot conclude from the data.

At lunch, Jeno Sokolowski (Columbia) spoke about accreting white dwarfs in orbit around red giant stars. I realized during her talk that we can potentially generate a catalog of enormous numbers of these from our work (with Anna Ho) on *LAMOST*.

The International Mathematical Union (with the assistance of the Friends of the International Mathematical Union and The World Academy of Sciences, and supported by Ian Agol, Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Richard Taylor, and myself) has just launched the Graduate Breakout Fellowships, which will offer highly qualified students from developing countries a full scholarship to […]

The International Mathematical Union (with the assistance of the Friends of the International Mathematical Union and The World Academy of Sciences, and supported by Ian Agol, Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Richard Taylor, and myself) has just launched the Graduate Breakout Fellowships, which will offer highly qualified students from developing countries a full scholarship to study for a PhD in mathematics at an institution that is also located in a developing country. Nominations for this fellowship (which should be from a sponsoring mathematician, preferably a mentor of the nominee) have just opened (with an application deadline of June 22); details on the nomination process and eligibility requirements can be found at this page.

Filed under: advertising Tagged: Breakout Fellowship

In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number […]

In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).

Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.

Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space of continuous functions on that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm ), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.

I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:

*Proof:* Suppose one had two norms on such that and were both Banach spaces with topologies stronger than . Now consider the graph of the identity function from the Banach space to the Banach space . This graph is closed; indeed, if is a sequence in this graph that converged in the product topology to , then converges to in norm and hence in , and similarly converges to in norm and hence in . But limits are unique in the Hausdorff topology , so . Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from to ; similarly for the inverse. Thus the norms are equivalent as claimed.

By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if is infinite dimensional, one can use a Hamel basis of to construct a linear bijection on that is unbounded with respect to a given Banach space norm , and which can then be used to give an inequivalent Banach space structure on .

One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a *canonical* choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. is the space of Schwartz functions on ) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on given by the usual Schwartz space seminorms).

Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm stronger than this topology that makes this space a Banach space. For, if there were, then letting be the standard basis of , the series would have to converge in , and hence pointwise, to an element of , but the only available pointwise limit for this series lies outside of . But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or -space) topology.

Filed under: 245B - Real analysis, expository, math.FA Tagged: Banach spaces, closed graph theorem, strong topology, weak topology

For the last 2.5 days I've been at the PQI2016: Quantum Challenges symposium. It's been a very fun meeting, bringing together talks spanning physical chemistry, 2d materials, semiconductor and oxide structures, magnetic systems, plasmonics, cold atoms, and quantum information. Since the talks are all going to end up streamable online from the PQI website, I'll highlight just a couple of things that I learned rather than trying to summarize everything.

- If you can make a material such that the dielectric permittivity \( \epsilon \equiv \kappa \epsilon_{0} \) is
*zero*over some frequency range, you end up with a very odd situation. The phase velocity of EM waves at that frequency would go to infinity, and the in-medium wavelength at that frequency would therefore become infinite. Everything in that medium (at that frequency) would be in the near-field of everything else. See here for a paper about what this means for transmission of EM waves through such a region, and here for a review. - Screening of charge and therefore carrier-carrier electrostatic interactions in 2d materials like transition metal dichalcogenides varies in a complicated way with distance. At short range, screening is pretty effective (logarithmic with distance, basically the result you'd get if you worried about the interaction potential from an infinitely long charged rod), and at longer distances the field lines leak out into empty space, so the potential falls like \(1/\epsilon_{0}r\). This has a big effect on the binding of electrons and holes into excitons in these materials.
- There are a bunch of people working on unconventional transistor designs, including devices based on band-to-band tunneling between band-offset 2d materials.
- In a discussion about growth and shapes of magnetic domains in a particular system, I learned about the Wulff construction, and this great paper by Conyers Herring on why crystal take the shapes that they do.
- After a public talk by Michel Devoret, I think I finally have some sense of the fundamental differences between the Yale group's approach to quantum computing and the John Martinis/Google group's approach. This deserves a longer post later.
- Oxide interfaces continue to show interesting and surprising properties - again, I hope to say more later.
- On a more science-outreach note, I learned about an app called Periscope (basically part of twitter) that allows people to do video broadcasting from their phones. Hat tip to Julia Majors (aka Feynwoman) who pointed this out to me and that it's becoming a platform for a lot of science education work.

Can we build up space-time from discrete entities? |

Dear Noa:

Discretization is a common procedure to deal with infinities. Since quantum mechanics relates large energies to short (wave) lengths, introducing a shortest possible distance corresponds to cutting off momentum integrals. This can remove infinites that come in at large momenta (or, as the physicists say “in the UV”).

Such hard cut-off procedures were quite common in the early days of quantum field theory. They have since been replaced with more sophisticated regulation procedures, but these don’t work for quantum gravity. Hence it lies at hand to use discretization to get rid of the infinities that plague quantum gravity.Lorentz-invariance is the symmetry of Special Relativity; it tells us how observables transform from one reference frame to another. Certain types of observables, called “scalars,” don’t change at all. In general, observables do change, but they do so under a well-defined procedure that is by the application of Lorentz-transformations.We call these “covariant.” Or at least we should. Most often invariance is conflated with covariance in the literature.

(To be precise, Lorentz-covariance isn’t the full symmetry of Special Relativity because there are also translations in space and time that should maintain the laws of nature. If you add these, you get Poincaré-invariance. But the translations aren’t so relevant for our purposes.)

Lorentz-transformations acting on distances and times lead to the phenomena of Lorentz-contraction and time-dilatation. That means observers at relative velocities to each other measure different lengths and time-intervals. As long as there aren’t any interactions, this has no consequences. But once you have objects that can interact, relativistic contraction has measurable consequences.

Heavy ions for example, which are collided in facilities like RHIC or the LHC, are accelerated to almost the speed of light, which results in a significant length contraction in beam direction, and a corresponding increase in the density. This relativistic squeeze has to be taken into account to correctly compute observables. It isn’t merely an apparent distortion, it’s a real effect.

Now consider you have a regular cubic lattice which is at rest relative to you. Alice comes by in a space-ship at high velocity, what does she see? She doesn’t see a cubic lattice – she sees a lattice that is squeezed into one direction due to Lorentz-contraction. Who of you is right? You’re both right. It’s just that the lattice isn’t invariant under the Lorentz-transformation, and neither are any interactions with it.

The lattice can therefore be used to define a preferred frame, that is a particular reference frame which isn’t like any other frame, violating observer independence. The easiest way to do this would be to use the frame in which the spacing is regular, ie your restframe. If you compute any observables that take into account interactions with the lattice, the result will now explicitly depend on the motion relative to the lattice. Condensed matter systems are thus generally not Lorentz-invariant.

A Lorentz-contraction can convert any distance, no matter how large, into another distance, no matter how short. Similarly, it can blue-shift long wavelengths to short wavelengths, and hence can make small momenta arbitrarily large. This however runs into conflict with the idea of cutting off momentum integrals. For this reason approaches to quantum gravity that rely on discretization or analogies to condensed matter systems are difficult to reconcile with Lorentz-invariance.

So what, you may say, let’s just throw out Lorentz-invariance then. Let us just take a tiny lattice spacing so that we won’t see the effects. Unfortunately, it isn’t that easy. Violations of Lorentz-invariance, even if tiny, spill over into all kinds of observables even at low energies.

A good example is vacuum Cherenkov radiation, that is the spontaneous emission of a photon by an electron. This effect is normally – ie when Lorentz-invariance is respected – forbidden due to energy-momentum conservation. It can only take place in a medium which has components that can recoil. But Lorentz-invariance violation would allow electrons to radiate off photons even in empty space. No such effect has been seen, and this leads to very strong bounds on Lorentz-invariance violation.

And this isn’t the only bound. There are literally dozens of particle interactions that have been checked for Lorentz-invariance violating contributions with absolutely no evidence showing up. Hence, we know that Lorentz-invariance, if not exact, is respected by nature to extremely high precision. And this is very hard to achieve in a model that relies on a discretization.

Having said that, I must point out that not every quantity of dimension length actually transforms as a distance. Thus, the existence of a fundamental length scale is not a priori in conflict with Lorentz-invariance. The best example is maybe the Planck length itself. It has dimension length, but it’s defined from constants of nature that are themselves frame-independent. It has units of a length, but it doesn’t transform as a distance. For the same reason string theory is perfectly compatible with Lorentz-invariance even though it contains a fundamental length scale.

The tension between discreteness and Lorentz-invariance appears always if you have objects that transform like distances or like areas or like spatial volumes. The Causal Set approach therefore is an exception to the problems with discreteness (to my knowledge the only exception). The reason is that Causal Sets are a randomly distributed collection of (unconnected!) points with a four-density that is constant on the average. The random distribution prevents the problems with regular lattices. And since points and four-volumes are both Lorentz-invariant, no preferred frame is introduced.

It is remarkable just how difficult Lorentz-invariance makes it to reconcile general relativity with quantum field theory. The fact that no violations of Lorentz-invariance have been found and the insight that discreteness therefore seems an ill-fated approach has significantly contributed to the conviction of string theorists that they are working on the only right approach. Needless to say there are some people who would disagree, such as probably Carlo Rovelli and Garrett Lisi.

Either way, the absence of Lorentz-invariance violations is one of the prime examples that I draw upon to demonstrate that it is possible to constrain theory development in quantum gravity with existing data. Everyone who still works on discrete approaches must now make really sure to demonstrate there is no conflict with observation.

Thanks for an interesting question!