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I'm giving the last lecture on my mini-tour for The Big Picture tonight at the Natural History Museum here in Los Angeles. If you can't make it, here's a decent substitute: video of the talk I gave last week at Google headquarters in Mountain View.
I don't think I've quite worked out all the kinks in this talk, but you get the general idea. My biggest regret was that I didn't have the time to trace the flow of free energy from the Sun to photosynthesis to ATP to muscle contractions. It's a great demonstration of how biological organisms are maintained through the creation of entropy.
I had a conversation today with Boris Leistedt about his work to build a latent model of galaxy SEDs and get template-space-marginalized photometric redshifts. I proposed that he instantiate the latent variables as observables (like rest-frame colors or line strengths or something); this will help the model break degeneracies and sensibly order the templates in the latent space. That is, it should regularize or simplify the model for inference. That's just an intuition. But it might also help people who have drunk less of our Kool Aid to understand!
In a QFT, the order of insertions in a correlator matters (unless we fix an ordering like time ordering). How is that represented on the bulk side?
A capset in the vector space over the finite field of three elements is a subset of that does not contain any lines , where and . A basic problem in additive combinatorics (discussed in one of the very first posts on this blog) is to obtain good upper and lower bounds for the maximal size of a capset in .
Trivially, one has . Using Fourier methods (and the density increment argument of Roth), the bound of was obtained by Meshulam, and improved only as late as 2012 to for some absolute constant by Bateman and Katz. But in a very recent breakthrough, Ellenberg (and independently Gijswijt) obtained the exponentially superior bound , using a version of the polynomial method recently introduced by Croot, Lev, and Pach. (In the converse direction, a construction of Edel gives capsets as large as .) Given the success of the polynomial method in superficially similar problems such as the finite field Kakeya problem (discussed in this previous post), it was natural to wonder that this method could be applicable to the cap set problem (see for instance this MathOverflow comment of mine on this from 2010), but it took a surprisingly long time before Croot, Lev, and Pach were able to identify the precise variant of the polynomial method that would actually work here.
The proof of the capset bound is very short (Ellenberg's and Gijswijt's preprints are both 3 pages long, and Croot-Lev-Pach is 6 pages), but I thought I would present a slight reformulation of the argument which treats the three points on a line in symmetrically (as opposed to treating the third point differently from the first two, as is done in the Ellenberg and Gijswijt papers; Croot-Lev-Pach also treat the middle point of a three-term arithmetic progression differently from the two endpoints, although this is a very natural thing to do in their context of ). The basic starting point is this: if is a capset, then one has the identity
for all , where is the Kronecker delta function, which we view as taking values in . Indeed, (1) reflects the fact that the equation has solutions precisely when are either all equal, or form a line, and the latter is ruled out precisely when is a capset.
To exploit (1), we will show that the left-hand side of (1) is "low rank" in some sense, while the right-hand side is "high rank". Recall that a function taking values in a field is of rank one if it is non-zero and of the form for some , and that the rank of a general function is the least number of rank one functions needed to express as a linear combination. More generally, if , we define the rank of a function to be the least number of "rank one" functions of the form
for some and some functions , , that are needed to generate as a linear combination. For instance, when , the rank one functions take the form , , , and linear combinations of such rank one functions will give a function of rank at most .
It is a standard fact in linear algebra that the rank of a diagonal matrix is equal to the number of non-zero entries. This phenomenon extends to higher dimensions:
Lemma 1 (Rank of diagonal hypermatrices) Let , let be a finite set, let be a field, and for each , let be a coefficient. Then the rank of the function
Proof: We induct on . As mentioned above, the case follows from standard linear algebra, so suppose now that and the claim has already been proven for .
It is clear that the function (2) has rank at most equal to the number of non-zero (since the summands on the right-hand side are rank one functions), so it suffices to establish the lower bound. By deleting from those elements with (which cannot increase the rank), we may assume without loss of generality that all the are non-zero. Now suppose for contradiction that (2) has rank at most , then we obtain a representation
for some sets of cardinalities adding up to at most , and some functions and .
Consider the space of functions that are orthogonal to all the , in the sense that
for all . This space is a vector space whose dimension is at least . A basis of this space generates a coordinate matrix of full rank, which implies that there is at least one non-singular minor. This implies that there exists a function in this space which is non-zero on some subset of of cardinality at least .
If we multiply (3) by and sum in , we conclude that
The right-hand side has rank at most , since the summands are rank one functions. On the other hand, from induction hypothesis the left-hand side has rank at least , giving the required contradiction.
On the other hand, we have the following (symmetrised version of a) beautifully simple observation of Croot, Lev, and Pach:
Lemma 2 On , the rank of the function is at most , where
Proof: Using the identity for , we have
The right-hand side is clearly a polynomial of degree in , which is then a linear combination of monomials
with with
In particular, from the pigeonhole principle, at least one of is at most .
Consider the contribution of the monomials for which . We can regroup this contribution as
where ranges over those with , is the monomial
and is some explicitly computable function whose exact form will not be of relevance to our argument. The number of such is equal to , so this contribution has rank at most . The remaining contributions arising from the cases and similarly have rank at most (grouping the monomials so that each monomial is only counted once), so the claim follows.
Upon restricting from to , the rank of is still at most . The two lemmas then combine to give the Ellenberg-Gijswijt bound
All that remains is to compute the asymptotic behaviour of . This can be done using the general tool of Cramer's theorem, but can also be derived from Stirling's formula (discussed in this previous post). Indeed, if , , for some summing to , Stirling's formula gives
where is the entropy function
We then have
where is the maximum entropy subject to the constraints
A routine Lagrange multiplier computation shows that the maximum occurs when
and is approximately , giving rise to the claimed bound of .
Remark 3 As noted in the Ellenberg and Gijswijt papers, the above argument extends readily to other fields than to control the maximal size of aubset of that has no non-trivial solutions to the equation , where are non-zero constants that sum to zero. Of course one replaces the function in Lemma 2 by in this case.
Remark 4 This symmetrised formulation suggests that one possible way to improve slightly on the numerical quantity by finding a more efficient way to decompose into rank one functions, however I was not able to do so (though such improvements are reminiscent of the Strassen type algorithms for fast matrix multiplication).
Remark 5 It is tempting to see if this method can get non-trivial upper bounds for sets with no length progressions, in (say) . One can run the above arguments, replacing the function
this leads to the bound where
Unfortunately, is asymptotic to and so this bound is in fact slightly worse than the trivial bound ! However, there is a slim chance that there is a more efficient way to decompose into rank one functions that would give a non-trivial bound on . I experimented with a few possible such decompositions but unfortunately without success.
Remark 6 Return now to the capset problem. Since Lemma 1 is valid for any field , one could perhaps hope to get better bounds by viewing the Kronecker delta function as taking values in another field than , such as the complex numbers . However, as soon as one works in a field of characteristic other than , one can adjoin a cube root of unity, and one now has the Fourier decomposition
Moving to the Fourier basis, we conclude from Lemma 1 that the function on now has rank exactly , and so one cannot improve upon the trivial bound of by this method using fields of characteristic other than three as the range field. So it seems one has to stick with (or the algebraic completion thereof).
Thanks to Jordan Ellenberg and Ben Green for helpful discussions.
This month, a bunch of 'logic hackers' have been seeking to determine the precise boundary between the knowable and the unknowable. The challenge has been around for a long time. But only now have people taken it up with the kind of world-wide teamwork that the internet enables.
A Turing machine is a simple model of a computer. Imagine a machine that has some finite number of states, say N states. It's attached to a tape, an infinitely long tape with lots of squares, with either a 0 or 1 written on each square. At each step the machine reads the number where it is. Then, based on its state and what it reads, it either halts, or it writes a number, changes to a new state, and moves either left or right.
The tape starts out with only 0's on it. The machine starts in a particular 'start' state. It halts if it winds up in a special 'halt' state.
The Busy Beaver Game is to find the Turing machine with N states that runs as long as possible and then halts.
The number BB(N) is the number of steps that the winning machine takes before it halts.
In 1961, Tibor Radó introduced the Busy Beaver Game and proved that the sequence BB(N) is uncomputable. It grows faster than any computable function!
A few values of BB(N) can be computed, but there's no way to figure out all of them.
As we increase N, the number of Turing machines we need to check increases faster than exponentially: it's
Of course, many could be ruled out as potential winners by simple arguments. But the real problem is this: it becomes ever more complicated to determine which Turing machines with N states never halt, and which merely take a huge time to halt.
Indeed, matter what axiom system you use for math, as long as it has finitely many axioms and is consistent, you can never use it to correctly determine BB(N) for more than some finite number of cases.
So what do people know about BB(N)?
For starters, BB(0) = 0. At this point I should admit that people don't count the halt state as one of our N states. This is just a convention. So, when we consider BB(0), we're considering machines that only have a halt state. They instantly halt.
Next, BB(1) = 1.
Next, BB(2) = 6.
Next, BB(3) = 21. This was proved in 1965 by Tibor Radó and Shen Lin.
Next, BB(4) = 107. This was proved in 1983 by Allan Brady.
Next, BB(5). Nobody knows what BB(5) equals!
The current 5-state busy beaver champion was discovered by Heiner Marxen and Jürgen Buntrock in 1989. It takes 47,176,870 steps before it halts. So, we know
People have looked at all the other 5-state Turing machines to see if any does better. But there are 43 machines that do very complicated things that nobody understands. It's believed they never halt, but nobody has been able to prove this yet.
We may have hit the wall of ignorance here… but we don't know.
That's the spooky thing: the precise boundary between the knowable and the unknowable is unknown. It may even be unknowable… but I'm not sure we know that.
Next, BB(6). In 1996, Marxen and Buntrock showed it's at least 8,690,333,381,690,951. In June 2010, Pavel Kropitz proved that
You may wonder how he proved this. Simple! He found a 6-state machine that runs for
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steps and then halts!
Of course, I'm just kidding when I say this was simple. The machine is easy enough to describe, but proving it takes exactly this long to run takes real work! You can read about such proofs here:
• Pascal Michel, The Busy Beaver Competition: a historical survey.
I don't understand them very well. All I can say at this point is that many of the record-holding machines known so far are similar to the famous Collatz conjecture. The idea there is that you can start with any positive integer and keep doing two things:
• if it's even, divide it by 2;
• if it's odd, triple it and add 1.
The conjecture is that this process will always eventually reach the number 1. Here's a graph of how many steps it takes, as a function of the number you start with:
Nice pattern! But this image shows how it works for numbers up to 10 million, and you'll see it doesn't usually take very long for them to reach 1. Usually less than 600 steps is enough!
So, to get a Turing machine that takes a long time to halt, you have to take this kind of behavior and make it much more long and drawn-out. Conversely, to analyze one of the potential winners of the Busy Beaver Game, people must take that long and drawn-out behavior and figure out a way to predict much more quickly when it will halt.
Next, BB(7). In 2014, someone who goes by the name Wythagoras showed that
It's fun to prove lower bounds on BB(N). For example, in 1964 Milton Green constructed a sequence of Turing machines that implies
Here I'm using Knuth's up-arrow notation, which is a recursively defined generalization of exponentiation, so for example
where there are threes in that tower.
But it's also fun to seek the smallest N for which we can prove BB(N) is unknowable! And that's what people are making lots of progress on right now.
Sometime in April 2016, Adam Yedidia and Scott Aaronson showed that BB(7910) cannot be determined using the widely accepted axioms for math called ZFC: that is, Zermelo—Fraenkel set theory together with the axiom of choice. It's a great story, and you can read it here:
• Scott Aaronson, The 8000th Busy Beaver number eludes ZF set theory: new paper by Adam Yedidia and me, Shtetl-Optimized, 3 May 2016.
• Adam Yedidia and Scott Aaronson, A relatively small Turing machine whose behavior is independent of set theory, 13 May 2016.
Briefly, Yedidia created a new programming language, called Laconic, which lets you write programs that compile down to small Turing machines. They took an arithmetic statement created by Harvey Friedman that's equivalent to the consistency of the usual axioms of ZFC together with a large cardinal axiom called the 'stationary Ramsey property', or SRP. And they created a Turing machine with 7910 states that seeks a proof of this arithmetic statement using the axioms of ZFC.
Since ZFC can't prove its own consistency, much less its consistency when supplemented with SRP, their machine will only halt if ZFC+SRP is inconsistent.
Since most set theorists believe ZFC+SRP is consistent, this machine probably doesn't halt. But we can't prove this using ZFC.
In short: if the usual axioms of set theory are consistent, we can never use them to determine the value of BB(7910).
The basic idea is nothing new: what's new is the explicit and rather low value of the number 7910. Poetically speaking, we know the unknowable starts here… if not sooner.
However, this discovery set off a wave of improvements! On the Metamath newsgroup, Mario Carneiro and others started 'logic hacking', looking for smaller and smaller Turing machines that would only halt if ZF—that is, Zermelo–Fraenkel set theory, without the axiom of choice—is inconsistent.
By just May 15th, Stefan O'Rear seems to have brought the number down to 1919. He found a Turing machine with just 1919 states that searches for an inconsistency in the ZF axioms. Interestingly, this turned out to work better than using Harvey Friedman's clever trick.
Thus, if O'Rear's work is correct, we can only determine BB(1919) if we can determine whether ZF set theory is consistent. However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.
For details, see:
• Stefan O'Rear, A Turing machine Metamath verifier, 15 May 2016.
I haven't checked his work, but it's available on GitHub.
What's the point of all this? At present, it's mainly just a game. However, it should have some interesting implications. It should, for example, help us better locate the 'complexity barrier'.
I explained that idea here:
• John Baez, The complexity barrier, Azimuth, 28 October 2011.
Briefly, while there's no limit on how much information a string of bits—or any finite structure—can have, there's a limit on how much information we can prove it has!
This amount of information is pretty low, perhaps a few kilobytes. And I believe the new work on logic hacking can be used to estimate it more accurately!
Morning sunlight illuminated John Preskill's lecture notes. The notes concern Caltech's quantum-computation course, Ph 219. I'm TAing (the teaching assistant for) Ph 219. I previewed lecture material one sun-kissed Sunday.
Pasadena sunlight spilled through my window. So did the howling of a dog that's deepened my appreciation for Billy Collins's poem "Another reason why I don't keep a gun in the house." My desk space warmed up, and I unbuttoned my jacket. I underlined a phrase, braided my hair so my neck could cool, and flipped a page.
I flipped back. The phrase concerned a mathematical statement called "the Yang-Baxter relation." A sunbeam had winked on in my mind: The Yang-Baxter relation described my hair.
The Yang-Baxter relation belongs to a branch of math called "topology." Topology resembles geometry in its focus on shapes. Topologists study spheres, doughnuts, knots, and braids.
Topology describes some quantum physics. Scientists are harnessing this physics to build quantum computers. Alexei Kitaev largely dreamed up the harness. Alexei, a Caltech professor, is teaching Ph 219 this spring.1 His computational scheme works like this.
We can encode information in radio signals, in letters printed on a page, in the pursing of one's lips as one passes a howling dog's owner, and in quantum particles. Imagine three particles on a tabletop.
Consider pushing the particles around like peas on a dinner plate. You could push peas 1 and 2 until they swapped places. The swap represents a computation, in Alexei's scheme.2
The diagram below shows how the peas move. Imagine slicing the figure into horizontal strips. Each strip would show one instant in time. Letting time run amounts to following the diagram from bottom to top.
Imagine swapping peas 1 and 3.
Humor me with one more swap, an interchange of 2 and 3.
Congratulations! You've modeled a significant quantum computation. You've also braided particles.
Let's recap: You began with peas 1, 2, and 3. You swapped 1 with 2, then 1 with 3, and then 2 with 3. The peas end up ordered oppositely the way they began—end up ordered as 3, 2, 1.
You could, instead, morph 1-2-3 into 3-2-1 via a different sequence of swaps. That sequence, or braid, appears below.
Congratulations! You've begun proving the Yang-Baxter relation. You've shown that each braid turns 1-2-3 into 3-2-1.
The relation states also that 1-2-3 is topologically equivalent to 3-2-1: Imagine standing atop pea 2 during the 1-2-3 braiding. You'd see peas 1 and 3 circle around you counterclockwise. You'd see the same circling if you stood atop pea 2 during the 3-2-1 braiding.
That Sunday morning, I looked at John's swap diagrams. I looked at the hair draped over my left shoulder. I looked at John's swap diagrams.
"Yang-Baxter relation" might sound, to nonspecialists, like a mouthful of tweed. It might sound like a sneeze in a musty library. But an eight-year-old could grasp the half the relation. When I braid my hair, I pass my left hand over the back of my neck. Then, I pass my right hand over. But I could have passed the right hand first, then the left. The braid would have ended the same way. The braidings would look identical to a beetle hiding atop what had begun as the middle hunk of hair.
I tried to keep reading John's lecture notes, but the analogy mushroomed. Imagine spinning one pea atop the table.
A 360° rotation returns the pea to its initial orientation. You can't distinguish the pea's final state from its first. But a quantum particle's state can change during a 360° rotation. Physicists illustrate such rotations with corkscrews.
Like the corkscrews formed as I twirled my hair around a finger. I hadn't realized that I was fidgeting till I found John's analysis.
I gave up on his lecture notes as the analogy sprouted legs.
I've never mastered the fishtail braid. What computation might it represent? What about the French braid? You begin French-braiding by selecting a clump of hair. You add strands to the clump while braiding. The addition brings to mind particles created (and annihilated) during a topological quantum computation.
Ancient Greek statues wear elaborate hairstyles, replete with braids and twists. Could you decode a Greek hairdo? Might it represent the first 18 digits in pi? How long an algorithm could you run on Rapunzel's hair?
Call me one bobby pin short of a bun. But shouldn't a scientist find inspiration in every fiber of nature? The sunlight spilling through a window illuminates no less than the hair spilling over a shoulder. What grows on a quantum physicist's head informs what grows in it.
1Alexei and John trade off on teaching Ph 219. Alexei recommends the notes that John wrote while teaching in previous years.
2When your mother ordered you to quit playing with your food, you could have objected, "I'm modeling computations!"
I've had this piece by Rick Borchelt on "science literacy" and this one by Paige Brown Jarreau on "echo chambers" open in tabs for… months. I keep them around because I have thoughts on the general subject, but I keep not writing them up because I suspect that what I want to say won't be read much, and I find it frustrating to put a lot of work into a blog post only to be greeted by crickets chirping.
But, now I find myself in a position where I sort of need to have a more thought-out version of the general argument. So I'm going to do a kind of slapdash blog post working this out as I type, and hopefully end up where I need to be, whether or not anyone else pays any attention.
So. The general thrust of both Borchelt and Jarreau's pieces is pretty similar: a lot of work in "science communication" seems to be misdirected or ineffective. The audience for science blogs and web sites and the rest is drawn from the same limited pool of people who actively seek that stuff out. Most of the rest of the public isn't looking for information about science, and thus, they're not getting it. Which is generally a cue for much hand-wringing among the science-communication crowd over how we're failing, and need to Do Better.
But over the last few years, I've started to wonder whether that's really as big a problem as all the deeply concerned blog posts I've read seem to think. And the reason for that is Beyoncé.
It's not anything that Beyoncé herself did, just the fact that I'm aware of her. I don't own any of her music, and I'm not sure I've ever listened to a complete song of hers, as occasional snippets have been enough to confirm that it's really not my thing. Nevertheless, I know of her, and have a generally positive impression, because news about her manages to impinge on my awareness in a variety of indirect ways– performing at the Super Bowl, bits of gossip on the pop-music station I listen to when SteelyKid's in the car (they don't regularly play her stuff, but they talk about her a bunch), or various science-y people on Twitter gushing about her dropping a new album, etc.
Beyoncé is just the most positive example of a general category of people I don't have any particular reason to care about who I am nonetheless vaguely informed about. I think of this general phenomenon as "stochastic awareness of pop culture." I don't have any systematic knowledge of Beyoncé or the various Kardashians, but I know who they are and a bit about them because that information randomly shows up in front of me. Which is more or less inevitable, because there are a lot of people out there who care very deeply about the activities of these individuals, and pump an enormous amount of effort into generating stories about them. And the end result is that even though her music is not my thing, I have a hazy sense of her place in the pop-culture firmament, and a generally positive impression.
And hand-wringing blog posts aside, I think science communication could do a lot worse than operating on this same basic model. That is, we generate a lot of content about science that is primarily consumed by people who already care about the subject, in the same way that legions of reporters generate endless stories and thinkpieces about Beyoncé and other celebrities. And some fraction of that content will, from time to time, randomly end up impinging on the awareness of people who aren't actively seeking information about science, leaving them with the same kind of stochastic awareness of science news that I have about celebrity culture.
Most of the time, this involves big, splashy stories– LIGO detecting gravitational waves, or the Pluto fly-by, and that kind of thing. In the pop-culture analogy, these are basically like Beyoncé performing at the Super Bowl. But there's also a lot of connections that are essentially random. My favorite personal example of this is when my Forbes blog post about friction, inspired by a silly episode where I didn't lose my phone off the roof of my car, wound up as a story in the Daily Mail. On a typical day, I'm pretty sure the overlap between my blog readership and the readership of the Daily Mail is negligible, but for essentially random reasons, this story ended up being put in front of a lot of people who wouldn't actively seek it out.
From a communications and policy perspective, the hope is that when these stories land in front of people, they spark a "Hey, that's cool…" sort of reaction. Ideally, this might prompt people to learn a bit more about the specific random topic, by reading other articles, or striking up conversations at work, etc. I can't say how effective this is for random phone-on-the-car stories, but it works for big news events– whenever NASA holds a press conference, I can expect a few questions about it at Starbucks the next day, from the regulars who know I'm a scientist. And hopefully those leave a generally positive impression about science as something that's pretty cool and thus worth a bit of money.
I would argue that the implication of the Borchelt and Jarreau posts I linked at the start of this post is that this is essentially what we're already doing. That is, the work people who write about science (or make videos, etc.) are doing mostly ends up in front of an audience who already care about that subject. In the same way that most of what celebrity-culture reporters write about Beyoncé ends up in front of people who already care deeply about pop music. But I think those posts, and a lot of other writing about this, sort of underplay the effect of the occasions when, for random reasons, science news ends up in front of pop-music fans.
This isn't an argument against doing science communication, though I'm sure it will be taken that way by some. It's not even an argument against trying to find better and more effective ways to communicate science. On the contrary, I think both of those things are essential– the more science content we put out there, the better the chance that something breaks through, and if we can figure out and put into practice techniques for making stories land more effectively, we can hopefully boost the impact of those stories that do break out. I think we've seen some real progress on the latter, actually– NASA comes in for some ribbing about the number of press conferences they hold, but their PR people are genuinely good at crafting their messages, and the LHC has done a brilliant job of getting public attention for really abstract stuff.
So I don't mean this as a "Science Communication: You're Doing It Wrong" article. Instead, it's an "Everybody take a deep breath and try to calm down" article. Yes, we're mostly talking to ourselves, but "mostly" isn't "only," and I'm not so sure that stochastic public awareness of science is a major crisis that we need to wring our hands over endlessly.
(The obvious counterargument to my position takes the form "Yes, but science news is Important, while celebrity gossip is just trash." Which is true to a point, but this is yet another area where science is not in any way unique. Pretty much any field of study that has even a slight connection to public policy has the same issues of stochastic public awareness of their subject, which is more Important than whatever Beyoncé is up to these days. And it's been that way for decades (at minimum; Borchelt offers some documentation in the case of science) without the world coming to an end. So, you know, maybe it's not that big a problem.
(I also find that I'm becoming less comfortable with declarations that this issue or that is What Really Matters, because there's a kind of fundamental elitism to the whole business that rubs me the wrong way. At the end of the day, what matters to the general public is what they say matters to them, and if that doesn't align with the elite consensus, it's on us to convince them otherwise. This is maybe a sign that I've been in academia too long, and the secondhand smoke of "postmodernism" is rotting my brain, but whatever…)
A whole bunch of physics posts over at Forbes so far this month:
—Recent Physics Books: Gravitational Waves and Brief Lessons: Short reviews of Janna Levin's Black Hole Blues and Carlo Rovelli's Seven Brief Lessons on Physics.
—The Real Reasons Quantum Entanglement Doesn't Allow Faster-Than-Light Communication: Expanding on and correcting some stuff I didn't like about Ethan "Starts With A Bang" Siegel's take on entanglement as a communications tool.
— Why Does The Rising sun Change Color?: I watched a bunch of sunrises on the cruise, which led to me scribbling equations, and then a blog post.
— How Quantum Entanglement Can Help You Understand Many-Worlds: Entanglement is weird, but serves as a nice concrete system for talking about how decoherence hides quantum behavior without requiring "collapse" into a single well-defined state.
— How Even Research That Sounds Silly Has Value: Kind of a weird thinkpiece, because that's the sort of mood I was in.
— Why Do Physicists Want A "Nuclear Clock"?: A story about a key step toward making a clock based on a nuclear isomer transition in thorium-229 caught my eye, so I tried to explain why that's cool.
— Historians and Astronomers Share These Scientific Methods In Common: A neat project based on digitizing ads about fugitive slaves reminded me of some of the things astronomers do, so I wrote about the parallel.
As usual, quite a range of stuff. I'll be at DAMOP all next week, which may or may not allow time for on-site blogging, but will almost certainly give me some topics to write on when I get back.
The most recent issue of the Notices includes a review by Slava Gerovitch of a book by Amir Alexander called Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. As the reviewer presents it, one of the main points of the book is that science was advanced the most by the people who studied and worked with infinitesimals despite their apparent formal inconsistency. The following quote is from the end of the review:
If… maintaining the appearance of infallibility becomes more important than exploration of new ideas, mathematics loses its creative spirit and turns into a storage of theorems. Innovation often grows out of outlandish ideas, but to make them acceptable one needs a different cultural image of mathematics — not a perfectly polished pyramid of knowledge, but a freely growing tree with tangled branches.
The reviewer makes parallels to more recent situations such as quantum field theory and string theory, where the formal mathematical justification may be lacking but the physical theory is meaningful, fruitful, and made correct predictions, even for pure mathematics. However, I couldn't help thinking of recent examples entirely within pure mathematics as well, and particularly in some fields of interest around here.
Here are a few; feel free to suggest others in the comments (or to take issue with mine).
Informal arguments in higher category theory. For example, Lurie's original paper On infinity topoi lacked a rigorous formal foundation, but contained many important insights. Because quasicategories had already been invented, he was able to make the ideas rigorous in reasonably short order; but I think it's fair to say the price is a minefield of technical lemmas. Nowadays one finds people wanting to say "we work with -categories model-independently" to avoid all the technicalities, but it's unclear whether this quite makes sense. (Although I have some hope now that a formal language closer to the informal one may come out of the Riehl-Verity theory of -cosmoi.)
String diagrams for monoidal categories. Joyal and Street's original paper "The geometry of tensor calculus" carefully defined string diagrams as topological graphs and proved that any labeled string diagram could be interpreted in a monoidal category. But since then, string diagrams have proven so useful that many people have invented variants of them that apply to many different kinds of monoidal categories, and in many (perhaps most) cases they proceed to use them without a similar justifying theorem. Kate and I proved the justifying theorem for our string diagrams for bicategories with shadows, but we didn't even try it with our string diagrams for monoidal fibrations.
Combining higher category with string diagrams, we have the recent "graphical proof assistant" Globular, which formally works with a certain kind of semistrict -category for . It's known that semistrict 3-categories (Gray-categories) suffice to model all weak 3-categories, but no such theorem is yet known for 4-categories. So officially, doing a proof about 4-categories in Globular tells you nothing more than that it's true about semistrict 4-categories, and I suspect that few naturally-ocurring 4-categories are naturally semistrict. However, such an argument clearly has meaning and applicability much more generally.
And, of course, there is homotopy type theory. Plenty of it is completely rigorous, of course (and even formally verified in a computer), but I'm thinking particularly of its conjectural higher-categorical semantics. Pretty much everyone agrees that HoTT should be an internal language for -topoi, but with present technology this depends on an initiality theorem for models of type theories in general that is universally believed to be true but is very fiddly to prove correctly and has only been written down carefully in one special case. Moreover, even granting the initiality theorem there are various slight mismatches between the formal theories in current use and what we can construct in higher toposes to model them, e.g. the universes are not strict enough and the HITs are too big. Nevertheless, this relationship has been very fruitful to both sides of the subject already (the type theory and the category theory).
The title of this post is a reference to a classic remark by Thoreau:
"If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them."
Briefly: it seems to me that the idea of the Croot-Lev-Pach paper I posted about yesterday can indeed be used to give a new bound on the size of subsets of F_3^n with no three-term arithmetic progression! Such a set has size at most (2.756)^n. (There's actually a closed form for the constant, I think, but I haven't written it down yet.)
Here's the preprint. It's very short. I'll post this to the arXiv in a day or two, assuming I (or you) don't find anything wrong with it, so comment if you have comments!
Update: Busy few days of administrative stuff and travel, sorry for not having updated the preprint yet, will try to finish it today. One note, already observed below in the comments: you get a similar bound for subsets of (F_q)^n free of solutions to (ax+by+cz=0) for any (a,b,c) with a+b+c=0; the cap set case is q=3, (a,b,c) = (1,1,1).
Update 2: Dion Gijswijt and I will be submitting this result as a joint paper, which will amalgamate the presentations of our essentially identical arguments. Dion carried out his work independently of mine at around the same time, and the idea should be credited to both of us.
And what they want, apparently, is 470-page treatises on the scientific and philosophical underpinnings of naturalism. To appear soon in the Newspaper of Record:
Happy also to see great science books like Lab Girl and Seven Brief Lessons on Physics make the NYT best-seller list. See? Science isn't so scary at all.
Having sent my cosmology inference draft to various friendlies for them to beat it up, I returned to other priorities. I have the goal to finish two more papers before my sabbatical ends. The first would be something (in the Data Analysis Recipes series) about why you should (or shouldn't) release your data and code. I keep making the same arguments over and over in person and by email, so I should write them down once and for all! This is exactly why I started the fitting-a-line document o-so-many years ago. The second is a paper on diffraction imaging with very few photons. I booted up both of these projects today.
On the first, I made a draft table of all the things that come in on the pro and con sides for releasing data and code. There are lots of overlaps, and lots of things appear in both the pro and the con column! For example, documentation: This is a con, because it is a burden, but a pro because you are encouraged to do it (and benefit from it). And it applies to both data and code.
On the second, I worked through the mathematics of the Ewald sphere, in preparation for generalizing my code so that it doesn't have to work in the small-angle limit.
Here it is. Enjoy! (But sorry, no new questions right now.)
Noted grouchy person John Horgan has found a new way to get people mad at him on the Internet, via a speech-turned-blog-post taking organized Skeptic groups to task for mostly going after "soft targets". This has generated lots of angry blog posts in response, and a far greater number of people sighing heavily and saying "There Horgan goes again…"
If you want to read only one counter to Horgan's piece to get caught up, you could do a lot worse than reading Daniel Loxton's calm and measured response. Loxton correctly notes that Horgan's comments are nothing especially unique, just a variant of an argument that you find everywhere:
I've spent much of my career confronting the common argument that skeptics should not perform the service skeptics do best, but instead tackle other subjects we may not be qualified to address. It's a head scratcher, honestly. "You have specialized expertise in X, but I think X is trivial. Why don't you specialize in Y, because I think Y is important?" Nobody ever says this to Shakespeare scholars or doctors or plumbers. ("Dear 'fire fighters,' fight fires less and solve more murders"?) Seemingly everyone says it to skeptics.
There are only two minor points where I disagree with Loxton. One is the claim that this is primarily deployed only against skeptics, because the general tactic is everywhere. I get occasional comments and emails of the form "Why are you wasting time writing about arcane quantum physics when climate change is so much more important?" The endless arguments defending "the humanities" in academia are another version of the same basic thing– "Why should students study English lit when computer coding is so much more important?" And there's even a sense in which much of the Democratic primary campaign has been dominated by this sort of thing– the arguments between Bernie Sanders supporters and Black Lives Matter activists, for example, basically boil down to each side thinking that the other is too focused on an issue that is not as important as their own primary concern.
So, skeptics have a lot of company in fending off "Your issue is trivial, you should spend more time on what I find most important."
The other tiny disagreement I have is that I would slightly expand the qualifications justifying a decision to work on X rather than Y. That is, I don't think it's just a matter of specialized knowledge, but also a question of temperament. I don't spend a whole lot of time battling quantum kookery– a rich source of targets both hard and soft– not because I lack specialized knowledge, but because I don't have the right sort of personality to be good at it.
It's not that I'm not bothered by charlatans trying to profit from misrepresentations of physics– on the contrary, I'm a little too bothered by it. I do occasionally write about this sort of thing, but it's very difficult for me to do it without becoming snide. It's sort of cathartic to vent about on occasion, but mostly not particularly productive– when I go back to stuff that I write in that mode, I generally don't like the way I sound.
And it's absolutely not in any way sustainable for me. One of the most notable thing about the skeptical fight is that it's neverending. No debunking of Bigfoot, or Ancient Aliens, or quantum crackpottery is ever definitive– the folks on the other side always come back for more. There are two ways to deal with this: you either draw from a bottomless well of righteous indignation, a la Orac, or have a similarly deep reservoir of patience, as Loxton seems to.
I can't really do either of those. I can be patient long enough to give a reasonably gracious reply to the nutty questions I get after public lectures, but that's exhausted pretty quickly. And while I can get angry about this stuff at times, I can't keep it up long enough to sustain me through the fifteenth round of the same stupid shit. I burn out, and that leads nowhere good.
Don't get me wrong– I'm not saying this to disparage Loxton or Orac or any of the other folks out there fighting the good fight. What they do is good and valuable, and I'm glad they're doing it. I'm also glad that I don't have to do it, because I just don't have the temperament.
But in the end, that's the fundamental problem with Horgan's provocation, and the similar arguments deployed by advocates of every Cause Y confronted with people who work on Issue X. It's not necessarily the case that someone who does good work on X will be well suited to help with Y. There's specialized knowledge involved in any of these issues, but also questions of personality and inclination. I'd do a lousy job of fighting kooks even within my field of expertise, let alone some other kind of "more important" political activism, because I don't have the personality for it.
At bottom, this is just the classic problem of specialization and division of labor in economics. Different people are good at different things, and making people do things they're not suited to will get you sub-optimal results. The best course is to have everyone work on the things they're good at: Orac does rage, Loxton does patience, I do "Hey, isn't quantum physics cool?" And Horgan pokes anthills with sticks.
This can be really hard to remember, especially when you're passionately attached to a particular thing. God knows, I do my share of grumbling about the overemphasis on particle physics and lack of attention for atomic and condensed-matter physics. But it's important to try to maintain perspective and recognize that just because you think Y is the most important thing in the world doesn't mean that the world would be improved by making people who are good at X work on Y instead.
Where do we draw the boundary between science and pseudoscience? It's is a question philosophers have debated for as long as there's been science – and last time I looked they hadn't made much progress. When you ask a sociologist their answer is normally a variant of: Science is what scientists do. So what do scientists do?You might have heard that scientists use what's called the scientific method, a virtuous cycle of generating and testing hypotheses which supposedly separates the good ideas from the bad ones. But that's only part of the story because it doesn't tell you where the hypotheses come from to begin with.
"The first thing I noticed was that every once in a while the classifier would spit something out as 'I don't know what category this is' and you'd look at it and it would be what we're calling this fringe stuff. That quite surprised me. How can this classifier that was tuned to figure out category be seemingly detecting quality?It doesn't surprise me much – you can see this happening in comment sections all over the place: The "insiders" can immediately tell who is an "outsider." Often it doesn't take more than a sentence or two, an odd expression, a term used in the wrong context, a phrase that nobody in the field would ever use. It is only consequential that with smart software you can tell insiders from outsiders even more efficiently than humans. According to Ginsparg:
"[Outliers] also show up in the stop word distribution, even if the stop words are just catching the style and not the content! They're writing in a style which is deviating, in a way. [...]
"What it's saying is that people who go through a certain training and who read these articles and who write these articles learn to write in a very specific language. This language, this mode of writing and the frequency with which they use terms and in conjunctions and all of the rest is very characteristic to people who have a certain training. The people from outside that community are just not emulating that. They don't come from the same training and so this thing shows up in ways you wouldn't necessarily guess. They're combining two willy-nilly subjects from different fields and so that gets spit out."
"We've actually had submissions to arXiv that are not spotted by the moderators but are spotted by the automated programme [...] All I was trying to do is build a simple text classifier and inadvertently I built what I call The Holy Grail of Crackpot Filtering."Trying to speak in the code of a group you haven't been part of at least for some time is pretty much impossible, much like it's impossible to fake the accent of a city you haven't lived in for some while. Such in-group and out-group demarcation is subject of much study in sociology, not specifically the sociology of science, but generally. Scientists are human and of course in-group and out-group behavior also shapes their profession, even though they like to deny it as if they were superhuman think-machines.
Martin-Löf type theory has been around for years, as have category theory, topos theory and homotopy theory. Bundle them all together within the package of homotopy type theory, and philosophy suddenly takes a lot more interest.
If you're looking for places to go to hear about this new interest, you are spoilt for choice:
For an event which delves back also to pre-HoTT days, try my
CFA: Foundations of Mathematical Structuralism
12-14 October 2016, Munich Center for Mathematical Philosophy, LMU Munich
In the course of the last century, different general frameworks for the foundations of mathematics have been investigated. The orthodox approach to foundations interprets mathematics in the universe of sets. More recently, however, there have been other developments that call into question the whole method of set theory as a foundational discipline. Category-theoretic methods that focus on structural relationships and structure-preserving mappings between mathematical objects, rather than on the objects themselves, have been in play since the early 1960s. But in the last few years they have found clarification and expression through the development of homotopy type theory. This represents a fascinating development in the philosophy of mathematics, where category-theoretic structural methods are combined with type theory to produce a foundation that accounts for the structural aspects of mathematical practice. We are now at a point where the notion of mathematical structure can be elucidated more clearly and its role in the foundations of mathematics can be explored more fruitfully.
The main objective of the conference is to reevaluate the different perspectives on mathematical structuralism in the foundations of mathematics and in mathematical practice. To do this, the conference will explore the following research questions: Does mathematical structuralism offer a philosophically viable foundation for modern mathematics? What role do key notions such as structural abstraction, invariance, dependence, or structural identity play in the different theories of structuralism? To what degree does mathematical structuralism as a philosophical position describe actual mathematical practice? Does category theory or homotopy type theory provide a fully structural account for mathematics?
Confirmed Speakers:
Call for Abstracts:
We invite the submission of abstracts on topics related to mathematical structuralism for presentation at the conference. Abstracts should include a title, a brief abstract (up to 100 words), and a full abstract (up to 1000 words), blinded for peer review. Authors should send their abstracts (in pdf format), together with their name, institutional affiliation and current position to firstname.lastname@example.com. We will select up to five submissions for presentation at the conference. The conference language is English.
Dates and Deadlines:
Sometimes blog posts about recent breakthroughs can be useful because they convey the main ideas of a proof without getting bogged down in the technical details. But the recent solution of the cap-set problem by Jordan Ellenberg, and independently and fractionally later by Dion Gijswijt, both making crucial use of an amazing lemma of Croot, Lev and Pach that was made public a week or so before, does not really invite that kind of post, since the papers are so short, and the ideas so transparent, that it's hard to know how a blog post can explain them more clearly.
But as I've got a history with this problem, including posting about it on this blog in the past, I feel I can't just not react. So in this post and a subsequent one (or ones) I want to do three things. The first is just to try to describe my own personal reaction to these events. The second is more mathematically interesting. As regular readers of this blog will know, I have a strong interest in the question of where mathematical ideas come from, and a strong conviction that they always result from a fairly systematic process — and that the opposite impression, that some ideas are incredible bolts from the blue that require "genius" or "sudden inspiration" to find, is an illusion that results from the way mathematicians present their proofs after they have discovered them.
From time to time an argument comes along that appears to present a stiff challenge to my view. The solution to the cap-set problem is a very good example: it's easy to understand the proof, but the argument has a magic quality that leaves one wondering how on earth anybody thought of it. I'm referring particularly to the Croot-Lev-Pach lemma here. I don't pretend to have a complete account of how the idea might have been discovered (if any of Ernie, Seva or Peter, or indeed anybody else, want to comment about this here, that would be extremely welcome), but I have some remarks.
The third thing I'd like to do reflects another interest of mine, which is avoiding duplication of effort. I've spent a little time thinking about whether there is a cheap way of getting a Behrend-type bound for Roth's theorem out of these ideas (and I'm not the only one). Although I wasn't expecting the answer to be yes, I think there is some value in publicizing some of the dead ends I've come across. Maybe it will save others from exploring them, or maybe, just maybe, it will stimulate somebody to find a way past the barriers that seem to be there.
There's not actually all that much to say here. I just wanted to comment on a phenomenon that's part of mathematical life: the feeling of ambivalence one has when a favourite problem is solved by someone else. The existence of such a feeling is hardly a surprise, but slightly more interesting are the conditions that make it more or less painful. For me, an extreme example where it was not at all painful was Wiles's solution of Fermat's Last Theorem. I was in completely the wrong area of mathematics to have a hope of solving that problem, so although I had been fascinated by it since boyhood, I could nevertheless celebrate in an uncomplicated way the fact that it had been solved in my lifetime, something that I hadn't expected.
Towards the other end of the spectrum for me personally was Tom Sanders's quasipolynomial version of the Bogolyubov-Ruzsa lemma (which was closely related to his bound for Roth's theorem). That was a problem I had worked on very hard, and some of the ideas I had had were, it turned out, somewhat in the right direction. But Tom got things to work, with the help of further ideas that I had definitely not had, and by the time he solved the problem I had gone for several years without seriously working on it. So on balance, my excitement at the solution was a lot greater than the disappointment that that particular dream had died.
The cap-set problem was another of my favourite problems, and one I intended to return to. But here I feel oddly un-disappointed. The main reason is that I know that if I had started work on it again, I would have continued to try to push the Fourier methods that have been so thoroughly displaced by the Lev-Croot-Pach lemma, and would probably have got nowhere. So the discovery of this proof has saved me from wasting a lot of time at some point in the future. It's also an incredible bonus that the proof is so short and easy to understand. I could almost feel my brain expanding as I read Jordan Ellenberg's preprint and realized that here was a major new technique to add to the toolbox. Of course, the polynomial method is not new, but somehow this application of it, at least for me, feels like one where I can make some headway with understanding why it works, rather than just gasping in admiration at each new application and wondering how on earth anyone thought of it.
That brings me neatly on to the next theme of this post. From now on I shall assume familiarity with the argument as presented by Jordan Ellenberg, but here is a very brief recap.
The key to it is the lemma of Croot, Lev and Pach (very slightly modified), which states that if and is a polynomial of degree in variables such that for every pair of distinct elements , then is non-zero for at most values of , where is the dimension of the space of polynomials in of degree at most .
Why does this help? Well, the monomials we consider are of the form where each . The expected degree of a random such monomial is , and for large the degree is strongly concentrated about its mean. In particular, if we choose , then the probability that a random monomial has degree greater than is exponentially small, and the probability that a random monomial has degree less than is also exponentially small.
Therefore, the dimension of the space of polynomials of degree at most (for this ) is at least , while the dimension of the space of polynomials of degree at most is at most . Here is some constant less than 1. It follows that if is a set of density greater than we can find a polynomial of degree that vanishes everywhere on and doesn't vanish on . Furthermore, if has density a bit bigger than this — say , we can find a polynomial of degree that vanishes on and is non-zero at more than points of . Therefore, by the lemma, it cannot vanish on all with distinct elements of , which implies that there exist distinct such that for some .
Now let us think about the Croot-Lev-Pach lemma. It is proved by a linear algebra argument: we define a map , where is a certain vector space over of dimension , and we also define a bilinear form on , with the property that for every . Then the conditions on translate into the condition that for all distinct . But if is non-zero at more than points in , that gives us such that if and only if , which implies that are linearly independent, which they can't be as they all live in the -dimensional space .
The crucial thing that makes this lemma useful is that we have a huge space of functions — of almost full dimension — each of which can be represented this way with a very small .
The question I want to think about is the following. Suppose somebody had realized that they could bound the size of an AP-free set by finding an almost full-dimensional space of functions, each of which had a representation of the form , where took values in a low-dimensional vector space . How might they have come to realize that polynomials could do the job? Answering this question doesn't solve the mystery of how the proof was discovered, since the above realization seems hard to come by: until you've seen it, the idea that almost all functions could be represented very efficiently like that seems somewhat implausible. But at least it's a start.
Let's turn the question round. Suppose we know that has the property that for every , with taking values in a -dimensional space. That is telling us that if we think of as a matrix — that is, we write for — then that matrix has rank . So we can ask the following question: given a matrix that happens to be of the special form (where the indexing variables live in ), under what circumstances can it possibly have low rank? That is, what about makes have low rank?
We can get some purchase on this question by thinking about how operates as a linear map on functions defined on . Indeed, we have that if is a function defined on (I'm being a bit vague for the moment about where takes its values, though the eventual answer will be ), then we have the formula . Now has rank if and only if the functions of the form form a -dimensional subspace. Note that if is the function , we have that . Since every is a linear combination of delta functions, we are requiring that the translates of should span a subspace of dimension . Of course, we'd settle for a lower dimension, so it's perhaps more natural to say at most . I won't actually write that, but it should be understood that it's what I basically mean.
What kinds of functions have the nice property that their translates span a low-dimensional subspace? And can we find a huge space of such functions?
The answer that occurs most naturally to me is that characters have this property: if is a character, then every translate of is a multiple of , since . So if is a linear combination of characters, then its translates span a -dimensional space. (So now, just to be explicit about it, my functions are taking values in .)
Moreover, the converse is true. What we are asking for is equivalent to asking for the convolutions of with other functions to live in a -dimensional subspace. If we take Fourier transforms, we now want the pointwise products of with other functions to live in a -dimensional subspace. Well, that's exactly saying that takes non-zero values. Transforming back, that gives us that needs to be a linear combination of characters.
But that's a bit of a disaster. If we want an -dimensional space of functions such that each one is a linear combination of at most characters, we cannot do better than to take . The proof is the same as one of the arguments in Ellenberg's preprint: in an -dimensional space there must be at least active coordinates, and then a random element of the space is on average non-zero on at least of those.
So we have failed in our quest to make exponentially close to and exponentially close to zero.
But before we give up, shouldn't we at least consider backtracking and trying again with a different field of scalars? The complex numbers didn't work out for us, but there is one other choice that stands out as natural, namely .
So now we ask a question that's exactly analogous to the question we asked earlier: what kinds of functions have the property that they and their translates generate a subspace of dimension ?
Let's see whether the characters idea works here. Are there functions with the property that ? No there aren't, or at least not any interesting ones, since that would give us that for every , which implies that is constant (and because , that constant has to be 0 or 1).
OK, let's ask a slightly different question. Is there some fairly small space of functions from to that is closed under taking translates? That is, we would like that if belongs to the space, then for each the function also belongs to the space.
One obvious space of functions with this property is linear maps. There aren't that many of these — just an -dimensional space of them (or -dimensional if we interpret "linear" in the polynomials sense rather than the vector-spaces sense) — sitting inside the -dimensional space of all functions from to .
It's not much of a stretch to get from here to noticing that polynomials of degree at most form another such space. For example, we might think, "What's the simplest function I can think of that isn't linear?" and we might then go for something like . That and its translates generate the space of all quadratic polynomials that depend on only. Then we'd start to spot that there are several spaces of functions that are closed under translation. Given any monomial, it and its translates generate the space generated by all smaller monomials. So for example the monomial and its translates generate the space of polynomials of the form . So any down-set of monomials defines a subspace that is closed under translation.
I think, but have not carefully checked, that these are in fact the only subspaces that are closed under translation. Let me try to explain why. Given any function from to , it must be given by a polynomial made out of cube-free monomials. That's simply because the dimension of the space of such polynomials is . And I think that if you take any polynomial, then the subspace that it and its translates generate is generated by all the monomials that are less than a monomial that occurs in with a non-zero coefficient.
Actually no, that's false. If I take the polynomial , then every translate of it is of the form . So without thinking a bit more, I don't have a characterization of the spaces of functions that are closed under translation. But we can at least say that polynomials give us a rich supply of them.
I'm starting this section a day after writing the sections above, and after a good night's sleep I have clarified in my mind something I sort of knew already, as it's essential to the whole argument, which is that the conjectures that briefly flitted across my mind two paragraphs ago and that turned out to be false absolutely had to be false. Their falsity is pretty much the whole point of what is going on. So let me come to that now.
Let me call a subspace closed if it is closed under translation. (Just to be completely explicit about this, by "translation" I am referring to operations of the form , which take a function to the function .) Note that the sum of two closed subspaces is closed. Therefore, if we want to find out what closed subspaces are like, we could do a lot worse than thinking about the closed subspaces generated by a single function, which it now seems good to think of as a polynomial.
Unfortunately, it's not easy to illustrate what I'm about to say with a simple example, because simple examples tend to be too small for the phenomenon to manifest itself. So let us argue in full generality. Let be a polynomial of degree at most . We would like to understand the rank of the matrix , which is equal to the dimension of the closed subspace generated by , or equivalently the subspace generated by all functions of the form .
At first sight it looks as though this subspace could contain pretty well all linear combinations of monomials that are dominated by monomials that occur with non-zero coefficients in . For example, consider the 2-variable polynomial . In this case we are trying to work out the dimension of the space spanned by the polynomials
These live in the space spanned by six monomials, so we'd like to know whether the vectors of the form span the whole of or just some proper subspace. Setting we see that we can generate the standard basis vectors and . Setting it's not hard to see that we can also get and . And setting we see that we can get the fourth and sixth coordinates to be any pair we like. So these do indeed span the full space. Thus, in this particular case one of my false conjectures from earlier happens to be true.
Let's see why it is false in general. The argument is basically repeating the proof of the Croot-Lev-Pach lemma, but using that proof to prove an equivalent statement (a bound for the rank of the closed subspace generated by ) rather than the precise statement they proved. (I'm not claiming that this is a radically different way of looking at things, but I find it slightly friendlier.)
Let be a polynomial. One thing that's pretty clear, and I think this is why I got slightly confused yesterday, is that for every monomial that's dominated by a monomial that occurs non-trivially in we can find some linear combination of translates such that occurs with a non-zero coefficient. So if we want to prove that these translates generate a low-dimensional space, we need to show that there are some heavy-duty linear dependences amongst these coefficients. And there are! Here's how the proof goes. Suppose that has degree at most . Then we won't worry at all about the coefficients of the monomials of degree at most : sure, these generate a subspace of dimension (that's the definition of , by the way), but unless is very close to , that's going to be very small.
But what about the coefficients of the monomials of degree greater than ? This is where the linear dependences come in. Let be such a monomial. What can we say about its coefficient in the polynomial ? Well, if we expand out and write it as a linear combination of monomials, then the coefficient of will work out as a gigantic polynomial in . However, and this is the key point, this "gigantic" polynomial will have degree at most . That is, for each such monomial , we have a polynomial of degree at most such that gives the coefficient of in the polynomial . But these polynomials all live in the -dimensional space of polynomials of degree at most , so we can find a spanning subset of them of size at most . In other words, we can pick out at most of the polynomials , and all the rest are linear combinations of those ones. This is the huge linear dependence we wanted, and it shows that the projection of the closed subspace generated by to the monomials of degree at least is also at most .
So in total we get that and its translates span a space of dimension at most , which for suitable is much much smaller than . This is what I am referring to when I talk about a "rank miracle".
Note that we could have phrased the entire discussion in terms of the rank of . That is, we could have started with the thought that if is a function defined on such that whenever are distinct elements of , and for at least points , then the matrix would have rank at least , which is the same as saying that and its translates span a space of dimension at least . So then we would be on the lookout for a high-dimensional space of functions such that for each function in the class, and its translates span a much lower-dimensional space. That is what the polynomials give us, and we don't have to mention a funny non-linear function from to a vector space .
I still haven't answered the question of whether the rank miracle is a miracle. I actually don't have a very good answer to this. In the abstract, it is a big surprise that there is a space of functions of dimension that's exponentially close to the maximal dimension such that for every single function in that space, the rank of the matrix is exponentially small. (Here "exponentially small/close" means as a fraction of .) And yet, once one has seen the proof, it begins to feel like a fairly familiar concentration of measure argument: it isn't a surprise that the polynomials of degree at most form a space of almost full dimension and the polynomials of degree at most form a space of tiny dimension. And it's not completely surprising (again with hindsight) that because in the expansion of you can't use more than half the degree for both and , there might be some way of arguing that the translates of live in a subspace of dimension closer to .
This post has got rather long, so this seems like a good place to cut it off. To be continued.
I've just uploaded to the arXiv my paper "Equivalence of the logarithmically averaged Chowla and Sarnak conjectures", submitted to the Festschrift "Number Theory – Diophantine problems, uniform distribution and applications" in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) conjectures in the two-point case. In that paper, the estimate
as was demonstrated, where was any positive integer and denoted the Liouville function. The proof proceeded using a method I call the "entropy decrement argument", which ultimately reduced matters to establishing a bound of the form
whenever was a slowly growing function of . This was in turn established in a previous paper of Matomaki, Radziwill, and myself, using the recent breakthrough of Matomaki and Radziwill.
It is natural to see to what extent the arguments can be adapted to attack the higher-point cases of the logarithmically averaged Chowla conjecture (ignoring for this post the more general Elliott conjecture for other bounded multiplicative functions than the Liouville function). That is to say, one would like to prove that
as for any fixed distinct integers . As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form
for a slowly growing function of and some fixed (in fact we can take for ), where is the (normalised) local Gowers uniformity norm. (In the case , , this becomes the Fourier-uniformity conjecture discussed in this previous post.) If one then applied the (now proven) inverse conjecture for the Gowers norms, this estimate is in turn equivalent to the more complicated looking assertion
where the supremum is over all possible choices of nilsequences of controlled step and complexity (see the paper for definitions of these terms).
The main novelty in the paper (elaborating upon a previous comment I had made on this blog) is to observe that this latter estimate in turn follows from the logarithmically averaged form of Sarnak's conjecture (discussed in this previous post), namely that
whenever is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in (1) means that we need a little bit more; roughly speaking, we need the class of nilsequences of a given step and complexity to have "uniformly zero entropy" in some sense.
On the other hand, it was already known (see previous post) that the Chowla conjecture implied the Sarnak conjecture, and similarly for the logarithmically averaged form of the two conjectures. Putting all these implications together, we obtain the pleasant fact that the logarithmically averaged Sarnak and Chowla conjectures are equivalent, which is the main result of the current paper. There have been a large number of special cases of the Sarnak conjecture worked out (when the deterministic sequence involved came from a special dynamical system), so these results can now also be viewed as partial progress towards the Chowla conjecture also (at least with logarithmic averaging). However, my feeling is that the full resolution of these conjectures will not come from these sorts of special cases; instead, conjectures like the Fourier-uniformity conjecture in this previous post look more promising to attack.
It would also be nice to get rid of the pesky logarithmic averaging, but this seems to be an inherent requirement of the entropy decrement argument method, so one would probably have to find a way to avoid that argument if one were to remove the log averaging.
It's one of those days where none of the stuff I probably ought to be writing seems even slightly appealing, so instead I'm going to do something frivolous and morale-boosting, namely think out loud about an imaginary course. Despite being on sabbatical, I do still check my work email, and have caught the edges of a new round of arguments about whether we're providing enough "Gen Ed" science courses pitched at non-majors. The hardest part of this is always meeting the "science with lab" component, because those courses are pretty resource-instensive, and we have a limited set of them that we run through. Which has occasionally led me to speculate about what we could do to offer another science-with-lab course for non-majors.
The one semi-serious idea that's stuck comes from a joking tweet I made some time back, when I said I was going to invent a "physics of sports" class to justify making the college pay for me to go to the Sloan Sports Conference. Which, like a lot of good jokes, sort of stuck in the back of my head, and every now and then I run across something that I mentally add to the imaginary syllabus. Which sort of looks like this, in roughly the order that they'd go in:
Topic 1: Kinematics: Position, velocity, and acceleration, and the relation between them. Basics of video analysis using Tracker to look at things like parabolic motion of thrown objects, acceleration of sprinters, etc.
Labs: Make and analyze video of simple motions. Look at existing video to check physics, a la Rhett's Angry Birds stuff.
Topic 2: Momentum: Conservation of momentum in collisions of simple objects– billiard balls, etc.
Labs: Video analysis of simple mostly-elastic collisions.
Topic 3: Energy: Get into it with deviations from ideal elastic collisions– energy loss in bouncing balls, etc. Potential and kinetic energy, energy loss to thermal motion, etc.
Labs: Video analysis of motion with energy loss: bouncing balls slowing to a stop, projectiles with significant air resistance, etc.
Topic 4: Rotational Motion/ Angular Momentum: Start with energy in bouncing footballs, etc. Look at conservation of angular momentum in ice skaters, gymnasts, etc.
Labs: Video analysis of motion of spinning things.
Topic 5: Basic Probability and Statistics: Different types of averages, their pros and cons. Some discussion of uncertainties.
Labs: Flipping coins, rolling dice, etc.
Topic 6: More Advanced Statistics: Why do all these people on ESPN go on about "advanced metrics" all the time?
Labs: "Hot hand" experiment?
Topic 7: Final Project: some sort of student-chosen investigation to end the term. Either a pro sports incident that can be analyzed on video, or making a video of something that can be analyzed.
Typed out like that, this seems like a good deal of fun, but also a shitload of work. There are a whole host of reasons why this probably won't happen, of course, but it has served its basic mood-elevating purpose for the morning, so hooray for that. And if you have suggestions of topics to add, or grant funds you'd be willing to direct my way to actually do this, well, you know where the comments are…
I've supervised a lot of great 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 a reasonable 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:
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 H2O and CO2 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 nth 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:
We also know some lower bounds:
See Heiner Marxen's page or the Googology Wiki (which somehow I only learned about today) for more information.
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 I'd personally have independent reasons to think about (but is that just a sign of my limited perspective?). 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 and by Davis, Matijasevich, and Robinson (we used the latter; an earlier version of this post incorrectly stated that we used the Lagarias result).
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, 32=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 other 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!
My friend Dennis The recently gave a new description of the Lie algebra of (as well as all the other complex simple Lie algebras, except ) as the symmetries of a system of partial differential equations. Even better, when he writes down his PDE explicitly, the exceptional Jordan algebra makes an appearance, as we will see.
This is a story with deep roots: it goes back to two very different models for the Lie algebra of , one due to Cartan and one due to Engel, which were published back-to-back in 1893. Dennis figured out how these two results are connected, and then generalized the whole story to nearly every simple Lie algebra, including .
Let's begin with that model of due to Cartan: the Lie algebra is formed by the infinitesimal symmetries of the system of PDE What does it mean to be an infintesimal symmetry of a PDE? To understand this, we need to see how PDE can be realized geometrically, using jet bundles.
A jet bundle over is a bundle whose sections are given by holomorphic functions and their partials, up to some order. Since we have a 2nd order PDE, we need the 2nd jet bundle: This is actually the trivial bundle whose total space is , but we label the coordinates suggestively: The bundle projection just picks out .
For the moment, , and so on are just the names of some extra coordinates and have nothing to do with derivatives. To relate them, we choose some distinguished 1-forms on , called the contact 1-forms, spanned by holomorphic combinations of These are chosen so that, if our suggestively named variables really were partials, these 1-forms would vanish.
For any holomorphic function we get a section of , called the prolongation of . It simply takes those variables that we named after the partial derivatives seriously, and gives us the actual partial derivatives of in those slots: Conversely, an arbitrary section of is the prolongation of some if and only if it annihilates the contact 1-forms. Since contact 1-forms are spanned by , and , it suffices that: Such sections are called holonomic. This correspondence between prolongations and holonomic sections is the key to thinking about jet bundles.
Our PDE carves out a submanifold of . Solutions correspond to local holonomic sections that land in . In general, PDE give us submanifolds of jet spaces.
The external symmetries of our PDE are those diffeomorphisms of that send contact 1-forms to contact 1-forms and send to itself. The infinitesimal external symmetries are vector fields that preserve and the contact 1-forms. There are also things called internal symmetries, but I won't need them here.
So now we're ready for:
Amazing theorem 1. The infinitesimal external symmetries of our PDE is the Lie algebra .
Like I said above, Dennis takes this amazing theorem of Cartan and connects it to an amazing theorem of Engel, and then generalizes the whole story to nearly all simple complex Lie algebras. Here's Engel's amazing theorem:
Amazing theorem 2. is the Lie algebra of infinitesimal contact transformations on a 5-dim contact manifold preserving a field of twisted cubic varieties.
This theorem lies at the heart of the story, so let me explain what it's saying. First, it requires us to become acquainted with contact geometry, the odd-dimensional cousin of symplectic geometry. A contact manifold is a -dimensional manifold with a contact distribution on it. This is a smoothly-varying family of -dimensional subspaces of each tangent space , satisfying a certain nondegeneracy condition.
In Engel's theorem, is 5-dimensional, so each is 4-dimensional. We can projectivize each to get a 3-dimensional projective space over each point. Our field of twisted cubic varieties is a curve in each of these projective spaces, the image of a cubic map: This gives us a curve in each , and taken together this is our field of twisted cubic varieties, . Engel gave explicit formulas for a contact structure on with a twisted cubic field whose symmetries are , and you can find these formulas in Dennis's paper.
How are these two theorems related? The secret is to go back to thinking about jet spaces, except this time, we'll start with the 1st jet space: This comes equipped with a space of contact 1-forms, spanned by a single 1-form: And now we see where contact 1-forms get their name: this contact 1-form defines a contact structure on , given by .
Many of you may know Darboux's theorem in symplectic geometry, which says that any two symplectic manifolds of the same dimension look the same locally. In contact geometry, the analogue of Darboux's theorem holds, and goes by the name of Pfaff's theorem. By Pfaff's theorem, there's an open set in which is contactomorphic to an open set in with Engel's contact structure. And we can use this map to transfer our twisted cubic field to , or at least an open subset of it. This gives us a twisted cubic field on , one that continues to have symmetry.
We are getting tantalizingly close to a PDE now. We have a jet space , with some structure on it. We just lack a submanifold of that jet space. Our twisted cubic field gives us a curve in each , not in itself.
To these ingredients, add a bit of magic. Dennis found a natural construction that takes our twisted cubic field and gives us a submanifold of a space that, at least locally, looks like , and hence describes a PDE. This PDE is the PDE.
It works like this. Our contact 1-form endows each with a symplectic structure, . Starting with our contact structure, , this symplectic structure is only defined up to rescaling, because determines only up to rescaling. Nonetheless, it makes sense to look for subspaces of that are Lagrangian: subspaces of maximal dimension on which vanishes. The space of all Lagrangian subspaces of is called the Lagrangian-Grassmannian, , and we can form a bundle whose fiber over each point is . It turns out is locally the same as , complete the with latter's complement of contact 1-forms.
Dennis's construction takes and gives us a submanifold of , as follows. Remember, each is a curve in . The tangent space to a point is thus a line in the projective space , and this corresponds to 2-dimensional subspace of the 4-dimensional contact space . This subspace turns out to be Lagrangian! Thus, points of give us points of , and letting and vary, we get a submanifold of . Locally, this is our PDE.
Dennis then generalizes this story to all simple Lie algebras besides . For simple Lie groups other than those in the and series, there is a homogenous space with a natural contact structure that has a field of twisted varieties living on it, called the field of "sub-adjoint varieties". The same construction that worked for now gives PDE for these. The and cases take more care.
Better yet, Dennis builds on work of Landsberg and Manivel to get explicit descriptions of all these PDE in terms of cubic forms on Jordan algebras! Landsberg and Manivel describe the field of sub-adjoint varieties using these cubic forms. For , the Jordan algebra in question is the complex numbers with the cubic form
Given any Jordan algebra with a cubic form on it, first polarize : and then cook up a PDE for a function as follows: where , and I've used the indices , , and for coordiantes in , 0 for the coordinate in . For , this gives us the PDE which is clearly equivalent to the PDE we wrote down earlier. Note that this PDE is determined entirely by the cubic form - the product on our Jordan algebra plays no role.
Now we're ready for Dennis's amazing theorem.
Amazing theorem 3. Let , the exceptional Jordan algebra, and be the cubic form on given by the determinant. Then the following PDE on has external symmetry algebra .
Thanks to Dennis The for explaining his work to me, and for his comments on drafts of this post.
(-2) Today's Bonus Announcement: Stefan O'Rear says that his Turing machine to search for contradictions in ZFC is now down to 1919 states. If verified, this is an important milestone: our upper bound on the number of Busy Beaver values that are knowable in standard mathematics is now less than the number of years since the birth of Christ (indeed, even since the generally-accepted dates for the writing of the Gospels).
Stefan also says that his Not-Quite-Laconic system has yielded a 1008-state Turing machine to search for counterexamples to the Riemann Hypothesis, improving on our 5372 states.
(-1) Another Bonus Announcement: Great news, everyone! Using a modified version of Adam Yedidia's Laconic language (which he calls NQL, for Not Quite Laconic), Stefan O'Rear has now constructed a 5349-state Turing machine that directly searches for contradictions in ZFC (or rather in Metamath, which is known to be equivalent to ZFC), and whose behavior is therefore unprovable in ZFC, assuming ZFC is consistent. This, of course, improves on my and Adam's state count by 2561 states—but it also fixes the technical issue with needing to assume a large cardinal axiom (SRP) in order to prove that the TM runs forever. Stefan promises further state reductions in the near future.
In other news, Adam has now verified the 43-state Turing machine by Jared S that halts iff there's a counterexample to Goldbach's Conjecture. The 27-state machine by code golf addict is still being verified.
(0) Bonus Announcement: I've had half a dozen "Ask Me Anything" sessions on this blog, but today I'm trying something different: a Q&A session on Quora. The way it works is that you vote for your favorite questions; then on Tuesday, I'll start with the top-voted questions and keep going down the list until I get tired. Fire away! (And thanks to Shreyes Seshasai at Quora for suggesting this.)
(1) When you announce a new result, the worst that can happen is that the result turns out to be wrong, trivial, or already known. The best that can happen is that the result quickly becomes obsolete, as other people race to improve it. With my and Adam Yedidia's work on small Turing machines that elude set theory, we seem to be heading for that best case. Stefan O'Rear wrote a not-quite-Laconic program that just searches directly for contradictions in a system equivalent to ZFC. If we could get his program to compile, it would likely yield a Turing machine with somewhere around 6,000-7,000 states whose behavior was independent of ZFC, and would also fix the technical problem with my and Adam's machine Z, where one needed to assume a large-cardinal axiom called SRP to prove that Z runs forever. While it would require a redesign from the ground up, a 1,000-state machine whose behavior eludes ZFC also seems potentially within reach using Stefan's ideas. Meanwhile, our 4,888-state machine for Goldbach's conjecture seems to have been completely blown out of the water: first, a commenter named Jared S says he's directly built a 73-state machine for Goldbach (now down to 43 states); second, a commenter named "code golf addict" claims to have improved on that with a mere 31 states (now down to 27 states). These machines are now publicly posted, but still await detailed verification.
(2) My good friend Jonah Sinick cofounded Signal Data Science, a data-science summer school that will be running for the second time this summer. They operate on an extremely interesting model, which I'm guessing might spread more widely: tuition is free, but you pay 10% of your first year's salary after finding a job in the tech sector. He asked me to advertise them, so—here!
(3) I was sad to read the news that Uber and Lyft will be suspending all service in Austin, because the city passed an ordinance requiring their drivers to get fingerprint background checks, and imposing other regulations that Uber and Lyft argue are incompatible with their model of part-time drivers. The companies, of course, are also trying to send a clear message to other cities about what will happen if they don't get the regulatory environment they want. To me, the truth of the matter is that Uber/Lyft are like the web, Google, or smartphones: clear, once-per-decade quality-of-life advances that you try once, and then no longer understand how you survived without. So if Austin wants to maintain a reputation as a serious, modern city, it has no choice but to figure out some way to bring these companies back to the negotiating table. On the other hand, I'd also say to Uber and Lyft that, even if they needed to raise fares to taxi levels to comply with the new regulations, I expect they'd still do a brisk business!
For me, the "value proposition" of Uber has almost nothing to do with the lower fares, even though they're lower. For me, it's simply about being able to get from one place to another without needing to drive and park, and also without needing desperately to explain where you are, over and over, to a taxi dispatcher who sounds angry that you called and who doesn't understand you because of a combination of language barriers and poor cellphone reception and your own inability to articulate your location. And then wondering when and if your taxi will ever show up, because the dispatcher couldn't promise a specific time, or hung up on you before you could ask them. And then embarking on a second struggle, to explain to the driver where you're going, or at least convince them to follow the Google Maps directions. And then dealing with the fact that the driver has no change, you only have twenties and fifties, and their little machine that prints receipts is out of paper so you can't submit your trip for reimbursement either.
So yes, I really hope Uber, Lyft, and the city of Austin manage to sort this out before Dana and I move there! On the other hand, I should say that there's another part of the new ordinance—namely, requiring Uber and Lyft cars to be labeled—that strikes me as an unalloyed good. For if there's one way in which Uber is less convenient than taxis, it's that you can never figure out which car is your Uber, among all the cars stopping or slowing down near you that look vaguely like the one in the app.
Sabbatical is an unreal experience. In case the loyal reader was wondering How awesome is your job?, let me add to the list of awesome the fact of having a year (every seven, in principle) with no teaching or committee duties, in which I can do whatever I want! One piece of evidence that this is happening right now is that I just finished the third (zeroth draft) first-author paper of this academic year, which is about six times as many first-author papers I complete in a normal year (yes, my normal rate is 0.5 first-author per year; thankfully my junior colleagues are far more productive and permit my co-authorship).
Of course these three first-author papers are not really done: On one I need to respond to referee, on one I am waiting for a bit of work from collaborators, and this new one is still very, very rough. But it is fully drafted, from end to end. The subject is: How we compare cosmological simulations to cosmological data, and the incorrect inferences we might be drawing by the wrongness of all this.
Getting caught up to today, a bunch of pictures from the past week (250/366 was the photo of Emmy's memorial shrub):
251/366: Wagons Ho!
Last weekend we had gorgeous spring weather, so the kids were playing outside a bunch. They ended up getting out the wooden wagon SteelyKid got for her birthday some years back, and giving each other rides. This inevitably resulted in me being asked to tow them around the neighborhood, but they did return the favor…
(Photo by Kate.) They're very strong.
Just a cell-phone snap of a display in the hall at the JCC, so not a great picture. It does, however, commemorate two things: 1) SteelyKid being honored for being a good kid at her after-school program, and 2) the bug-eyed face she's doing when asked to smile for a picture these days. This is entirely deliberate on her part, for the record– she knows it's a goofy face, and delights in making it whenever anybody tries to take a picture.
253/366: Lead Off
SteelyKid is playing rec softball this spring, which gives me an excuse to take a lot of telephoto pictures, because baseball variants take a loooooong time. This is probably my favorite from the batch of photos I got at her very first game on Tuesday.
254/366: At Bat
Of course, I can't post a softball picture without showing her at the bat. Where she's the Ted Williams of the second-grade set– the longest at-bat she's had in two games has gone three pitches.
255/366: Climb High, Climb Far
The park where the softball games are played has this big Platonic solid with a web of ropes inside for kids to climb on. When we arrived on Tuesday, The Pip promptly climbed to the little platform in the center, then demanded in a slightly whiny voice "Dad, why can't I climb higher than this?" I was a little frazzled trying to identify the correct field for SteelyKid's game and herd her in that direction, so I distractedly replied "What? You don't have to stop there. Climb as high as you like. Go nuts."
So, yeah, that got a little scary. Notice Kate in the lower right, for scale.
256/366: Sakura
I realize that in a lot of the country, peak flowering-tree season passed a month or more ago, but remember, we're in the Northeast. So we just had pretty cherry blossoms this week on the small tree in our front hard.
|Einstein and Lorentz.
[Image: Wikipedia]
The longer answer is that the invariance of the speed of light features prominently in the popular explanations of special relativity for historic reasons, not for technical reasons. Einstein was lead to special relativity contemplating what it would be like to travel with light, and then tried to find a way to accommodate an observer's motion with the invariance of the speed of light. But the derivation of special relativity is much more general than that, and it is unnecessary to postulate that the speed of light is invariant.Special relativity is really just physics in Minkowski space, that is the 4-dimensional space-time you obtain after promoting time from a parameter to a coordinate. Einstein wanted the laws of physics to be the same for all inertial observers in Minkowski-space, ie observers moving at constant velocity. If you translate this requirement into mathematics, you are lead to ask for the symmetry transformations in Minkowski-space. These transformations form a group – the Poincaré-group – from which you can read off all the odd things you have heard of: time-dilatation, length-contraction, relativistic mass, and so on.
Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.
A (real) manifold can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space . On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:
Theorem 1 (Whitney embedding theorem) Let be a compact manifold. Then there exists an embedding from to a Euclidean space .
In fact, if is -dimensional, one can take to equal , which is often best possible (easy examples include the circle which embeds into but not , or the Klein bottle that embeds into but not ). One can also relax the compactness hypothesis on to second countability, but we will not pursue this extension here. We give a "cheap" proof of this theorem below the fold which allows one to take equal to .
A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:
Theorem 2 (Nash embedding theorem) Let be a compact Riemannian manifold. Then there exists a isometric embedding from to a Euclidean space .
In order to obtain the isometric embedding, the dimension has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound
is attained, which I believe is still the record for large . (In the converse direction, one cannot do better than , basically because this is the number of degrees of freedom in the Riemannian metric .) Nash's original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).
I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of , but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in is that it allows one to freely exploit the very useful tool of pairing together two maps , to form a combined map that can be closer to an embedding or an isometric embedding than the original maps . This lets one perform a "divide and conquer" strategy in which one first starts with the simpler problem of constructing some "partial" embeddings of and then pairs them together to form a "better" embedding.
In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.
— 1. The Whitney embedding theorem —
To prove the Whitney embedding theorem, we first prove a weaker version in which the embedding is replaced by an immersion:
Theorem 3 (Weak Whitney embedding theorem) Let be a compact manifold. Then there exists an immersion from to a Euclidean space .
Proof: Our objective is to construct a map such that the derivatives are linearly independent in for each . For any given point , we have a coordinate chart from some neighbourhood of to . If we set to be multiplied by a suitable cutoff function supported near , we see that is an immersion in a neighbourhood of . Pairing together finitely many of the and using compactness, we obtain the claim.
Now we upgrade the immersion from the above theorem to an embedding by further use of pairing. First observe that as is smooth and compact, an embedding is nothing more than an immersion that is injective. Let be an immersion. Let be the set of pairs of distinct points such that ; note that this set is compact since is an immersion (and so there is no failure of injectivity when is near the diagonal). If is empty then is injective and we are done. If contains a point , then by pairing with some scalar function that separates and , we can replace by another immersion (in one higher dimension ) such that a neighbourhood of and a neighbourhood of get mapped to disjoint sets, thus effectively removing an open neighbourhood of from . Repeating these procedures finitely many times, using the compactness of , we end up with an immersion which is injective, giving the Whitney embedding theorem.
At present, the embedding of an -dimensional compact manifold could be extremely high dimensional. However, if , then it is possible to project from to by the random projection trick (discussed in this previous post). Indeed, if one picks a random element of the unit sphere, and then lets be the (random) orthogonal projection to the hyperplane orthogonal to , then it is geometrically obvious that will remain an embedding unless either is of the form for some distinct , or lies in the tangent plane to at for some . But the set of all such excluded is of dimension at most (using, for instance, the Hausdorff notion of dimension), and so for almost every in will avoid this set. Thus one can use these projections to cut the dimension down by one for ; iterating this observation we can end up with the final value of for the Whitney embedding theorem.
Remark 4 The Whitney embedding theorem for is more difficult to prove. Using the random projection trick, one can arrive at an immersion which is injective except at a finite number of "double points" where meets itself transversally (think of projecting a knot in randomly down to ). One then needs to "push" the double points out of existence using a device known as the "Whitney trick".
— 2. Reduction to a local isometric embedding theorem —
We now begin the proof of the Nash embedding theorem. In this section we make a series of reductions that reduce the "global" problem of isometric embedding a compact manifold to a "local" problem of turning a near-isometric embedding of a torus into a true isometric embedding.
We first make a convenient (though not absolutely necessary) reduction: in order to prove Theorem 2, it suffices to do so in the case when is a torus (equipped with some metric which is not necessarily flat). Indeed, if is not a torus, we can use the Whitney embedding theorem to embed (non-isometrically) into some Euclidean space , which by rescaling and then quotienting out by lets one assume without loss of generality that is some submanifold of a torus equipped with some metric . One can then use a smooth version of the Tietze extension theorem to extend the metric smoothly from to all of ; this extended metric will remain positive definite in some neighbourhood of , so by using a suitable (smooth) partition of unity and taking a convex combination of with the flat metric on , one can find another extension of to that remains positive definite (and symmetric) on all of , giving rise to a Riemannian torus . Any isometric embedding of this torus into will induce an isometric embedding of the original manifold , completing the reduction.
The main advantage of this reduction to the torus case is that it gives us a global system of (periodic) coordinates on , so that we no longer need to work with local coordinate charts. Also, one can easily use Fourier analysis on the torus to verify the ellipticity properties of the Laplacian that we will need later in the proof. These are however fairly minor conveniences, and it would not be difficult to continue the argument below without having first reduced to the torus case.
Henceforth our manifold is assumed to be the torus equipped with a Riemannian metric , where the indices run from to . Our task is to find an injective map which is isometric in the sense that it obeys the system of partial differential equations
for , where denotes the usual dot product on . Let us write this equation as
where is the symmetric tensor
The operator is a nonlinear differential operator, but it behaves very well with respect to pairing:
We can use (2) to obtain a number of very useful reductions (at the cost of worsening the eventual value of , which as stated in the introduction we will not be attempting to optimise). First we claim that we can drop the injectivity requirement on , that is to say it suffices to show that every Riemannian metric on is of the form for some map into some Euclidean space . Indeed, suppose that this were the case. Let be any (not necessarily isometric) embedding (the existence of which is guaranteed by the Whitney embedding theorem; alternatively, one can use the usual exponential map to embed into ). For small enough, the map is short in the sense that pointwise in the sense of symmetric tensors (or equivalently, the map is a contraction from to ). For such an , we can write for some Riemannian metric . If we then write for some (not necessarily injective) map , then from (2) we see that ; since inherits its injectivity from the component map , this gives the desired isometric embedding.
Call a metric on good if it is of the form for some map into a Euclidean space . Our task is now to show that every metric is good; the relation (2) tells us that the sum of any two good metrics is good.
In order to make the local theory work later, it will be convenient to introduce the following notion: a map is said to be free if, for every point , the vectors , and the vectors , are all linearly independent; equivalently, given a further map , there are no dependencies whatsoever between the scalar functions , and , . Clearly, a free map into is only possible for , and this explains the bulk of the formula (1) of the best known value of .
For any natural number , the "Veronese embedding" defined by
can easily be verified to be free. From this, one can construct a free map by starting with an arbitrary immersion and composing it with the Veronese embedding (the fact that the composition is free will follow after several applications of the chain rule).
Given a Riemannian metric , one can find a free map which is short in the sense that , by taking an arbitrary free map and scaling it down by some small scaling factor . This gives us a decomposition
for some Riemannian metric .
The metric is clearly good, so by (2) it would suffice to show that is good. What is easy to show is that is approximately good:
Proposition 5 Let be a Riemannian metric on . Then there exists a smooth symmetric tensor on with the property that is good for every .
Proof: Roughly speaking, the idea here is to use "tightly wound spirals" to capture various "rank one" components of the metric , the point being that if a map "oscillates" at some high frequency with some "amplitude" , then is approximately equal to the rank one tensor . The argument here is related to the technique of convex integration, which among other things leads to one way to establish the -principle of Gromov.
By the spectral theorem, every positive definite tensor can be written as a positive linear combination of symmetric rank one tensors for some vector . By adding some additional rank one tensors if necessary, one can make this decomposition stable, in the sense that any nearby tensor is also a positive linear combination of the . One can think of as the gradient of some linear function . Using compactness and a smooth partition of unity, one can then arrive at a decomposition
for some finite , some smooth scalar functions (one can take to be linear functions on small coordinate charts, and to basically be cutoffs to these charts).
For any and , consider the "spiral" map defined by
Direct computation shows that
and the claim follows by summing in (using (2)) and taking .
The claim then reduces to the following local (perturbative) statement, that shows that the property of being good is stable around a free map:
Theorem 6 (Local embedding) Let be a free map. Then is good for all symmetric tensors sufficiently close to zero in the topology.
Indeed, assuming Theorem 6, and with as in Proposition 5, we have good for small enough. By (2) and Proposition 5, we then have good, as required.
The remaining task is to prove Theorem 6. This is a problem in perturbative PDE, to which we now turn.
— 3. Proof of local embedding —
We are given a free map and a small tensor . It will suffice to find a perturbation of that solves the PDE
We can expand the left-hand side and cancel off to write this as
where the symmetric tensor-valued first-order linear operator is defined (in terms of the fixed free map ) as
To exploit the free nature of , we would like to write the operator in terms of the inner products and . After some rearranging using the product rule, we arrive at the representation
Among other things, this allows for a way to right-invert the underdetermined linear operator . As is free, we can use Cramer's rule to find smooth maps for (with ) that is dual to in the sense that
where denotes the Kronecker delta. If one then defines the linear zeroth-order operator from symmetric tensors to maps by the formula
then direct computation shows that for any sufficiently regular . As a consequence of this, one could try to use the ansatz and transform the equation (3) to the fixed point equation
One can hope to solve this equation by standard perturbative techniques, such as the inverse function theorem or the contraction mapping theorem, hopefully exploiting the smallness of to obtain the required contraction. Unfortunately we run into a fundamental loss of derivatives problem, in that the quadratic differential operator loses a degree of regularity, and this loss is not recovered by the operator (which has no smoothing properties).
We know of two ways around this difficulty. The original argument of Nash used what is now known as the Nash-Moser iteration scheme to overcome the loss of derivatives by replacing the simple iterative scheme used in the contraction mapping theorem with a much more rapidly convergent scheme that generalises Newton's method; see this previous blog post for a similar idea. The other way out, due to Gunther, is to observe that can be factored as
where is a zeroth order quadratic operator , so that (3) can be written instead as
and using the right-inverse , it now suffices to solve the equation
(compare with (4)), which can be done perturbatively if is indeed zeroth order (e.g. if it is bounded on Hölder spaces such as ).
It remains to achieve the desired factoring (5). We can bilinearise as , where
The basic point is that when is much higher frequency than , then
which can be approximated by applied to some quantity relating to the vector field ; similarly if is much higher frequency than . One can formalise these notions of "much higher frequency" using the machinery of paraproducts, but one can proceed in a slightly more elementary fashion by using the Laplacian operator and its (modified) inverse operator (which is easily defined on the torus using the Fourier transform, and has good smoothing properties) as a substitute for the paraproduct calculus. We begin by writing
The dangerous term here is . Using the product rule and symmetry, we can write
The second term will be "lower order" in that it only involves second derivatives of , rather than third derivatives. As for the higher order term , the main contribution will come from the terms where is higher frequency than (since the Laplacian accentuates high frequencies and dampens low frequencies, as can be seen by inspecting the Fourier symbol of the Laplacian). As such, we can profitably use the approximation (7) here. Indeed, from the product rule we have
Putting all this together, we obtain the decomposition
If we then use Cramer's rule to create smooth functions dual to the in the sense that
then we obtain the desired factorisation (5) with
Note that is the smoothing operator applied to quadratic expressions of up to two derivatives of . As such, one can show using elliptic (Schauder) estimates to show that is Lipschitz continuous in the Holder spaces for (with the Lipschitz constant being small when has small norm); this together with the contraction mapping theorem in the Banach space is already enough to solve the equation (6) in this space if is small enough. This is not quite enough because we also need to be smooth; but it is possible (using Schauder estimates and product Hölder estimates) to establish bounds of the form
for any (with implied constants depending on but independent of ), which can be used (for small enough) to show that the solution constructed by the contraction mapping principle lies in for any (by showing that the iterates used in the construction remain bounded in these norms), and is thus smooth.
When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology. However, there are some nice things one can do in an electronic medium, such as this blog. Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.
Suppose we are in the classical (Kolmogorov) framework of probability theory, in which one has a probability space representing all possible states . One can make a distinction between deterministic quantities that do not depend on the state , and random variables (or stochastic variables) that do depend (in some measurable fashion) on the state . (As discussed in this previous post, it is often helpful to adopt a perspective that suppresses the sample space as much as possible, but we will not do so for the current discussion.)
One can visualise the distinction as follows. If I pick a deterministic integer between and , say , then this fixes the value of for the rest of the discussion:
However, if I pick a random integer uniformly from (e.g. by rolling a fair die), one can think of as a quantity that keeps changing as one flips from one state to the next:
Here, I have "faked" the randomness by looping together a finite number of images, each of which is depicting one of the possible values could take. As such, one may notice that the above image eventually repeats in an endless loop. One could presumably write some more advanced code to render a more random-looking sequence of 's, but the above imperfect rendering should hopefully suffice for the sake of illustration.
Here is a ("faked" rendering of) a random variable that also takes values in , but is non-uniformly distributed, being more biased towards smaller values than larger values:
For continuous random variables, taking values for instance in with some distribution (e.g. uniform in a square, multivariate gaussian, etc.) one could display these random variables as a rapidly changing dot wandering over ; if one lets some "afterimages" of previous dots linger for some time on the screen, one can begin to see the probability density function emerge in the animation. This is unfortunately beyond my ability to quickly whip up as an image; but if someone with a bit more programming skill is willing to do so, I would be very happy to see the result :).
The operation of conditioning to an event corresponds to ignoring all states in the sample space outside of the event. For instance, if one takes the previous random variable , and conditions to the event , one gets the conditioned random variable
One can use the animation to help illustrate concepts such as independence or correlation. If we revert to the unconditioned random variable
and let be an independently sampled uniform random variable from , one can sum the variables together to create a new random variable , ranging in :
(In principle, the above images should be synchronised, so that the value of stays the same from line to line at any given point in time. Unfortunately, due to internet lag, caching, and other web artefacts, you may experience an unpleasant delay between the two. Closing the page, clearing your cache and returning to the page may help.)
If on the other hand one defines the random variable to be , then has the same distribution as (they are both uniformly distributed on , but now there is a very strong correlation between and , leading to completely different behaviour for :
As you know I love the affine cap problem: how big can a subset of (Z/3Z)^n be that contains no three elements summing to 0 — or, in other words, that contains no 3-term arithmetic progression? The best upper bounds, due to Bateman and Katz, are on order 3^n / n^(1+epsilon). And I think it's fair to say that all progress on this problem, since Meshulam's initial results, have come from Fourier-analytic arguments.
So I'm charmed by this paper of Ernie Croot, Vsevolod Lev, and Peter Pach which proves a much stronger result for A = (Z/4Z)^n: a subset with no 3-term arithmetic progression has size at most c^n for c strictly less than 4. Better still (for an algebraic geometer) the argument has no harmonic analysis at all, but proceeds via the polynomial method!
This is surprising for two reasons. First, it's hard to make the polynomial method work well for rings, like Z/4Z, that aren't fields; extending our knowledge about additive combinatorics to such settings is a long-standing interest of mine. Second, the polynomial method over finite fields usually works in the "fixed dimension large field" regime; problems like affine cap, where the base ring is fixed and the dimension are growing, have so far been mostly untouched.
As for the first issue, here's the deal. This looks like a problem over Z/4Z but is really a problem over F_2, because the condition for being a 3-term AP
a – 2b + c = 0
has a 2 in it. In other words: the two outer terms have to lie in the same coset of 2A, and the middle term is only determined up to 2A.
So CLP recast the problem as follows. Let S be a large subset of A with no 3-term AP. Let V be 2A, which is an n-dimensional vector space over F_2. For each v in V, there's a coset of V consisting of the solutions to 2a = v, and we can let S_v be the intersection of S with this coset.
We want to make this a problem about V, not about A. So write T_v for a translate of S_v by some element of the coset, so T_v now sits in V. Which element? Doesn't matter!
We can now write the "no 3-term AP" condition strictly in terms of these subsets of V. Write (T_v – T_v)^* for the set of differences between distinct elements of T_v. Write U for the set of v in V such that T_v is nonempty. Then the union over all v in U of
(T_v – T_v)^* + v
is disjoint from U.
I leave it as an exercise to check the equivalence.
Now we have a combinatorial question about vector spaces over F_2; we want to show that, under the condition above, the sum of |T_v| over all v in U can't be too large.
This is where the polynomial method comes in! CLP show that (over any field, not just F_2), a polynomial of low degree vanishing on (T_v – T_v)^* has to vanish at 0 as well; this is Lemma 1 in their paper. So write down a polynomial P vanishing on V – U; by dimension considerations we can choose one which doesn't vanish on all of V. (This uses the fact that the squarefree monomials of degree up to d are linearly independent functions on F_2^n.) If U is big, we can choose P to have lowish degree.
Since P vanishes on V-U, P has to vanish on (T_v – T_v)^* + v for all v. Since P has low degree, it has to vanish on v too, for all v. But then P vanishes everywhere, contrary to our assumption.
The magic of the paper is in Lemma 1, in my view, which is where you really see the polynomial method applied in this unusual fixed-field-large-dimension regime. Let me say a vague word about how it works. (The actual proof is less than a page, by the way, so I'm not hiding much!) Let P be your polynomial and d its degee. You send your vector space into a subvariety of a much larger vector space W via degree-d Veronese embedding F_d. In fact you do this twice, writing
V x V -> W x W.
Now if P is your polynomial of degree-d, you can think of P(v_1 – v_2) as a bilinear form <,> on W x W. Suppose S is a subset of V such that P(s_1 – s_2) vanishes for all distinct s_1, s_2 in S. That means
<F_d(s_1), F_d(s_2)> = 0
for all distinct s_1, s_2 in S. On the other hand,
<F_d(s_1), F_d(s_1)>
doesn't depend on s_1; it just takes the value P(0). So if P(0) is not equal to 0, you have |S| vectors of nonzero norm which are mutually orthogonal under this bilinear form, and so there can be at most dim W of these, and that's the bound on |S| you need.
This is very slick and I hope the idea is more generally applicable!
Boris Leistedt dropped in today and we discussed his methods to build a physically possible model of galaxy spectral energy distributions and therefore photometric redshifts, but with an exceedingly flexible model. His method is brilliant because it is entirely data-driven (no fixed templates) and yet it respects the physics of special relativity (the Doppler shift), which the machine-learning methods do not.
He made the amusing point that his method can be trained with a training set that contains literally a single galaxy with a spectroscopic redshift! That is, if you even have only one single redshift, you can put photometric redshifts (with, admittedly, large error bars) on all the other photometric galaxies! That is a property that no other data-driven method has. The point is that if you have multi-wavelength data on a single galaxy with a redshift, you can make rough predictions about how other galaxies would look at other redshifts.
His real breakthrough is the idea of using Gaussian processes to put priors on the spectral energy distributions (templates): If the SEDs are drawn from a Gaussian process, then all of the photometry (which consists of linear projections of the SEDs) is also drawn from a Gaussian process. We discussed the magic of all of this.
I also read a proposal by Daniela Huppenkothen, and wrote words in my inference-of-variances paper.
For my inference-of-variances project, I got ABC working. It is delivering a correct posterior, even at a reasonable (finite) distance threshold. All is good! I put a figure showing this into the nascent paper.
Such a great colloquium last week by Peter Sarnak, this year's Hilldale Lecturer, on his paper with Bourgain and Gamburd. My only complaint is that he promised to talk about the mapping class group and then barely did! So I thought I'd jot down what their work has to do with mapping class groups and spaces of branched covers.
Let E be a genus 1 Riemann surface — that is, a torus — and O a point of E. Then pi_1(E-O) is just a free group on two generators, whose commutator is (the conjugacy class of) a little loop around the puncture. If G is a group, a G-cover of E branched only at O is thus a map from pi_1(E-O) to G, which is to say a pair (a,b) of elements of G. Well, such a pair considered up to conjugacy, since we didn't specify a basepoint for our pi_1. And actually, we might as well just think about the surjective maps, which is to say the connected G-covers.
Let's focus on the case G = SL_2(Z). And more specifically on those maps where the puncture class is sent to a matrix of trace -2. Here's an example: we can take
You can check that in this case the puncture class has trace -2; that is, it is the negative of a unipotent matrix. Actually, I gotta be honest, these matrices don't generate SL_2(Z); they generate a finite-index subgroup H of SL_2(Z), its commutator.
Write S for the set of all conjugacy classes of pairs (a,b) of matrices which generate H and have commutator with trace -2. It turns out that this set is the set of integral points of an affine surface called the Markoff surface: namely, if we take x = Tr(a)/3, y = Tr(b)/3, and z = Tr(ab)/3, then the three traces obey the relation
and indeed every solution to this equation corresponds to an element of S.
So the integral points on the Markoff surface are acted on by an infinite discrete group. Which if you just look at the equation seems like kind of a ridiculous miracle. But in the setting of H-covers is very natural. Because there's a natural group acting on S: namely, the mapping class group Γ of type (1,1). This group's whole purpose in life is to act on the fundamental group of a once-punctured torus! (For readers unfamiliar with mapping class groups, I highly recommend Benson Farb and Dan Margalit's wonderful textbook.) So you start with a surjection from pi_1(E-O) to H, you compose with the action of Γ, and you get a new homomorphism. The action of Γ on pi_1(E-O) is only outer, but that's OK, because we're only keeping track of conjugacy classes of homomorphisms from pi_1(E-O) to H.
So Γ acts on S; and now the lovely theorem is that this action is transitive.
I don't want to make this mapping class group business sound more abstract than it is. Γ isn't a mystery group; it acts on H_1(E-O), a free abelian group of rank 2, which gives a map from Γ to SL_2(Z), which turns out to be an isomorphism. What's more, the action of Γ on pairs (a,b) is completely explicit; the standard unipotent generators of SL_2(Z) map to the moves
(a,b) -> (ab,b)
(a,b) -> (a,ab)
(Sanity check: each of these transformations preserves the conjugacy class of the commutator of a and b.)
Sarnak, being a number theorist, is interested in strong approximation: are the integral solutions of the Markoff equation dense in the adelic solutions? In particular, if I have a solution to the Markoff equation over F_p — which is to say, a pair (a,b) in SL_2(F_p) with the right commutator — can I lift it to a solution over Z?
Suppose I have a pair (a,b) which lifts to a pair (a,b). We know (a,b) = g(a_0,b_0) for some g in Γ. Thus (a,b) = g(a_0,b_0). In other words, if strong approximation is true, Γ acts transitively on the set S_p of Markoff solutions mod p. And this is precisely what Bourgain, Gamburd, and Sarnak conjecture. (In fact, they conjecture more: that the Cayley-Schreier graph of this action is an expander, which is kind of a quantitative version of an action being transitive.) One reason to believe this: if we replace F_p with C, we replace S with the SL_2(C) character variety of pi_1(E-O), and Goldman showed long ago that the action of mapping class groups on complex character varieties of fundamental groups was ergodic; it mixes everything around very nicely.
Again, I emphasize that this is on its face a question of pure combinatorial group theory. You want to know if you can get from any pair of elements in SL_2(F_p) with negative-unipotent commutator to any other via the two moves above. You can set this up on your computer and check that it holds for lots and lots of p (they did.) But it's not clear how to prove this transitivity for all p!
They're not quite there yet. But what they can prove is that the action of Γ on S_p has a very big orbit, and has no very small orbits.
Now that G is the finite group SL_2(F_p), we're in my favorite situation, that of Hurwitz spaces. The mapping class group Γ is best seen as the fundamental group of the moduli stack M_{1,1} of elliptic curves. So an action of Γ on the finite set S_p is just a cover H_p of M_{1,1}. It is nothing but the Hurwitz space parametrizing maps (f: X -> E) where E is an elliptic curve and f an SL_2(F_p)-cover branched only at the origin. What Bourgain, Gamburd, and Sarnak conjecture is that H_p is connected.
If you like, this is a moduli space of curves with nonabelian level structure as in deJong and Pikaart. Or, if you prefer (and if H_p is actually connected) it is a noncongruence modular curve corresponding to the stabilizer of an element of S_p in Γ = SL_2(Z). This stabilizer is in general going to be a noncongruence subgroup, except it is a congruence subgroup in the more general sense of Thurston.
This seems like an interesting family of algebraic curves! What, if anything, can we say about it?
One of a series of quick posts on the six sections of my book The Big Picture — Cosmos, Understanding, Essence, Complexity, Thinking, Caring.
Chapters in Part Six, Caring:
In this final section of the book, we take a step back to look at the journey we've taken, and ask what it implies for how we should think about our lives. I intentionally kept it short, because I don't think poetic naturalism has many prescriptive advice to give along these lines. Resisting the temptation to give out a list of "Ten Naturalist Commandments," I instead offer a list of "Ten Considerations," things we can keep in mind while we decide for ourselves how we want to live.
A good poetic naturalist should resist the temptation to hand out commandments. "Give someone a fish," the saying goes, "and you feed them for a day. Teach them to fish, and you feed them for a lifetime." When it comes to how to lead our lives, poetic naturalism has no fish to give us. It doesn't even really teach us how to fish. It's more like poetic naturalism helps us figure out that there are things called "fish," and perhaps investigate the various possible ways to go about catching them, if that were something we were inclined to do. It's up to us what strategy we want to take, and what to do with our fish once we've caught them.
There are nevertheless some things worth saying, because there are a lot of untrue beliefs to which we all tend to cling from time to time. Many (most?) naturalists have trouble letting go of the existence of objective moral truths, even if they claim to accept the idea that the natural world is all that exists. But you can't derive ought from is, so an honest naturalist will admit that our ethical principles are constructed rather than derived from nature. (In particular, I borrow the idea of "Humean constructivism" from philosopher Sharon Street.) Fortunately, we're not blank slates, or computers happily idling away; we have aspirations, desires, preferences, and cares. More than enough raw material to construct workable notions of right and wrong, no less valuable for being ultimately subjective.
Of course there are also incorrect beliefs on the religious or non-naturalist side of the ledger, from the existence of divinely-approved ways of being to the promise of judgment and eternal reward for good behavior. Naturalists accept that life is going to come to an end — this life is not a dress rehearsal for something greater, it's the only performance we get to give. The average person can expect a lifespan of about three billion heartbeats. That's a goodly number, but far from limitless. We should make the most of each of our heartbeats.
The finitude of life doesn't imply that it's meaningless, any more than obeying the laws of physics implies that we can't find purpose and joy within the natural world. The absence of a God to tell us why we're here and hand down rules about what is and is not okay doesn't leave us adrift — it puts the responsibility for constructing meaningful lives back where it always was, in our own hands.
Here's a story one could imagine telling about the nature of the world. The universe is a miracle. It was created by God as a unique act of love. The splendor of the cosmos, spanning billions of years and countless stars, culminated in the appearance of human beings here on Earth — conscious, aware creatures, unions of soul and body, capable of appreciating and returning God's love. Our mortal lives are part of a larger span of existence, in which we will continue to participate after our deaths.
It's an attractive story. You can see why someone would believe it, and work to reconcile it with what science has taught us about the nature of reality. But the evidence points elsewhere.
Here's a different story. The universe is not a miracle. It simply is, unguided and unsustained, manifesting the patterns of nature with scrupulous regularity. Over billions of years it has evolved naturally, from a state of low entropy toward increasing complexity, and it will eventually wind down to a featureless equilibrium condition. We are the miracle, we human beings. Not a break-the-laws-of-physics kind of miracle; a miracle in that it is wondrous and amazing how such complex, aware, creative, caring creatures could have arisen in perfect accordance with those laws. Our lives are finite, unpredictable, and immeasurably precious. Our emergence has brought meaning and mattering into the world.
That's a pretty darn good story, too. Demanding in its own way, it may not give us everything we want, but it fits comfortably with everything science has taught us about nature. It bequeaths to us the responsibility and opportunity to make life into what we would have it be.
I do hope people enjoy the book. As I said earlier, I don't presume to be offering many final answers here. I do think that the basic precepts of naturalism provide a framework for thinking about the world that, given our current state of knowledge, is overwhelmingly likely to be true. But the hard work of understanding the details of how that world works, and how we should shape our lives within it, is something we humans as a species have really only just begun to tackle in a serious way. May our journey of discovery be enlivened by frequent surprises!
One of a series of quick posts on the six sections of my book The Big Picture — Cosmos, Understanding, Essence, Complexity, Thinking, Caring.
Chapters in Part Five, Thinking:
Even many people who willingly describe themselves as naturalists — who agree that there is only the natural world, obeying laws of physics — are brought up short by the nature of consciousness, or the mind-body problem. David Chalmers famously distinguished between the "Easy Problems" of consciousness, which include functional and operational questions like "How does seeing an object relate to our mental image of that object?", and the "Hard Problem." The Hard Problem is the nature of qualia, the subjective experiences associated with conscious events. "Seeing red" is part of the Easy Problem, "experiencing the redness of red" is part of the Hard Problem. No matter how well we might someday understand the connectivity of neurons or the laws of physics governing the particles and forces of which our brains are made, how can collections of such cells or particles ever be said to have an experience of "what it is like" to feel something?
These questions have been debated to death, and I don't have anything especially novel to contribute to discussions of how the brain works. What I can do is suggest that (1) the emergence of concepts like "thinking" and "experiencing" and "consciousness" as useful ways of talking about macroscopic collections of matter should be no more surprising than the emergence of concepts like "temperature" and "pressure"; and (2) our understanding of those underlying laws of physics is so incredibly solid and well-established that there should be an enormous presumption against modifying them in some important way just to account for a phenomenon (consciousness) which is admittedly one of the most subtle and complex things we've ever encountered in the world.
My suspicion is that the Hard Problem won't be "solved," it will just gradually fade away as we understand more and more about how the brain actually does work. I love this image of the magnetic fields generated in my brain as neurons squirt out charged particles, evidence of thoughts careening around my gray matter. (Taken by an MEG machine in David Poeppel's lab at NYU.) It's not evidence of anything surprising — not even the most devoted mind-body dualist is reluctant to admit that things happen in the brain while you are thinking — but it's a vivid illustration of how closely our mental processes are associated with the particles and forces of elementary physics.
The divide between those who doubt that physical concepts can account for subjective experience and those who are think it can is difficult to bridge precisely because of the word "subjective" — there are no external, measurable quantities we can point to that might help resolve the issue. In the book I highlight this gap by imagining a dialogue between someone who believes in the existence of distinct mental properties (M) and a poetic naturalist (P) who thinks that such properties are a way of talking about physical reality:
M: I grant you that, when I am feeling some particular sensation, it is inevitably accompanied by some particular thing happening in my brain — a "neural correlate of consciousness." What I deny is that one of my subjective experiences simply is such an occurrence in my brain. There's more to it than that. I also have a feeling of what it is like to have that experience.
P: What I'm suggesting is that the statement "I have a feeling…" is simply a way of talking about those signals appearing in your brain. There is one way of talking that speaks a vocabulary of neurons and synapses and so forth, and another way that speaks of people and their experiences. And there is a map between these ways: when the neurons do a certain thing, the person feels a certain way. And that's all there is.
M: Except that it's manifestly not all there is! Because if it were, I wouldn't have any conscious experiences at all. Atoms don't have experiences. You can give a functional explanation of what's going on, which will correctly account for how I actually behave, but such an explanation will always leave out the subjective aspect.
P: Why? I'm not "leaving out" the subjective aspect, I'm suggesting that all of this talk of our inner experiences is a very useful way of bundling up the collective behavior of a complex collection of atoms. Individual atoms don't have experiences, but macroscopic agglomerations of them might very well, without invoking any additional ingredients.
M: No they won't. No matter how many non-feeling atoms you pile together, they will never start having experiences.
P: Yes they will.
M: No they won't.
P: Yes they will.
I imagine that close analogues of this conversation have happened countless times, and are likely to continue for a while into the future.
Charles Ehresmann's complete works are now available for free here:
There are 630 pages on algebraic topology and differential geometry, 800 pages on local structures and ordered categories, and their applications to topology, 900 pages on structured categories and quotients and internal categories and fibrations, and 850 pages on sketches and completions and sketches and monoidal closed structures.
That's 3180 pages!
On top of this, more issues of the journal he founded, Cahiers de Topologie et Géométrie Différentielle Catégoriques, will become freely available online.
Andrée Ehresmann announced this magnificent gift to the world on the category theory mailing list, writing:
We are pleased to announce that the issues of the Cahiers de Topologie et Géométrie Différentielle Catégoriques, from Volume L (2009) to LV (2014) included, are now freely downloadable from the internet site of the Cahiers:
through the hyperlink to Recent Volumes.
In the future the issues of the Cahiers will become freely available on the site of the Cahiers two years after their paper publication. We recall that papers published up to Volume XLIX are accessible on the NUMDAM site.
Moreover, the 7 volumes of Charles Ehresmann: Oeuvres complètes et commentées (edited by A. Ehresmann from 1980-83 as Supplements to the Cahiers) are now also freely downloadable from the site
These 2 sites are included in the site of Andrée Ehresmann
and they can also be accessed through hyperlinks on its first page.
Andree Ehresmann, Marino Gran and Rene Guitart,
Chief-Editors of the Cahiers
A couple of songs I want to remember I like.
"Jaltarang" is an old record by Ramrao Parsatwar. The jaltarang is a percussion instrument consisting of bowls partially filled with water. This comes to me via radiooooo, which I heartily recommend.
Something about this instrumental reminds me of EEK's amazing "Trinity," — I don't know what to say about this one except it comes from Egypt and it sounds kind of like video game music but from a video game too fun to humanly play.
But as you know I mostly listen to square Anglophone guitar-based indie so here's a gem in that vein, "Waste the Alphabet," by Dick Diver, like the EEK track a 2015 release. Vocals a little like John Vanderslice, bass that sounds like Athens, GA in 1982, sort of New Zealandy guitar (this is actually from Australia.) Hits all my spots.
One of a series of quick posts on the six sections of my book The Big Picture — Cosmos, Understanding, Essence, Complexity, Thinking, Caring.
Chapters in Part Four, Complexity:
One of the most annoying arguments a scientist can hear is that "evolution (or the origin of life) violates the Second Law of Thermodynamics." The idea is basically that the Second Law says things become more disorganized over time, but the appearance of life represents increased organization, so what do you have to say about that, Dr. Smarty-Pants?
This is a very bad argument, since the Second Law only says that entropy increases in closed systems, not open ones. (Otherwise refrigerators would be impossible, since the entropy of a can of Diet Coke goes down when you cool it.) The Earth's biosphere is obviously an open system — we get low-entropy photons from the Sun, and radiate high-entropy photons back to the universe — so there is manifestly no contradiction between the Second Law and the appearance of complex structures.
As right and true as that response is, it doesn't quite address the question of why complex structures actually do come into being. Sure, they can come into being without violating the Second Law, but that doesn't quite explain why they actually do. In Complexity, the fourth part of The Big Picture, I talk about why it's very natural for such a thing to happen. This covers the evolution of complexity in general, as well as specific questions about the origin of life and Darwinian natural selection. When it comes to abiogenesis, there's a lot we don't know, but good reason to be optimistic about near-term progress.
In 2000, Gretchen Früh-Green, on a ship in the mid-Atlantic Ocean as part of an expedition led by marine geologist Deborah Kelley, stumbled across a collection of ghostly white towers in the video feed from a robotic camera near the ocean floor deep below. Fortunately they had with them a submersible vessel named Alvin, and Kelley set out to explore the structure up close. Further investigation showed that it was just the kind of alkaline vent formation that Russell had anticipated. Two thousand miles east of South Carolina, not far from the Mid-Atlantic Ridge, the Lost City hydrothermal vent field is at least 30,000 years old, and may be just the first known example of a very common type of geological formation. There's a lot we don't know about the ocean floor.
The chemistry in vents like those at Lost City is rich, and driven by the sort of gradients that could reasonably prefigure life's metabolic pathways. Reactions familiar from laboratory experiments have been able to produce a number of amino acids, sugars, and other compounds that are needed to ultimately assemble RNA. In the minds of the metabolism-first contingent, the power source provided by disequilibria must come first; the chemistry leading to life will eventually piggyback upon it.
Albert Szent-Györgyi, a Hungarian physiologist who won the Nobel Prize in 1937 for the discovery of Vitamin C, once offered the opinion that "Life is nothing but an electron looking for a place to rest." That's a good summary of the metabolism-first view. There is free energy locked up in certain chemical configurations, and life is one way it can be released. One compelling aspect of the picture is that it's not simply working backwards from "we know there's life, how did it start?" Instead, its suggesting that life is the solution to a problem: "we have some free energy, how do we liberate it?"
Planetary scientists have speculated that hydrothermal vents similar to Lost City, might be abundant on Jupiter's moon Europa or Saturn's moon Enceladus. Future exploration of the Solar System might be able to put this picture to a different kind of test.
A tricky part of this discussion is figuring out when it's okay to say that a certain naturally-evolved organism or characteristic has a "purpose." Evolution itself has no purpose, but according to poetic naturalism it's perfectly okay to ascribe purposes to specific things or processes, as long as that kind of description actually provides a useful way of talking about the higher-level emergent behavior.
Very happy to see that the L-functions and Modular Forms Database is now live!
When I was a kid we looked up our elliptic curves in Cremona's tables, on paper. Then William Stein created the Modular Forms Database (you can still go there but it doesn't really work) and suddenly you could look at the q-expansions of cusp forms in whatever weight and level you wanted, up to the limits of what William had computed.
The LMFDB is a sort of massively souped up version of Cremona and Stein, put together by a team of dozens and dozens of number theorists, including too many friends of mine to name individually. And it's a lot more than what the title suggests: the incredibly useful Jones-Roberts database of local fields is built in; there's a database of genus 2 curves over Q with small conductor; it even has Maass forms! I've been playing with it all night. It's like an adventure playground for number theorists.
One of my first trips through Stein's database came when I was a postdoc and was thinking about Ljunggren's equation y^2 + 1 = 2x^4. This equation has a large solution, (13,239), which has to do with the classical identity
It turns out, as I explain in an old survey paper, that the existence of such a large solution is "explained" by the presence of a certain weight-2 cuspform in level 1024 whose mod-5 Galois representation is reducible.
With the LMFDB, you can easily wander around looking for more such examples! For instance, you can very easily ask the database for non-CM elliptic curves whose mod-7 Galois representation is nonsurjective. Among those, you can find this handsome curve of conductor 1296, which has very large height relative to its conductor. Applying the usual Frey curve trick you can turn this curve into the Diophantine oddity
3*48383^2 – (1915)^3 = 2^13.
Huh — I wonder whether people ever thought about this Diophantine problem, when can the difference between a cube and three times a square be a power of 2? Of course they did! I just Googled
48383 Diophantine
and found this paper of Stanley Rabinowitz from 1978, which finds all solutions to that problem, of which this one is the largest.
Now whether you can massage this into an arctan identity, that I don't know!
With each passing decade, mathematics grows substantially. As it grows, mathematicians are forced to become more specialized — in the sense of knowing a smaller fraction of the whole — and the time and effort needed to get to the frontier of what is known, and perhaps to contribute to it, increases. One might think that this process will eventually mean that nobody is prepared to make the effort any more, but fortunately there are forces that work in the opposite direction. With the help of the internet, it is now far easier to find things out, and this makes research a whole lot easier in important ways.
It has long been a conviction of mine that the effort-reducing forces we have seen so far are just the beginning. One way in which the internet might be harnessed more fully is in the creation of amazing new databases, something I once asked a Mathoverflow question about. I recently had cause (while working on a research project with a student of mine, Jason Long) to use Sloane's database in a serious way. That is, a sequence of numbers came out of some calculations we did, we found it in the OEIS, that gave us a formula, and we could prove that the formula was right. The great thing about the OEIS was that it solved an NP-ish problem for us: once the formula was given to us, it wasn't that hard to prove that it was correct for our sequence, but finding it in the first place would have been extremely hard without the OEIS.
I'm saying all this just to explain why I rejoice that a major new database was launched today. It's not in my area, so I won't be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.
Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms. This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function and the discriminant modular form , but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.
As of today, the "L-functions and modular forms database" is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops. An article by John Cremona summarising the purpose of the database can be found here.
(Thanks to Andrew Sutherland and Kiran Kedlaya for the information.)
Infinity is a very strange concept. Like alien spores floating down from the sky, large infinities can come down and contaminate the study of questions about ordinary finite numbers! Here's an example.
A shelf is a set with a binary operation that distributes over itself:
There are lots of examples, the simplest being any group, where we define
They have a nice connection to knot theory, which you can see here if you think hard:
My former student Alissa Crans, who invented the term 'shelf', has written a lot about them, starting here:
• Alissa Crans, Lie 2-Algebras, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004.
I could tell you a long and entertaining story about this, including the tale of how shelves got their name. But instead I want to talk about something far more peculiar, which I understand much less well. There's a strange connection between shelves, extremely large infinities, and extremely large finite numbers! It was first noticed by a logician named Richard Laver in the late 1980s, and it's been developed further by Randall Dougherty.
It goes like this. For each there's a unique shelf structure on the numbers such that
So, the elements of our shelf are
and so on, until we get to
However, we can now calculate
and so on. You should try it yourself for a simple example! You'll need to use the self-distributive law. It's quite an experience.
You'll get a list of numbers, but this list will not contain all the numbers Instead, it will repeat with some period
Here is where things get weird. The numbers form this sequence:
1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 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16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, …
It may not look like it, but the numbers in this sequence approach infinity!
… if we assume an extra axiom, which goes beyond the usual axioms of set theory, but so far seems consistent!
This axiom asserts the existence of an absurdly large cardinal, called an I3 rank-into-rank cardinal.
I'll say more about this kind of cardinal later. But, this is not the only case where a 'large cardinal axiom' has consequences for down-to-earth math, like the behavior of some sequence that you can define using simple rules.
On the other hand, Randall Dougherty has proved a lower bound on how far you have to go out in this sequence to reach the number 32.
And, it's an incomprehensibly large number!
The third Ackermann function is roughly 2 to the th power. The fourth Ackermann function is roughly 2 raised to itself times:
And so on: each Ackermann function is defined by iterating the previous one.
Dougherty showed that for the sequence to reach 32, you have to go at least
This is an insanely large number!
I should emphasize that if we use just the ordinary axioms of set theory, the ZFC axioms, nobody has proved that the sequence ever reaches 32. Neither is it known that this is unprovable if we only use ZFC.
So, what we've got here is a very slowly growing sequence… which is easy to define but grows so slowly that (so far) mathematicians need new axioms of set theory to prove it goes to infinity, or even reaches 32.
I should admit that my definition of the Ackermann function is rough. In reality it's defined like this:
And if you work this out, you'll find it's a bit annoying. Somehow the number 3 sneaks in:
where means raised to itself times,
where means with the number repeated times, and so on.
However, these irritating 3's scarcely matter, since Dougherty's number is so large… and I believe he could have gotten an even larger upper bound if he wanted.
Perhaps I'll wrap up by saying very roughly what an I3 rank-into-rank cardinal is.
In set theory the universe of all sets is built up in stages. These stages are called the von Neumann hierarchy. The lowest stage has nothing in it:
Each successive stage is defined like this:
where is the the power set of that is, the set of all subsets of For 'limit ordinals', that is, ordinals that aren't of the form we define
An I3 rank-into-rank cardinal is an ordinal such that admits a nontrivial elementary embedding into itself.
Very roughly, this means the infinity is so huge that the collection of sets that can be built by this stage can mapped into itself, in a one-to-one but not onto way, into a smaller collection that's indistinguishable from the original one when it comes to the validity of anything you can say about sets!
More precisely, a nontrivial elementary embedding of into itself is a one-to-one but not onto function
that preserves and reflects the validity of all statements in the language of set theory. That is: for any sentence in the language of set theory, this statement holds for sets if and only if holds.
I don't know why, but an I3 rank-into-rank cardinal, if it's even consistent to assume one exists, is known to be extraordinarily big. What I mean by this is that it automatically has a lot of other properties known to characterize large cardinals. It's inaccessible (which is big) and ineffable (which is bigger), and measurable (which is bigger), and huge (which is even bigger), and so on.
How in the world is this related to shelves?
The point is that if
are elementary embeddings, we can apply to any set in But in set theory, functions are sets too: sets of ordered pairs So, is a set. It's not an element of but all its subsets are, where So, we can define
Laver showed that this operation distributes over itself:
And, he showed that if we take one elementary embedding and let it generate a shelf by this this operation, we get the free shelf on one generator!
The shelf I started out describing, the numbers with
also has one generator namely the number 1. So, it's a quotient of the free shelf on one generator by one relation, namely the above equation.
That's about all I understand. I don't understand how the existence of a nontrivial elementary embedding of into itself implies that the function goes to infinity, and I don't understand Randall Dougherty's lower bound on how far you need to go to reach For more, read these:
• Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), 209–231.
• Richard Laver, On the algebra of elementary embeddings of a rank into itself, Adv. Math. 110 (1995), 334–346.
• Randall Dougherty and Thomas Jech, Finite left distributive algebras and embedding algebras, Adv. Math. 130 (1997), 201–241.
• Randall Dougherty, Critical points in an algebra of elementary embeddings, Ann. Pure Appl. Logic 65 (1993), 211–241.
• Randall Dougherty, Critical points in an algebra of elementary embeddings, II.
A professor of economics was escorted from an American Airlines flight and questioned by secret police after the woman in the next seat spotted him writing page after page of mysterious symbols. It's all over the internet. Press reports do not specify which differential equation it was.
Although his suspiciously mediterranean appearance may have contributed to his neighbour's paranoia, the professor has the privilege of not having an Arabic name and says he was treated with respect. He's Italian. The flight was delayed by an hour or two, he was allowed to travel, and no harm seems to have been done.
Unfortunately, though, this story is part of a genre. It's happening depressingly often in the US that Muslims (and occasionally others) are escorted off planes and treated like criminals on the most absurdly flimsy pretexts. Here's a story where some passengers were afraid of the small white box carried by a fellow passenger. It turned out to contain baklava. Here's one where a Berkeley student was removed from a flight for speaking Arabic, and another where a Somali woman was ejected because a flight attendant "did not feel comfortable" with her request to change seats. The phenomenon is now common enough that it has acquired a name: "Flying while Muslim".
Among the scientific disciplines, physics is unique: Concerned with the most fundamental entities, its laws must be respected in all other areas of science. While there are many emergent laws which are interesting in their own right – from neurobiology to sociology – there is no doubt they all have to be compatible with energy conservation. And the second law of thermodynamics. And quantum mechanics. And the standard model better be consistent with whatever you think are the neurological processes that make you "you." There's no avoiding physics.In his new book, The Big Picture Sean explains just why you can't ignore physics when you talk about extrasensory perception, consciousness, god, afterlife, free will, or morals. In the first part, Sean lays out what, to our best current knowledge, the fundamental laws of nature are, and what their relevance is for all other emergent laws. In the later parts he then goes through the consequences that follow from this.
One of the big problems with intermittent power sources like wind and solar is the difficulty of storing energy. But if we ever get a lot of electric vehicles, we'll have a lot of batteries—and at any time, most of these vehicles are parked. So, they can be connected to the power grid.
This leads to the concept of vehicle-to-grid or V2G. In a V2G system, electric vehicles can connect to the grid, with electricity flowing from the grid to the vehicle or back. Cars can help solve the energy storage problem.
Here's something I read about vehicle-to-grid systems in Sierra magazine:
At the University of Delaware, dozens of electric vehicles sit in a uniform row. They're part of an experiment involving BMW, power-generating company NRG, and PJM—a regional organization that moves electricity around 13 states and the District of Columbia—that's examining how electric vehicles can give energy back to the electricity grid.
It works like this: When the cars are idle (our vehicles typically sit 95 percent of the time), they're plugged in and able to deliver the electricity in their batteries back to the grid. When energy demand is high, they return electricity to the grid; when demand is low, they absorb electricity. One car doesn't offer much, but 30 of them is another story—worth about 300 kilowatts of power. Utilities will pay for this service, called "load leveling," because it means that they don't have to turn on backup power plants, which are usually coal or natural gas burners. And the EV owners get regular checks—approximately $2.50 a day, or about $900 a year.
It's working well, according to Willett Kempton, a longtime V2G guru and University of Delaware professor who heads the school's Center for Carbon-Free Power Integration: "In three years hooked up to the grid, the revenue was better than we thought. The project, which is ongoing, shows that V2G is viable. We can earn money from cars that are driven regularly."
V2G still has some technical hurdles to overcome, but carmakers—and utilities, too—want it to happen. In a 2014 report, Edison Electric Institute, the power industry's main trade group, called on utilities to promote EVs [electric vehicles], describing EV adoption as a "quadruple win" that would sustain electricity demand, improve customer relations, support environmental goals, and reduce utilities' operating costs.
Utilities appear to be listening. In Virginia and North Carolina, Dominion Resources is running a pilot project to identify ways to encourage EV drivers to only charge during off-peak demand. In California, San Diego Gas & Electric will be spending $45 million on a vehicle-to-grid integration system. At least 25 utilities in 14 states are offering customers some kind of EV incentive. And it's not just utilities—the Department of Defense is conducting V2G pilot programs at four military bases.
Paula DuPont-Kidd, a spokesperson for PJM, says V2G is especially useful for what's called "frequency regulation service"—keeping electricity transmissions at a steady 60 cycles per second. "V2G has proven its ability to be a resource to the grid when power is aggregated," she says. "We know it's possible. It just hasn't happened yet."
I wonder how much, exactly, this system would help.
My quote comes from here:
• Jim Motavalli, Siri, will connected vehicles be greener?, Sierra, May–June 2016.
Motavalli also discusses vehicle-to-vehicle connectivity and vehicle-to-building systems. The latter could let your vehicle power your house during a blackout—which seems of limited use to me, but maybe I don't get the point.
In general, it seems good to have everything I own have the ability to talk to all the rest. There will be security concerns. But as we move toward 'ecotechnology', our gadgets should become less obtrusive, less hungry for raw power, more communicative, and more intelligent.
It's been a year since the last entry here. So I could blog about the end of Planck, the first observation of gravitational waves, fatherhood, or the horror (comedy?) of the US Presidential election. Instead, it's going to be rock 'n' roll, though I don't know if that's because it's too important, or not important enough.
It started last year when I came across Christgau's A+ review of Wussy's Attica and the mentions of Sonic Youth, Nirvana and Television seemed compelling enough to make it worth a try (paid for before listening even in the streaming age). He was right. I was a few years late (they've been around since 2005), but the songs and the sound hit me immediately. Attica was the best new record I'd heard in a long time, grabbing me from the first moment, "when the kick of the drum lined up with the beat of [my] heart", in the words of their own description of the feeling of first listening to The Who's "Baba O'Riley". Three guitars, bass, and a drum, over beautiful screams from co-songwriters Lisa Walker and Chuck Cleaver.
And they just released a new record, Forever Sounds, reviewed in Spin Magazine just before its release:
To certain fans of Lucinda Williams, Crazy Horse, Mekons and R.E.M., Wussy became the best band in America almost instantaneously…
Indeed, that list nailed my musical obsessions with an almost google-like creepiness. Guitars, soul, maybe even some politics. Wussy makes me feel almost like the Replacements) did in 1985.
So I was ecstatic when I found out that Wussy was touring the UK, and their London date was at the great but tiny Windmill in Brixton, one of the two or three venues within walking distance of my flat (where I had once seen one of the other obsessions from that list, The Mekons). I only learned about the gig a couple of days before, but tickets were not hard to get: the place only holds about 150 people, but their were far fewer on hand that night — perhaps because Wussy also played the night before as part of the Walpurgis Nacht festival. But I wanted to see a full set, and this night they were scheduled to play the entire new Forever Sounds record. I admit I was slightly apprehensive — it's only a few weeks old and I'd only listened a few times.
But from the first note (and after a good set from the third opener, Slowgun) I realised that the new record had already wormed its way into my mind — a bit more atmospheric, less song-oriented, than Attica, but now, obviously, as good or nearly so. After the 40 or so minutes of songs from the album, they played a few more from the back catalog, and that was it (this being London, even after the age of "closing time", most clubs in residential neighbourhoods have to stop the music pretty early). Though I admit I was hoping for, say, a cover of "I Could Never Take the Place of Your Man", it was still a great, sloppy, loud show, with enough of us in the audience to shout and cheer (but probably not enough to make very much cash for the band, so I was happy to buy my first band t-shirt since, yes, a Mekons shirt from one of their tours about 20 years ago…). I did get a chance to thank a couple of the band members for indeed being the "best band in America" (albeit in London). I also asked whether they could come back for an acoustic show some time soon, so I wouldn't have to tear myself away from my family and instead could bring my (currently) seven-month old baby to see them some day soon.
They did say UK tours might be a more regular occurrence, and you can follow their progress on the Wussy Road Blog. You should just buy their records, support great music.
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 …
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
$$1+9\left( \frac{1}{4}\right) ^{4}+17\left( \frac{1\cdot5}{4\cdot8}\right)
^{4}+25\left( \frac{1\cdot5\cdot9}{4\cdot8\cdot12}\right) ^{4}+\cdots
=\frac{2^{3/2}}{\pi^{1/2}\Gamma\left( 3/4\right) ^{2}}$$
}}=\left( \sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}\right)
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:
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.
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.
"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.