Time flies! This week I'll finally finish saying what I did on my summer vacation. After my trip to Oslo I stayed in London, or more precisely Greenwich. While there, I talked with some good mathematicians and physicists: in particular, Minhyong Kim, Ray Streater, Andreas Döring and Chris Isham. I also went to a topology conference in Sheffield... and Eugenia Cheng explained some cool stuff on the train ride there. I want to tell you about all this before I forget.
Also, the Tale of Groupoidification has taken a shocking new turn: it's now becoming available as a series of videos.
But first, some miscellaneous fun stuff on math and astronomy.
Math: if you haven't seen a sphere turn inside out, you've got to watch this classic movie, now available for free online:
1) The Geometry Center, Outside in, http://video.google.com/videoplay?docid=-6626464599825291409
Astronomy: did you ever wonder where dust comes from? I'm not talking about dust bunnies under your bed - I'm talking about the dust cluttering our galaxy, which eventually clumps together to form planets and... you and me!
These days most dust comes from aging stars called asymptotic giant branch stars:
The sun will eventually become one of these. The story goes like this: first it'll keep burning until the hydrogen in its core is exhausted. Then it'll cool and become a red giant. Eventually helium at the core will ignite, and the Sun will heat up and shrink again... but its core will then become cluttered with even heavier elements, so it'll cool and expand once more, moving onto the "asymptotic giant branch". At this point it'll have a layered structure: heavier elements near the bottom, then a layer of helium, then hydrogen on the top:
(A similar fate awaits any star between 0.6 and 10 solar masses, though the details depend on the mass. For the more dramatic fate of heavier stars, see "week204".)
Anyway: this layered structure is unstable, so asymptotic giant branch stars pulse every 10 to 100 thousand years or so. And, they puff out dust! Stellar wind then blows this dust out into space.
A great example is the Red Rectangle:
2) Rungs of the Red Rectangle, Astronomy picture of the day, May 13, 2004, http://apod.nasa.gov/apod/ap040513.html
Here two stars 2300 light years from us are spinning around each other while pumping out a huge torus of icy dust grains and hydrocarbon molecules. It's not really shaped like a rectangle or X - it just looks that way. The scene is about 1/3 of a light year across.
Ciska Markwick-Kemper is an expert on dust. She's an astrophysicist at the University of Manchester. Together with some coauthors, she wrote a paper about the Red Rectangle:
3) F. Markwick-Kemper, J. D. Green, E. Peeters, Spitzer detections of new dust components in the outflow of the Red Rectangle, Astrophys. J. 628 (2005) L119-L122. Also available as arXiv:astro-ph/0506473.
They used the Spitzer Space Telescope - an infrared telescope on a satellite in earth orbit - to find evidence of magnesium and iron oxides in this dust cloud.
But, what made dust in the early Universe? It took about a billion years after the Big Bang for asymptotic giant branch stars to form. But we know there was a lot of dust even before then! We can see it in distant galaxies lit up by enormous black holes called "quasars", which pump out vast amounts of radiation as stuff falls into them.
Markwick-Kemper and coauthors have also tackled that question:
4) F. Markwick-Kemper, S. C. Gallagher, D. C. Hines and J. Bouwman, Dust in the wind: crystalline silicates, corundum and periclase in PG 2112+059, Astrophys. J. 668 (2007), L107-L110. Also available as arXiv:0710.2225.
They used spectroscopy to identify various kinds of dust in a distant galaxy: a magnesium silicate that geologists call "forsterite", a magnesium oxide called "periclase", and aluminum oxide, otherwise known as "corundum" - you may have seen it on sandpaper.
And, they hypothesize that these dust grains were formed in the hot wind emanating from the quasar at this galaxy's core!
So, besides being made of star dust, as in the Joni Mitchell song, you also may contain a bit of black hole dust.
Okay - now that we've got that settled, on to London!
Minhyong Kim is a friend I met back in 1986 when he was a grad student at Yale. After dabbling in conformal field theory, he became a student of Serge Lang and went into number theory. He recently moved to England and started teaching at University College, London. I met him there this summer, in front of the philosopher Jeremy Bentham, who had himself mummified and stuck in a wooden cabinet near the school's entrance.
If you're not into number theory, maybe you should read this:
5) Minhyong Kim, Why everyone should know number theory, available at http://www.ucl.ac.uk/~ucahmki/numbers.pdf
Personally I never liked the subject until I realized it was a form of geometry. For example, when we take an equation like this
x2 + y3 = 1
and look at the real solutions, we get a curve in the plane - a "real curve". If we look at the complex solutions, we get something bigger. People call it a "complex curve", because it's analogous to a real curve. But topologically, it's 2-dimensional.
If we use polynomial equations with more variables, we get higher-dimensional shapes called "algebraic varieties" - either real or complex. Either way, we can study these shapes using geometry and topology.
But in number theory, we might study the solutions of these equations in some other number system - for example in Z/p, meaning the integers modulo some prime p. At first glance there's no geometry involved anymore. After all, there's just a finite set of solutions! However, algebraic geometers have figured out how to apply ideas from geometry and topology, mimicking tricks that work for the real and complex numbers.
All this is very fun and mind-blowing - especially when we reach Grothendieck's idea of étale topology, developed around 1958. This is a way of studying "holes" in things like algebraic varieties over finite fields. Amazingly, it gives results that nicely match the results we get for the corresponding complex algebraic varieties! That's part of what the Weil conjectures say.
You can learn the details here:
6) J. S. Milne, Lectures on Étale Cohomology, available at http://www.jmilne.org/math/CourseNotes/math732.html
Anyway, I quizzed about Minhyong about one of the big mysteries that's been puzzling me lately. I want to know why the integers resemble a 3-dimensional space - and how prime numbers give something like "knots" in this space!
I made a small step toward explaining this back in "week205". There I sketched one of the basic ideas of algebraic geometry: every commutative ring, for example the integers or the integers modulo p, has a kind of space associated to it, called its "spectrum". We can think of elements of the commutative ring as functions on this space. I also explained why the process turning a commutative ring into a space is "contravariant". This implies that the obvious map from the integers to the integers modulo p
Z → Z/p
gives rise to a map going the other way between spectra:
Spec(Z/p) → Spec(Z)
In "week218" I reviewed an old argument saying that Spec(Z) is analogous to the complex plane, and that Spec(Z/p) is analogous to a point. From this viewpoint, primes gives something like points in a plane.
However, from a different viewpoint, primes give something like circles in a 3d space!
The easy thing to see is how Spec(Z/p) acts more like a circle than a point. In particular, its "étale topology" resembles the topology of a circle. Oversimplifying a bit, the reason is that just as the circle has one n-fold cover for each integer n > 0, so too does Spec(Z/p). To get the n-fold cover of the circle, you just wrap it around itself n times. To get the n-fold cover of Spec(Z/p), we take the spectrum of the field with pn elements, which is called Fpn. Z/p sits inside this larger field:
Z/p → Fpn
so by the contravariance I mentioned, we get a map going the other way:
Spec(Fpn) → Spec(Z/p)
which is our n-fold cover.
I should explain this in much more detail someday - it involves the relation between étale cohomology, Galois theory and covering spaces. I began tackling this in "week213", but I have a long way to go.
Anyway, the basic idea here is that each prime p gives a "circle" Spec(Z/p) sitting inside Spec(Z). But the really nonobvious part is that according to étale cohomology, Spec(Z) is 3-dimensional - and the different circles corresponding to different primes are linked!
I've been fascinated by this ever since I heard about it, but I got even more interested when I saw a draft of a paper by Kapranov and Smirnov. I got it from Thomas Riepe, who got it from Yuri Manin. There's a version on the web:
7) M. Kapranov and A. Smirnov, Cohomology determinants and reciprocity laws: number field case, available at http://wwwhomes.uni-bielefeld.de/triepe/F1.html
The analogies between number fields and function fields have been a long-time source of inspiration in arithmetic. However, one of the most intriguing problems in this approach, namely the problem of the absolute point, is still far from being satisfactorialy understood. The scheme Spec(Z), the final object in the category of schemes, has dimension 1 with respect to the Zariski topology and at least 3 with respect to the etale topology. This has generated a long-standing desire to introduce a more mythical object P, the "absolute point", with a natural morphism X → P given for any arithmetic scheme X [...]Even though I don't fully understand this, I can tell something big is afoot here. I think they're saying that because Spec(Z) is so big and fancy from the viewpoint of étale topology, there should be some mysterious kind of "point" that's much smaller than Spec(Z) - the "absolute point".
Anyway, in this paper the authors explain how the Legendre symbol of primes is analogous to the linking number of knots.
The Legendre symbol depends on two primes: it's 1 or -1 depending on whether or not the first is a square modulo the second. The linking number depends on two knots: it says how many times the first winds around the second.
The linking number stays the same when you switch the two knots. The Legendre symbol has a subtler symmetry when you switch the two primes: this symmetry is called quadratic reciprocity, and it has lots of proofs, starting with a bunch by Gauss - all a bit tricky.
I'd feel very happy if I truly understood why quadratic reciprocity reduces to the symmetry of the linking number when we think of primes as analogous to knots. Unfortunately, I'll need to think a lot more before I really get the idea. I got into number theory late in life, so I'm pretty slow at it.
This paper studies subtler ways in which primes can be "linked":
8) Masanori Morishita, Milnor invariants and Massey products for prime numbers, Compositio Mathematica 140 (2004), 69-83.
You may know the Borromean rings, a design where no two rings are linked in isolation, but all three are when taken together. Here the linking numbers are zero, but the linking can be detected by something called the "Massey triple product". Morishita generalizes this to primes!
But I want to understand the basics...
The secret 3-dimensional nature of the integers and certain other "rings of algebraic integers" seems to go back at least to the work of Artin and Verdier:
9) Michael Artin and Jean-Louis Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.
You can see it clearly here, starting in section 2:
10) Barry Mazur, Notes on the étale cohomology of number fields, Annales Scientifiques de l'Ecole Normale Superieure Ser. 4, 6 (1973), 521-552. Also available at http://www.numdam.org/numdam-bin/fitem?id=ASENS_1973_4_6_4_521_0
By now, a big "dictionary" relating knots to primes has been developed by Kapranov, Mazur, Morishita, and Reznikov. This seems like a readable introduction:
11) Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields, available as arXiv:math/0107210.
I need to study it. These might also be good - I haven't looked at them yet:
12) Masanori Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141-167.
Masanori Morishita, On analogies between knots and primes, Sugaku 58 (2006), 40-63.
After giving a talk on 2-Hilbert spaces at University College, I went to dinner with Minhyong and some folks including Ray Streater. Ray Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and All That". Like almost every mathematician who has seriously tried to understand quantum field theory, I've learned a lot from this book. So, it was fun meeting Streater, talking with him - and finding out he'd once been made an honorary colonel of the US Army to get a free plane trip to the Rochester Conference! This was a big important particle physics conference, back in the good old days.
He also described Geoffrey Chew's Rochester conference talk on the analytic S-matrix, given at the height of the bootstrap model fad. Wightman asked Chew: why assume from the start that the S-matrix was analytic? Why not try to derive it from simpler principles? Chew replied that "everything in physics is smooth". Wightman asked about smooth functions that aren't analytic. Chew thought a moment and replied that there weren't any.
What's the joke? Well, first of all, Wightman had already succeeded in deriving the analyticity of the S-matrix from simpler principles. Second, any good mathematician - but not necessarily every physicist, like Chew - will know examples of smooth functions that aren't analytic.
Anyway, Streater has just finished an interesting book on "lost causes" in physics: ideas that sounded good, but never panned out. Of course it's hard to know when a cause is truly lost. But a good pragmatic definition of a lost cause in physics is a topic that shouldn't be given as a thesis problem.
So, if you're a physics grad student and some professor wants you to work on hidden variable theories, or octonionic quantum mechanics, or deriving laws of physics from Fisher information, you'd better read this:
13) Ray F. Streater, Lost Causes in and Beyond Physics, Springer Verlag, Berlin, 2007.
(I like octonions - but I agree with Streater about not inflicting them on physics grad students! Even though all my students are in the math department, I still wouldn't want them working mainly on something like that. There's a lot of more general, clearly useful stuff that students should learn.)
I also spoke to Andreas Döring and Chris Isham about their work on topos theory and quantum physics. Andreas Döring lives near Greenwich, while Isham lives across the Thames in London proper. So, I talked to Döring a couple times, and once we visited Isham at his house.
I mainly mention this because Isham is one of the gurus of quantum gravity, profoundly interested in philosophy... so I was surprised, at the end of our talk, when he showed me into a room with a huge rack of computers hooked up to a bank of about 8 video monitors, and controls reminiscent of an airplane cockpit.
It turned out to be his homemade flight simulator! He's been a hobbyist electrical engineer for years - the kind of guy who loves nothing more than a soldering iron in his hand. He'd just gotten a big 750-watt power supply, since he'd blown out his previous one.
Anyway, he and Döring have just come out with a series of papers:
14) Andreas Döring and Christopher Isham, A topos foundation for theories of physics: I. Formal languages for physics, available as arXiv:quant-ph/0703060.
II. Daseinisation and the liberation of quantum theory, available as arXiv:quant-ph/0703062.
III. The representation of physical quantities with arrows, available as arXiv:quant-ph/0703064.
IV. Categories of systems, available as arXiv:quant-ph/0703066.
Though they probably don't think of it this way, you can think of their work as making precise Bohr's ideas on seeing the quantum world through classical eyes. Instead of talking about all observables at once, they consider collections of observables that you can measure simultaneously without the uncertainty principle kicking in. These collections are called "commutative subalgebras".
You can think of a commutative subalgebra as a classical snapshot of the full quantum reality. Each snapshot only shows part of the reality. One might show an electron's position; another might show it's momentum.
Some commutative subalgebras contain others, just like some open sets of a topological space contain others. The analogy is a good one, except there's no one commutative subalgebra that contains all the others.
Topos theory is a kind of "local" version of logic, but where the concept of locality goes way beyond the ordinary notion from topology. In topology, we say a property makes sense "locally" if it makes sense for points in some particular open set. In the Döring-Isham setup, a property makes sense "locally" if it makes sense "within a particular classical snapshot of reality" - that is, relative to a particular commutative subalgebra.
(Speaking of topology and its generalizations, this work on topoi and physics is related to the "étale topology" idea I mentioned a while back - but technically it's much simpler. The étale topology lets you define a topos of sheaves on a certain category. The Döring-Isham work just uses the topos of presheaves on the poset of commutative subalgebras. Trust me - while this may sound scary, it's much easier.)
Döring and Isham set up a whole program for doing physics "within a topos", based on existing ideas on how to do math in a topos. You can do vast amounts of math inside any topos just as if you were in the ordinary world of set theory - but using intuitionistic logic instead of classical logic. Intuitionistic logic denies the principle of excluded middle, namely:
In Döring and Isham's setup, if you pick a commutative subalgebra that contains the position of an electron as one of its observables, it can't contain the electron's momentum. That's because these observables don't commute: you can't measure them both simultaneously. So, working "locally" - that is, relative to this particular subalgebra - the statement
is neither true nor false! It's just not defined.
Their work has inspired this very nice paper:
15) Chris Heunen and Bas Spitters, A topos for algebraic quantum theory, available as arXiv:0709.4364.
so let me explain that too.
I said you can do a lot of math inside a topos. In particular, you can define an algebra of observables - or technically, a "C*-algebra".
By the Isham-Döring work I just sketched, any C*-algebra of observables gives a topos. Heunen and Spitters show that the original C*-algebra gives rise to a C*-algebra in this topos, which is commutative even if the original one was noncommutative! That actually makes sense, since in this setup each "local view" of the full quantum reality is classical.
So, they get this sort of picture:
I've been taking the "ambient topos" to be the familiar category of sets, but it could be something else.
What's really neat is that the Gelfand-Naimark theorem, saying commutative C*-algebras are always algebras of continuous functions on compact Hausdorff spaces, can be generalized to work within any topos. So, we get a space in our topos such that observables of the C*-algebra in the topos are just functions on this space.
I know this sounds technical if you're not into this stuff. But it's really quite wonderful. It basically means this: using topos logic, we can talk about a classical space of states for a quantum system! However, this space typically has "no global points" - that's called the "Kochen-Specker theorem". In other words, there's no overall classical reality that matches all the classical snapshots.
As you can probably tell, category theory is gradually seeping into this post, though I've been doing my best to keep it hidden. Now I want to say what Eugenia Cheng explained on that train to Sheffield. But at this point, I'll break down and assume you know some category theory - for example, monads.
If you don't know about monads, never fear! I defined them in "week89", and studied them using string diagrams in "week92". Even better, Eugenia Cheng and Simon Willerton have formed a little group called the Catsters - and under this name, they've put some videos about monads and string diagrams onto YouTube! This is a really great new use of technology. So, you should also watch these:
16) The Catsters, Monads, http://youtube.com/view_play_list?p=0E91279846EC843E
The Catsters, Adjunctions, http://youtube.com/view_play_list?p=54B49729E5102248
The Catsters, String diagrams, monads and adjunctions, http://youtube.com/view_play_list?p=50ABC4792BD0A086
A very famous monad is the "free abelian group" monad
F: Set → Set
which eats any set X and spits out the free abelian group on X, say F(X). A guy in F(X) is just a formal linear combination of guys in X, with integer coefficients.
Another famous monad is the "free monoid" monad
G: Set → Set
This eats any set X and spits out the free monoid on X, namely G(X). A guy in G(X) is just a formal product of guys in X.
Now, there's yet another famous monad, called the "free ring" monad, which eats any set X and spits out the free ring on this set. But, it's easy to see that this is just F(G(X))! After all, F(G(X)) consists of formal linear combinations of formal products of guys in X. But that's precisely what you find in the free ring on X.
But why is FG a monad? There's more to a monad than just a functor. A monad is really a kind of monoid in the world of functors from our category (here Set) to itself. In particular, since F is a monad, it comes with a natural transformation called a "multiplication":
m: FF => F
which sends formal linear combinations of formal linear combinations to formal linear combinations, in the obvious way. Similarly, since G is a monad, it comes with a natural transformation
n: GG => G
sending formal products of formal products to formal products. But how does FG get to be a monad? For this, we need some natural transformation from FGFG to FG!
There's an obvious thing to try, namely
mn FGFG ======> FFGG ======> FGwhere in the first step we switch G and F somehow, and in the second step we use m and n. But, how do we do the first step?
We need a natural transformation
d: GF => FG
which sends formal products of formal linear combinations to formal linear combinations of formal products. Such a thing obviously exists; for example, it sends
(x + 2y)(x - 3z)
xx + 2yx - 3xz - 6yz
It's just the distributive law!
Quite generally, to make the composite of monads F and G into a new monad FG, we need something that people call a "distributive law", which is a natural transformation
d: GF => FG
This must satisfy some equations - but you can work out those yourself. For example, you can demand that
FdG mn FGFG ======> FFGG ======> FGmake FG into a monad, and see what that requires. (Besides the "multiplication" in our monad, we also need the "unit", so you should also think about that - I'm ignoring it here because it's less sexy than the multiplication, but it's equally essential.)
However: all this becomes more fun with string diagrams! As the Catsters explain, and I explained in "week89", the multiplication m: FF => F can be drawn like this:
\ / \ / F\ F/ \ / \ / \ / \ / \ / |m | | | | | F| |And, it has to satisfy the associative law, which says we get the same answer either way when we multiply three things:
\ / / \ \ / \ / / \ \ / F\ /F F/ F\ F\ /F \/ / \ \/ m\ / \ /m \ / \ / F\ / \ /F \ / \ / |m |m | | | = | | | | | | | F| F| | |The multiplication n: GG => G looks similar to m, and it too has to satisfy the associative law.
How do we draw the distributive law d: FG => GF? Since it's a process of switching two things, we draw it as a braiding:
F\ /G \ / / / \ G/ \FI hope you see how incredibly cool this is: the good old distributive law is now a braiding, which pushes our diagrams into the third dimension!
Given this, let's draw the multiplication for our would-be monad FG, namely
FdG mn FGFG ======> FFGG ======> FGIt looks like this:
\ \ / / \ \ / / F\ G\ F/ /G \ \ / / \ \ / / \ \ / / \ / / \ / \ / |m |n | | | | | | | | | | F| |G | |Now, we want this multiplication to be associative! So, we need to draw an equation like this:
\ / / \ \ / \ / / \ \ / \ / / \ \ / \/ / \ \/ \ / \ / \ / \ / \ / \ / \ / \ / | | | | | = | | | | | | | | | | |but with the strands doubled, as above - I'm too lazy to draw this here. And then we need to find some nice conditions that make this associative law true. Clearly we should use the associative laws for m and n, but the "braiding" - the distributive law d: FG => GF - also gets into the act.
I'll leave this as a pleasant exercise in string diagram manipulation. If you get stuck, you can peek in the back of the book:
17) Wikipedia, Distibutive law between monads, http://en.wikipedia.org/wiki/Distributive_law_between_monads
The two scary commutative rectangles on this page are the "nice conditions" you need. They look nicer as string diagrams. One looks like this:
F\ G\ /G F\ G/ /G \ \ / \ / / \ |n \ / / \ / / / \ / = / \ / / / / / \ / / / \ \ / \ / \ \ / \ G/ \F |n \F / \ G| \In words:
If the pictures were made of actual string, this would be obvious!
The other condition is very similar. I'm too lazy to draw it, but it says
All this is very nice, and it goes back to a paper by Beck:
18) Jon Beck, Distributive laws, Lecture Notes in Mathematics 80, Springer, Berlin, 1969, pp. 119-140.
This isn't what Eugenia explained to me, though - I already knew this stuff. She started out by explaining something in a paper by Street:
19) Ross Street, The formal theory of monads, J. Pure Appl. Alg. 2 (1972), 149-168.
which is reviewed at the beginning here:
20) Steve Lack and Ross Street, The formal theory of monads II, J. Pure Appl. Alg. 175 (2002), 243-265. Also available at http://www.maths.usyd.edu.au/u/stevel/papers/ftm2.html
(Check out the cool string diagrams near the end!)
Street noted that we can talk about monads, not just in the 2-category of categories, but in any 2-category. I actually explained monads at this level of generality back in "week89". Indeed, for any 2-category C, there's a 2-category Mnd(C) of monads in C.
And, he noted that a monad in Mnd(C) is a pair of monads in C related by a distributive law!
That's already mindbogglingly beautiful. According to Eugenia, it's practically the last sentence in Street's paper. But in her new work:
21) Eugenia Cheng, Iterated distributive laws, available as arXiv:0710.1120.
she goes a bit further: she considers monads in Mnd(Mnd(C)), and so on. Here's the punchline, at least for today: she shows that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related by distributive laws satisfying the Yang-Baxter equation:
F\ G/ |H F| G\ /H \ / | | \ / / | | / / \ | | / \ / \ | \ / \ | \ / \ / | | / = / | | / \ / \ | | / \ / \ | \ / | | \ / \ / | | \ / / | | / / \ | | / \ H/ \G |F H| G/ \FThis is also just what you need to make the composite FGH into a monad!
By the way, the pathetic piece of ASCII art above is lifted from "week1", where I first explained the Yang-Baxter equation. That was back in 1993. So, it's only taken me 14 years to learn that you can derive this equation from considering monads in the category of monads in the category of monads in a 2-category.
You may wonder if this counts as progress - but Eugenia studies lots of examples of this sort of thing, so it's far from pointless.
Okay... finally, the Tale of Groupoidification. I'm a bit tired now, so instead of telling you more of the tale, let me just say the big news.
Starting this fall, James Dolan and I are running a seminar on geometric representation theory, which will discuss:
This is the Tale of Groupoidification in another guise.
Moreover, the Catsters have inspired me to make videos of this seminar! You can already find some here, along with course notes and blog entries where you can ask questions and talk about the material:
22) John Baez and James Dolan, Geometric representation theory seminar, http://math.ucr.edu/home/baez/qg-fall2007/
More will show up in due course. I hope you join the fun.
Addenda: I thank Eugenia Cheng for some corrections. Thomas Larsson points out that you can find some of Streater's "lost causes in physics" online:
23) Ray F. Streater, Various causes in physics and elsewhere, http://www.mth.kcl.ac.uk/~streater/causes.html
For the proof of the Gelfand-Naimark theorem inside a topos, see:
24) Bernhard Banachewski and Christopher J. Mulvey, A globalisation of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006), 62-103. Also available at http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf
They show that any commutative C*-algebra A in a Grothendieck topos is canonically isomorphic to the C*-algebra of continuous complex functions on the compact, completely regular locale that is its maximal spectrum (that is, the space of homomorphisms f: A → C). Conversely, they show any compact completely regular locale X gives a commutative C*-algebra consisting of continuous complex functions on X. Even better, they explain what all this stuff means.
Jordan Ellenberg sent me the following comments about knots and primes:
Here's one of Deninger's papers:
- In the viewpoint of Deninger, very badly oversimplified, Spec Z is to be thought of not just as a 3-manifold but as a 3-manifold with a flow, in which the primes are not just knots, but are precisely the closed orbits of the flow!
- One thing to keep in mind about the analogy is that "the complement of a knot or link in a 3-manifold" and "the complement of a prime or composite integer in Spec Z" (which is to say Spec Z[1/N]) are both "things which have fundamental groups," thanks to Grothendieck in the latter case. And much of the concrete part of the analogy (like the stuff about linking numbers) follows from this fact.
- On a similar note, a recent paper of Dunfield and Thurston which I like a lot, "Finite covers of random 3-manifolds," develops a model of "random 3-manifold" and shows that the behavior of the first homology of a random 3-manifold mod p is exactly the same as the predicted behavior of the mod p class group of a random number field under the Cohen--Lenstra heuristics. In other words, you should not think of Spec Z or Spec Z[1/N] as being anything like a particular 3-manifold -- better to think of the class of 3-manifolds as being like the class of number fields.
25) Christopher Deninger, Number theory and dynamical systems on foliated spaces, available as arXiv:math/0204110.
And here's the paper by Dunfield and Thurston:
26) Nathan M. Dunfield and William P. Thurston, Finite covers of random 3-manifolds, available as arXiv:math/0502567.
On the n-Category Café, a number theorist named James corrected some serious mistakes in the original version of this Week's Finds. Here are his remarks on why Spec(Z) is 3-dimensional:
So then why should there be the two dimensions of primes needed to make Spec(Z) three-dimensional? I don't think there is a pure-thought answer to this question. As you wrote, there is a scientific answer in terms of Artin-Verdier duality, which is pretty much the same as class field theory. There is also a pure-thought answer to an analogous question. Let me try to explain that.
Instead of considering Z, let's consider F[x], where F is a finite field. They are both principal ideal domains with finite residue fields, and this makes them behave very similarly, even on a deep level. I'll explain why F[x] is three-dimensional, and then by analogy we can hope Z is, too. Now F[x] is an F-algebra. In other words, X = Spec(F[x]) is a space mapping to S = Spec(F). I already explained why S is a circle from the point of view of the étale topology. So, if X is supposed to be three-dimensional, the fibers of this map better be two-dimensional. What are the fibers of this map? Well, what are the points of S? A point in the étale topology is Spec of some field with a trivial absolute Galois group, or in other words, an algebraically closed field (even better, a separably closed one). Therefore a étale point of S is the same thing as Spec of an algebraic closure F– of F. What then is the fiber of X over this point? It's Spec of the ring F–[x]. Now, this is just the affine line over an algebraically closed field, so we can figure out its cohomological dimension. The affine line over the complex numbers, another algebraically closed field, is a plane and therefore has cohomological dimension 2. Since étale cohomology is kind of the same as usual singular cohomology, the étale cohomological dimension of Spec(F–[x]) ought to be 2.
Therefore X looks like a 3-manifold fibered in 2-manifolds over Spec(F), which looks like a circle. Back to Spec(Z), we analogously expect it to look like a 3-manifold, but absent a (non-formal) theory of the field with one element, Z is not an algebra over anything. Therefore we expect Spec(Z) to be a 3-manifold, but not fibered over anything.
For more discussion, go to the n-Category Café.
© 2007 John Baez