I've been roaming around Europe this summer - first Paris, then Delphi and Olympia, then Greenwich, then Oslo, and now back to Greenwich. I'm dying to tell you about the Abel Symposium in Oslo. There were lots of cool talks about topological quantum field theory, homotopy theory, and motivic cohomology.
I especially want to describe Jacob Lurie and Ulrike Tillman's talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen's talks on string topology, Stephan Stolz's talk on cohomology and quantum field theory, and Fabien Morel's talk on A1-homotopy theory. But this stuff is sort of technical, and I usually try to start each issue of This Week's Finds with something you don't need a PhD to enjoy.
So, here's a tour of the Paris Observatory:
1) John Baez, Astronomical Paris, http://golem.ph.utexas.edu/category/2007/07/astronomical_paris.html
Back when England and France were battling to rule the world, each had a team of astronomers, physicists and mathematicians devoted to precise measurement of latitudes, longitudes, and times. The British team was centered at the Royal Observatory here in Greenwich. The French team was centered at the Paris Observatory, and it featured luminaries such as Cassini, Le Verrier and Laplace.
In "week175", written during an earlier visit to Greenwich, I mentioned a book on this battle:
2) Dava Sobel, Longitude, Fourth Estate Ltd., London, 1996.
It's a lot of fun, and I recommend it highly.
There's a lot more to say, though. The speed of light was first measured by Ole Romer at the Paris Observatory in 1676.
Later, Henri Poincaré worked for the French Bureau of Longitude. Among other things, he was the scientific secretary for its mission to Ecuador.
To keep track of time precisely all over the world, you need to think about the finite speed of light. This may have spurred Poincaré's work on relativity! Here's a good book that argues this case:
3) Peter Galison, Einstein's Clocks, Poincaré's Maps: Empires of Time, W. W. Norton, New York, 2003. Reviewed by Robert Wald in Physics Today at http://www.physicstoday.org/vol-57/iss-9/p57.html
I met Galison in Delphi, and it's clear he like to think about the impact of practical stuff on math and physics.
I was in Delphi for a meeting of "Thales and Friends":
4) Thales and Friends, http://www.thalesandfriends.org
This is an organization that's trying to bridge the gap between mathematics and the humanities. It's led by Apostolos Doxiadis, who is famous for this novel:
5) Apostolos Doxiadis, Uncle Petros and Goldbach's Conjecture, Bloomsbury, New York, 2000. Review by Keith Devlin at http://www.maa.org/reviews/petros.html
There's a lot I could say about this meeting, but I just want to advertise a forthcoming book by Doxiadis and a computer scientist friend of his. It's a comic book - sorry, I mean "graphic novel"! - about the history of mathematical logic from Russell to Goedel:
6) Apostolos Doxiadis and Christos Papadimitriou, Logicomix, to appear.
I saw a partially finished draft. I think it does a good job of explaining to nonmathematicians what the big deal was with mathematical logic around the turn of the last century... and how these ideas eventually led to computers. It's also a fun story.
If you're eager for summer reading and can't wait for Logicomix, you might try this other novel by Papadimitrou:
7) Christos Papadimitriou, Turing (a Novel about Computation), MIT Press, Boston, 2003.
It's a history of mathematics from the viewpoint of computer science, as told by a computer program named Turing to a lovelorn archaeologist. I haven't seen it yet.
Okay - enough fun stuff. On to the Abel Symposium!
8) Abel Symposium 2007, http://abelsymposium.no/2007
Actually this was a lot of fun too. A bunch of bigshots were there, including a bunch who didn't even give talks, like Eric Friedlander, Ib Madsen, Jack Morava, and Graeme Segal.
(My apologies to all the bigshots I didn't list.)
Speaking of bigshots, Vladimir Voevodsky gave a special surprise lecture on symmetric powers of motives. He wowed the audience not only with his mathematical powers but also his ability to solve a technical problem that had stumped all the previous speakers! The blackboards in the lecture hall were controlled electronically, by a switch. But, the blackboards only moved a few inches before stalling out. So, people had to keep hitting the switch over and over. It was really annoying, and it became the subject of running jokes. People would ask the speakers: "Can't you talk and press buttons at the same time?"
So, what did Voevodsky do? He lifted the blackboard by hand! He laughed and said "Russian solution". But, I think it's a great example of how he gets around problems by creative new approaches.
It really pleased me how many talks mentioned n-categories, and even used them to do exciting things. This seems quite new. In the old days, bigshots might think about n-categories, but they'd be embarrassed to actually mention them, since they had a reputation for being "too abstract".
In fact, Dan Freed alluded to this in his talk on topological quantum field theory. He said that every mathematician has an "n-category number". Your n-category number is the largest n such that you can think about n-categories for a half hour without getting a splitting headache.
When Freed first invented this concept, he felt pretty self-satisfied, since his n-category number was 1, while for most mathematicians it was 0. But lately, he says, other people's n-category numbers have been increasing, while his has stayed the same.
He said this makes him suspicious. In light of the scandals plaguing the Tour de France and American baseball, he suspects mathematicians are taking "category-enhancing substances"!
Freed shouldn't feel bad: he was among the first to introduce n-categories in the subject of topological quantum field theory! He gave a nice talk on this, clear and unpretentious, leading up to a conjecture for the 3-vector space that Chern-Simons theory assigns to a point.
That would make a great followup to these papers on the 2-vector space that Chern-Simons theory assigns to a circle:
9) Daniel S. Freed, The Verlinde algebra is twisted equivariant K-theory, available as arXiv:math/0101038.
Daniel S. Freed, Twisted K-theory and loop groups, available as arXiv:math/0206237.
Daniel S. Freed, Michael J. Hopkins and Constantin Teleman, Loop groups and twisted K-theory II, available as arXiv:math/0511232.
Daniel S. Freed, Michael J. Hopkins and Constantin Teleman, Twisted K-theory and loop group representations, available as arXiv:math/0312155.
In a similar vein, Jacob Lurie talked about his work with Mike Hopkins in which they proved a version of the "Baez-Dolan cobordism hypothesis" in dimensions 1 and 2. I'm calling it this because that's what Lurie called it in his title, and it makes me feel good.
You can read about this hypothesis here:
10) John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, J.Math.Phys. 36 (1995) 6073-6105 Also available as arXiv:q-alg/9503002.
It was an attempt to completely describe the algebraic structure of the n-category nCob, where:
and so on up to dimension n. Unfortunately, at the time we proposed it, little was known about n-categories above n = 3. For a more recent take on these ideas, see:
11) Eugenia Cheng and Nick Gurski, Towards an n-category of cobordisms, Theory and Applications of Categories 18 (2007), 274-302. Available at http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html
Lurie and Hopkins use a new trick: they redefine nCob to be a special sort of ∞-category. The idea is to use diffeomorphisms and homotopies between these as morphisms above dimension n. This gives an ∞-category version of nCob, where:
and so on for ever!
Since everything here is invertible above dimension n, this is called an "(∞,n)-category".
This sounds worse than an n-category, but it's okay for small n. In particular, (∞,1)-categories are pretty well understood by now. There are a bunch of different approaches, with scary names like "topological categories", "simplicial categories", "A∞ categories", "Segal categories", "complete Segal spaces", and "quasicategories". Luckily, all these approaches are known to be equivalent - see "week245" for some good introductory material by Julie Bergner and Andre Joyal. Joyal is now writing a book on this stuff.
Lurie is a real expert on (∞,1)-categories. In fact, starting as a grad student, he wrote a mammoth tome generalizing topos theory from categories to (∞,1)-categories:
12) Jacob Lurie, Higher topos theory, available as arXiv:math/0608040.
I'm sure Freed would suspect him of taking category-enhancing substances: his category number is infinite, and this book is 619 pages long! Then he went on to apply this stuff to algebraic geometry... and the world is still reeling. I was happy to discover that he's a nice guy, enthusiastic and friendly - not the terrifying fiend I expected.
Anyway, Lurie and Hopkins have worked out the precise structure of the (∞,1)-category version of 1Cob, and also the (∞,2)-category version of 2Cob. Unfortunately this work is not yet written up. But, they use results from this paper:
13) Soren Galatius, Ib Madsen, Ulrike Tillmann, Michael Weiss, The homotopy type of the cobordism category, available as arXiv:math/0605249.
And, Ulrike Tillmann gave a talk about this paper! It computes the "nerve" of the (∞,1)-category where:
The "nerve" is a trick for turning any sort of ∞-category into a space, or simplicial set. (See item J of "week117" for the nerve of a plain old category. This should give you the general idea.)
In her talk, she went further and computed the nerve of the (∞,k)-category where:
This is also joint work with the same coauthors, but it seems not to be written up yet, except for k = 1, where it's proved in the above paper. The cool thing about the new work is that it uses an idea familiar from higher category theory - a k-simplicial space - to give a rigorous description of the nerve of the above (∞,k)-category! Indeed, Tillmann told me she thinks of k-simplicial spaces as just a convenient way of dealing with higher categories.
Stephan Stolz's talk also involved cobordism n-categories, but I'll say more about that later.
Ralph Cohen and Dennis Sullivan both gave talks on string topology - a trick for studying a space by studying collections of loops in that space, and relating this to ideas from string theory.
String topology started when Chas and Sullivan took the ideas of string theory and applied them in a somewhat ethereal form to strings propagating in any manifold.
In full-fledged string theory, one of the main tools is "conformal field theory". In a CFT, if you have a state of n strings, and a Riemann surface going from n strings to m strings, you get a state of m strings.
A good way to get CFTs is to consider strings propagating on some manifold or other. Of course the manifold needs some sort of geometry, like a Riemannian metric, for your strings to know how to propagate.
But Chas and Sullivan figured out what you can do if the spacetime is a bare manifold, without any metric. Basically, you just need to stick the word "homology" in front of everything! This makes everything sufficiently floppy.
So, instead of considering actual loops in a manifold M, which form a space LM, they took the homology of LM and got a vector space or abelian group H(LM). Then, for each homology class C on the moduli space of Riemann surfaces that go from n circles to m circles, they got an operation with n inputs and m outputs:
Z(C): H(LM)⊗n → H(LM)⊗m
All these operations fit together into a slight generalization of an operad, called a "PROP".
If you don't remember what an "operad" is, give yourself twenty lashes with a wet noodle and review "week220". Suitably punished, you can then enjoy this:
14) Ralph Cohen and Alexander Voronov, Notes on string topology, available as arXiv:math/0503625.
Both PROPs and operads are defined near the beginning here. PROPs and operads are gadgets for describing operations with any number of inputs. Operads can only handle operations with one output. PROPS can handle operations with any number of outputs.
To see a more geometrical treatment of string topology, the way it looked before the operadchiks got ahold of it, try the original paper by Chas and Sullivan:
15) Moira Chas and Dennis Sullivan, String topology, available as arXiv:math/9911159.
Sullivan talked about some recent refinements of string topology which deal with the fact that the moduli space of Riemann surfaces has a "boundary", so it doesn't have a closed "top-dimensional homology class".
Cohen's talk described some cool relations between string topology and symplectic geometry! In physics we use symplectic manifolds to describe the space of states - the so-called "phase space" - of a classical system. So, if you have a loop in a symplectic manifold, it can describe a periodic orbit of some classical system. In particular, if we pick a periodic time-dependent Hamiltonian for this system, a loop will be a solution of Hamilton's equations iff it's a critical point for the "action".
But, we can also imagine letting loops move in the direction of decreasing action, following the "gradient flow". They'll trace out 2d surfaces which we can think of as string world-sheets! This is just what string topology studies, but now we can get "Morse theory" into the game: this studies a space (here LM) by looking at critical points of a function on this space, and its gradient flow.
So, we get a nice interaction between periodic orbits in phase space, and the string topology of that space, and Morse theory! For more, try this:
16) Ralph Cohen, The Floer homotopy type of the cotangent bundle, available as arXiv:math/0702852.
Next, let me say a bit about Stephan Stolz's talk. He spoke on his work with Peter Teichner, which is a very ambitious attempt to bring quantum field theory right into the heart of algebraic topology.
I discussed this in "week197". I said they were working on a wonderful analogy between quantum field theories and different flavors of cohomology. It's been published since then:
17) Stephan Stolz and Peter Teichner, What is an elliptic object? Available at http://math.berkeley.edu/~teichner/papers.html
Back then, the analogy looked like this:
1-dimensional supersymmetric QFTs complex K-theory 2-dimensional supersymmetric conformal QFTs elliptic cohomology
When I saw this, I tried to guess a generalization to higher dimensions.
There's an obvious guess for the right-hand column, since there's something called the "chromatic filtration", which is - very roughly - a list of cohomology theories. Complex K-theory is the 1st entry on this list, and elliptic cohomology is the 2nd! (For a lot more details, see "week149" and "week150".)
There's also an obvious guess for the left-hand column: n-dimensional supersymmetric QFTs of some sort!
The problem is the word "conformal" in the second row. In 2 dimensions, a conformal structure is a way of making spacetime look locally like the complex plane. This is great, because elliptic cohomology has a lot to do with complex analysis - or more precisely, elliptic curves and modular forms. But, it's not clear how one should generalize this to higher dimensions!
Luckily, thanks to a subsequent conversation with Witten, Stolz and Teichner realized that the partition function of a 2d supersymmetric QFT gives a modular form even if the QFT is not invariant under conformal transformations. This means we can remove the word "conformal" from the second row! For more details, try this:
18) Stephan Stolz and Peter Teichner, Super symmetric field theories and integral modular forms, preliminary version available at http://math.berkeley.edu/~teichner/papers.html
They've also gone back and added a 0th row to their chart. It's always wise to start counting at zero! Now the chart looks much nicer:
0-dimensional supersymmetric QFTs deRham cohomology 1-dimensional supersymmetric QFTs complex K-theory 2-dimensional supersymmetric QFTs elliptic cohomology
Yes, good old deRham cohomology is the 0th entry in the "chromatic filtration"! It's the least scary sort of cohomology theory, at least for physicists. They get scarier as we move down the chart.
Quantum field theory also gets scarier as we move down the chart - the infinities that plague quantum field theory tend to get worse in higher dimensions of spacetime. So, while we can dream about extensions of this chart, there's already plenty to handle here.
The most audacious idea in Stolz and Teichner's work is to take a manifold X and study the set of all n-dimensional QFT's "parametrized by X".
For X a point, such a thing is just an ordinary n-dimensional QFT. Roughly speaking, this is a gadget Z that assigns:
If you're a mathematician, you may know that M is really a "cobordism" from S to S', written M: S → S'. And if you're really cool, you'll know that cobordisms form a symmetric monoidal category nCob, and that Z should be a symmetric monoidal functor.
If you're a physicist, you'll know that S stands for "space" and "M" stands for "spacetime". All the stuff I'm describing should remind you of the definition of a "TQFT", except now our spaces and spacetimes have Riemannian metrics, because we're doing honest QFTs, not topological ones.
Given a spacetime M, we try to compute the operator Z(M) as a path integral; for example, an integral over all maps
f: M → T
where f is a "field" taking values in a "target space" T.
If this seems too scary, take n = 1. Then we've got a 1-dimensional quantum field theory, so we can take our spacetime M to be an interval. Then f is just a path in some space T. In this case the path integral is really an integral over all paths a particle could trace out in T. So, 1-dimensional quantum field theory is just ordinary quantum mechanics!
There are a lot of subtleties I'm skipping over here, both on the math and physics sides. But never mind - the really cool part is this generalization:
Roughly speaking, an n-dimensional QFT "parametrized by X" assigns:
If you're a mathematician, you may see we've switched to using cobordisms "over X". It's a straightforward generalization.
But what does it mean physically? Here the path integral picture is helpful. Now we're doing a path integral over all fields
f: M → T × X
where we demand that the second component of this function is
g: M → X
For example, if we've got a 1d QFT, we're letting a particle roam over T × X, but demanding that its X coordinates follow a specific path g.
So, we're doing a constrained path integral!
In heaven, everything physicists do can be made mathematically rigorous. Up there, knowing how to do these constrained path integrals would tell us how to do unconstrained path integrals: we'd just integrate over all choices of the path g. So, a QFT parametrized by X would automatically give us an ordinary QFT.
Now, an ordinary QFT is just a QFT parametrized by a point! So, if we use QFT(X) to mean the set of n-dimensional QFTs parametrized by X, we'd have a map
QFT(X) → QFT(point)
This is called "pushing forward to a point".
More generally, we could hope that any map
F: X → X'
gives a "pushforward" map
F*: QFT(X) → QFT(X')
Let's see if this makes any sense. In fact, I've been overlooking some important issues. An example will shed light on this.
Consider a 0-dimensional QFT parametrized by some manifold X. Let's call it Z. What is Z like, concretely?
For starters, notice that the only (-1)-dimensional manifold is the empty set. A 0-dimensional manifold "going from the empty set to the empty set" is just a set of points. Also, while I didn't mention it earlier, all manifolds in this game must be compact. So, this set of points must be finite.
If you now take the definition I wrote down and use that "symmetric monoidal functor" baloney, you'll see Z assigns a number to any finite set of points mapped into X. Furthermore, this assignment must be multiplicative. So, it's enough to know a number for each point in X. In short, our QFT is just a function:
Z: X → C
Now suppose we map X to a point:
F: X → point
What should the pushforward
F*: QFT(X) → QFT(point)
do to the function Z?
There's an obvious guess: we should integrate this function on X to get a number - that is, a function on a point. Indeed, that's what "path integration" should reduce to in this pathetically simple case: plain old integration!
Alas, there's no good way to integrate a function over X unless this manifold comes equipped with a measure. But, if X is compact, oriented and p-dimensional, we can integrate a p-form over X.
More generally, if we have a bundle
F: X → X'
with compact d-dimensional fibers, we can take a p-form on X and integrate it over the fibers to get a (p-d)-form on X'. This is how you "push forward" differential forms.
So, pushing forward is a bit subtler than I led you to believe at first. We should really talk about n-dimensional QFTs "of degree p" parametrized by X. Let's call the set of these
I won't define them, but for n = 0 they're just p-forms on X. Anyway: if we have a bundle
F: X → X'
with compact d-dimensional fibers, we can hope there's a pushforward map
F*: QFTp(X) → QFTp-d(X')
There should also be a pullback map
F*: QFTp(X') → QFTp(X)
This is a lot less tricky, and I'll let you figure out how it works.
I should warn you, I've been glossing over lots of important aspects of this work - like the role played by n-categories, and the role played by supersymmetry. Supersymmetry doesn't matter much for the broad conceptual picture I've been sketching. But, we need it for this analogy to work:
0-dimensional supersymmetric QFTs deRham cohomology 1-dimensional supersymmetric QFTs complex K-theory 2-dimensional supersymmetric QFTs elliptic cohomology
The idea is to impose an equivalence relation on supersymmetric QFTs, called "concordance", and try to show:
So far people have done this in the 0d and 1d cases. The 2d case is a major project, because it pushes the limits of what people can do with quantum field theory.
Why did I spend so much time talking about pushforwards of QFTs? Well, it's very important for defining invariants like the "fundamental class" of an oriented manifold, or the "A-hat genus" of a spin manifold, or the "Witten index" of a string manifold.
Here's how it goes, very roughly. Suppose X is a compact Riemannian manifold. Then the simplest n-dimensional QFT parametrized by X is the one where we take the target space T (mentioned a while back) to be just a point!
This parametrized QFT is called the "nonlinear σ model", for stupid historic reasons. All the fun happens when we push this QFT forwards to a point. Then we integrate over all the maps g: M → X. The result - usually called the "partition function" of the nonlinear σ model - should be an interesting invariant of X.
In the case n = 1, this trick gives the "A-hat genus" of X, but it only works when X is a spin manifold: we need this to define the 1d supersymmetric nonlinear σ-model.
In the case n = 2, this trick gives the "Witten genus" of X, but it only works when X is a string manifold: we need this to define the 2d supersymmetric nonlinear σ-model.
For more on the n = 1 case, see:
19) Henning Hohnhold, Peter Teichner and Stephan Stolz, From minimal geodesics to super symmetric field theories. In memory of Raoul Bott. Available at http://math.berkeley.edu/~teichner/papers.html
For the n = 2 case, see the papers I already listed.
(I'm confused about the case n = 0, for reasons having to do with the "degree" I mentioned earlier.)
Finally: the cool part, which I haven't even mentioned, is that we really need to describe n-dimensional QFTs using an n-category of cobordisms - not just a mere 1-category, as I sloppily said above.
This first gets exciting when we hit n = 2: you'll see a bunch of stuff about 2-categories (or technically, "bicategories") in the old Stolz-Teichner paper "What is an elliptic object", listed above.
In short: we're starting to see a unified picture where we study spaces by letting particles, strings, and their n-dimensional cousins roam around in these spaces. There are lots of slight variants: string topology, the Stolz-Teichner picture, and of course good old-fashioned topological quantum field theory. All of them have a lot to do with n-categories.
There's a lot more to say about all this... but luckily, there should be a proceedings of this conference, where you can read more. My own talk is here:
20) John Baez, Higher gauge theory and elliptic cohomology, http://math.ucr.edu/home/baez/abel/
It, too, is about studying spaces by letting strings roam around inside them!
But instead of summarizing my own talk, I want to say a bit about the other side of the symposium - the motivic cohomology side!
I'll only summarize a few basic definitions. I got these from the talks by Fabian Morel and Vladimir Voevodsky, and I want to write them down before I forget! For more, try these:
21) Fabian Morel and Vladimir Voevodsky, A1-homotopy theory of schemes, September 1998. Available at http://citeseer.ist.psu.edu/morel98suphomotopy.html
22) Vladimir Voevodsky (notes by Charles Weibel), Voevodsky's Seattle Lectures: K-theory and motivic cohomology. Available at http://citeseer.ist.psu.edu/249068.html
A1-homotopy theory is an attempt to do homotopy theory for algebraic geometry. In algebraic geometry we often work over a fixed field k, and the goal here is to create a category which contains smooth algebraic varieties over k as objects, but also other more general spaces, providing a sufficiently flexible category in which to do homotopy theory.
One of the simplest smooth algebraic varieties over k is the "affine line" A1. The algebraic functions on this line are just polynomials in one variable with coefficients in k. In A1-homotopy theory, we want to set up a context where we can use the affine line A1 to parametrize homotopies, much as we use the unit interval [0,1] in ordinary homotopy theory.
For this, people start by looking at Sm(k), the category of smooth algebraic varieties over k. Then, they consider the category of "simplicial presheaves" on Sm(k).
A simplicial presheaf on Sm(k) is just a functor
F: Sm(k)op → SimpSet
where SimpSet is the category of simplicial sets (see item C of "week115") We think of F as specifying some sort of space by telling us for each smooth algebraic variety X the simplicial set F(X) of all maps into this space.
To make this kind of abstract space work nicely, F(X) should depend "locally" on X. For this, we insist that given a cover of a variety X by varieties Ui, guys in F(X) are the same as guys in F(Ui) that agree on the intersections
Ui ∩ Uj.
Here "cover" means "cover in the Nisnevich topology" - that is, an étale cover such that every point being covered is the image of a point in the cover for which the covering map induces an isomorphism of residue fields.
If you've come this far, you may not be scared to hear that the Nisnevich topology is really a "Grothendieck topology" on Sm(k), and I'm really demanding that F be a "sheaf" with respect to this topology.
So, the kind of "space" we're studying is a simplicial sheaf on the category of smooth varieties over k with its Nisnevich topology. We call these category of these guys Space(k)
Just saying this already makes me feel smart. Just think how smart I'd feel if I knew why the Nisnevich topology was better than the good old etale topology!
Anyway, to do homotopy theory with these simplicial sheaves, we need to make Space(k) into a "model category". I should have explained model categories in some previous Week, but I've never gotten around to it, and right now is not the time. So, I'll just say one key thing.
The most important thing about a model category is that it's equipped with a collection of morphisms that act like homotopy equivalences. They're called "weak equivalences".
Already in ordinary topology, these weak equivalences are a slight generalization of homotopy equivalences. They're actually the same as homotopy equivalences when the spaces involved are nice; they're designed to work better for nasty spaces.
In A1 homotopy theory, the weak equivalences are generated by two kinds of morphisms:
Here X is any space in Space(k), and C(U) is the "Cech nerve" of the cover U.
This framework seems like a really cool blend of algebraic geometry and homotopy theory. But, to do homology theory in a good way we need to go a bit further, and introduce "motives".
However, I'm tired, and I bet you are too! Motives are a big idea, and it doesn't make sense to start talking about them now. So, some other day....
© 2007 John Baez