December 18, 2006

This Week’s Finds in Mathematical Physics (Week 242)

Posted by John Baez

In week242 of This Week’s Finds, see some incredible photos of Saturn’s rings taken by the Cassini spacecraft:

Hear about some of the other exciting space missions NASA may cancel to pay for an expensive plan to send canned primates to Mars. See the Sun in neutrinos. And learn about Jeffrey Morton’s new approach to topological quantum field theory using a double bicategory of cobordisms with corners!

Posted at December 18, 2006 8:31 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 242)

As all you grad students reading this know, applying for jobs is pretty scary the first time around

But, to help them out a bit, I’d like to talk about [my students’] work.

So that’s what I did wrong. I should have set my advisor up with a popular internet soapbox prior to submitting my dissertation. Curses, foiled again.

Posted by: John Armstrong on December 18, 2006 9:26 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

I see K. T. Chen appears in Jeff’s bibliography, though I couldn’t find his articles referred to in the text. Chen also featured prominently in Kapranov’s paper mentioned here. Do I get the feeling we ought to be learning about iterated integrals? Kreimer used them in his approach to Feynman diagram calculations. And Cartier spoke about them:

P. Cartier, An algebraic theory of iterated integrals

Abstract. We propose an algebraic theory of iterated integrals, a version of Chen’s classical results well suited to applications in algebraic geometry. Among the applications we shall mention:

* a broad generalization of the Bloch-Wigner function (due to my student Francis Brown);

* geometry of configuration spaces M(0, n);

* algebraic relations among multiple zeta values, and connections with integral representations of these numbers

Posted by: David Corfield on December 18, 2006 9:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Aha. Thanks for pointing that out. That is a vermiform-appendix of a bibliography entry, in the sense that it was introduced when I was expecting to use the category of smooth spaces, rather than manifolds, to work in. Then I decided to solve my problem in a different way. I’ll have to remove the reference, among other changes, before the paper is ready for a referee.

However, you may be correct.

Posted by: Jeff Morton on December 18, 2006 10:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Do I get the feeling we ought to be learning about iterated integrals?

As far as my expericence goes, Chen’s iterated integrals are the kind of structures that are encoded by smooth functors on strict $n$-categories of paths.

The “classical” iterated integral, involving iterations of just a single 1-form

(1)$W_A(\gamma(\sigma)) = P \exp(\int_\gamma A)$

is the “path ordered exponential” which describes smooth functors from 1-paths to a Lie group $G$.

As one goes to higher functors, more kinds of $p$-forms appear in the integrand.

2-functors from 2-paths into a 2-group come from a path ordered integral of a 1-form on loop space, which is itself an iterated integral, namely

(2)$\int_\gamma W_A(\sigma) (\mathrm{ev}^*B(\sigma)) \; d\sigma \,.$

Here $A$ takes values in a Lie algebra that acts on the Lie algebra that $B$ takes values in, and the expression is supposed to indicate the operation where we integrate the pullback of the 2-form $B$ over the loop $\gamma$, while continually parallel transporting it, using the path ordered exponential of the 1-form, to the origin of the loop.

This is discussed here.

There is a nice simple diagrammatic description behind this rather involved-looking iterated integral. It really expresses the limit of having lots of little squares inside the strict 2-group $G_2 = (H \to G)$ like this:

(3)$\array{ \bullet &\stackrel{1 + A(x\to y)}{\to}& \bullet &\stackrel{1 + A(y\to z)}{\to}& \bullet \\ \downarrow &\Downarrow 1+B(x)& \downarrow &\Downarrow 1 + B(y)& \downarrow \\ \bullet &\stackrel{1 + A(x'\to y')}{\to}& \bullet &\stackrel{1 + A(y'\to z')}{\to}& \bullet } \,.$

I don’t have a proof for this general statement, but I expect that what is going on is this:

A smooth $n$-functor is entirely defined by its derivatives at all identity $n$-morphisms. These derivatives can typically be expressed in terms of various $p$-forms. The functor’s value on a finite $n$-morphisms then is the result of composing lots of small $n$-morphisms, as indicated in the above diagram, each of which being describeable as the avaluation of some $p$-form.

Therefore, the value of the $n$-functor on a finite $n$-morphism is a an “iterated” sum of products of values of $p$-forms.

It’s like a generalization of the fundamental theorem of calculus from $n=1$ to arbitrary $n$.

Posted by: urs on December 18, 2006 12:12 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Do I get the feeling we ought to be learning about iterated integrals?

Yes, they’re good things. But let me sketch why Chen is famous for two things: iterated integrals and smooth spaces.

Jeff cites Chen glancingly for his work on smooth spaces. Unlike the category of smooth manifolds, the category of smooth spaces has pushouts. This means you can always compose cospans without trouble. Jeff wants to compose cobordisms, which are specially nice cospans of smooth spaces. This would be a snap in the world of smooth spaces.

Chen developed the category of smooth spaces not primarily so it would have all pushouts, but so it would be cartesian closed. This implies that the space of paths in a smooth space is automatically a smooth space. This helped Chen develop the theory of iterated integrals with a minimum of fuss and muss.

I first learned iterated integrals back in the 90’s when I started work on loop quantum gravity. Such integrals play a big role there, and indeed throughout gauge theory, whenever you’re doing calculations with “path-ordered exponentials” - also known as “holonomies”. You can see this clearly in Gambini and Pullin’s book:

• Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories, and Quantum Gravity, Cambridge U. Press, Cambridge, 1996.

When Urs and I started working on higher gauge theory, we needed both aspects of Chen’s work! We needed the iterated integrals to compute the holonomy of a 2-connection along a path of paths… and we needed smooth spaces, to make sure the space of paths in a smooth space was again a smooth space!

Posted by: John Baez on December 18, 2006 5:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Along with LISA and Constellation-X, don’t forget the other Beyond Einstein programs: the Joint Dark Energy Mission, the Inflation Probe, and the Black Hole Finder. Evidently only one of these projects will be funded, at the expense of the other four, thanks to the NASA budget crunch for unmanned science missions.

Posted by: Nathan Urban on December 18, 2006 1:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

OTOH funds for the Terrestrial Planet Finder and the Europa mission have been reinstated in June 2006.

Re: This Week’s Finds in Mathematical Physics (Week 242)

OTOH funds for the Terrestrial Planet Finder and the Europa mission have been reinstated in June 2006.

Hmm, interesting! As of November 2006, NASA continues to argue against continuing these programs. So, the situation doesn’t seem to be settled.

Posted by: John Baez on December 18, 2006 5:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Thanks for the info, Nathan and Torbjörn!

When researching this article I had some trouble finding the current state of all the different NASA programs. It was easy to find news articles, but with a constantly shifting situation, it’s hard to tell the latest articles from the outdated articles - and a lot of cancelled or indefinitely postponed NASA programs have webpages that make them look perfectly healthy… until you look at the dates.

Is there an easy place to overview all the programs and their funding history?

Posted by: John Baez on December 18, 2006 5:12 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Wouldn’t LISA be able to detect a message coded into the cosmic gravity-wave background, representing Cartan matrices as Hsu and Zee suggested for the CMB? I’d guess that anybody who had that background to play with could encode 100 kilobytes or so… .

Posted by: Blake Stacey on December 18, 2006 5:16 PM | Permalink | Reply to this

Gravitantional SETI; Re: This Week’s Finds in Mathematical Physics (Week 242)

Yes, Blake. But if you look further back in time, and to less academic sources, you’ll see the cover story:

Jonathan V. Post, “Star Power for Supersocieties”, Omni, April 1980 (1st popular article to predict giant black hole in the center of Milky Way galaxy; 1st popular discussion of J. Post invention “gravity wave telegraph”).

A Gravity Wave Telegraph has a sender throwing a coded sequence of big asteroids and little asteroids into a black hole. Once we have LISA, if the sender is nearby, we become a receiver.

Is there any earlier citation to graviton SETI?

Posted by: Jonathan Vos Post on December 19, 2006 1:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Concerning Jeffrey’s work on extended cobordisms categories, as well as other work with a similar aim, I am meaning to write something and ask something about the idea of describing the local version of such extended cobordisms. But I don’t at the moment find the time.

Here when saying “local version” I am thinking of considering an $n$-category of $n$-paths inside some $n$-cobordism, i.e. cutting all possible $(n-1)$-disk-shaped pieces out of the $n$-cobordism.

When considering TFTs on our cobordisms, we can then compute the propagation over such $(n-1)$-disk-shaped $n$-paths and then apply generalized tracing to glue parts of the boundaries of these disks to obtain the possibly topologically nontrivial shape of the full $n$-cobordism.

That might sound more obscure than it is. For $n=1$ this is in fact the more familar procedure:

When we want to compute the holonomy of a circle under a vector transport, we first cut the circle open to give it the shape of a 0-disk (an interval), then we compute the parallel transport

(1)$V_x \stackrel{\mathrm{tra}_\nabla(I)}{\to} V_x$

over that open interval, and then we insert the information that the ends of the interval are to be glued by taking the trace of the result:

In this string diagram notation the trace literally glues the ends where we cut our 0-disk open.

Something similar works for 2-cobordisms – for instance if we consider 2-vector transport over 2-paths. Locally trivialized, this will choose a dual triangulation of the 2-cobordisms and decorate it in a Frobenius algebra, as known from the state sum model description of 2-dimensional TFT #.

If the Frobenius algebra here lives internal to a braided monoidal category, we can braid various strands past each other and then glue them – analogous to the trace operation for $n=1$ above – with the strands coming from edges that are to be identitfied.

For instance for the trinion (the pair-of-pants), the result looks like this

The dotted lines indicate the topologically 2-disk-shaped 2-path obtained by taking the trinion and cutting it open. The solid lines inside these are the decoration obtained by applying a 2-vector transport to that and locally trivializing it. Solid lines which cross parts of the boundary of our 2-disk which need to be identified in order to glue to a trinion are then appropriately braided and glued.

The resulting string diagram is indeed the well-known diagram decorating the trinion in the familiar state sum models.

So, as I said, I would have liked to describe this in more detail, and to ask various questions about how people working on more “global” versions of extended cobordism transport perceive this. But I am lacking the time right now.

Posted by: urs on December 19, 2006 10:49 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

What can be said about the relationship between the approaches to weak n-categories according to the different shapes one opts for: globular, cubical, simplicial, opetopic, etc.? Is it that one wants them all to be developed, and shown to be equivalent in some sense, since each finds some area of application in which they are most convenient?

I seem to recall Frank Adams writing somewhere that he felt it a shame that the cubical approach to homology theory had been neglected as it had many advantages, but can’t remember what they were.

Posted by: David Corfield on December 19, 2006 12:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Massey’s book “Singular Homology Theory” does cubical homology theory. One reason that cubical is good is that the product of two cubes is a cube, so, if I remember correctly, you don’t have to worry about Eilenberg-Zilber type of things.

In the realm of gerbes, Cheeger-Simons groups were defined using cubical complexes, and that makes doing transgression to loop spaces much less hassle. If you do that with a simplicial approach you end up having to decompose the prism of the interval cross an $n$-simplex into $n+1$ $(n+1)$-simplices which is quite nifty but messier.

Posted by: Simon Willerton on December 19, 2006 1:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Another nice thing about cubes is that they admit a noncommutative discrete calculus that approaches that of smooth forms in the continuum limit. This is worked out in (among other places):

Discrete Differential Geometry on Causal Graphs
Eric Forgy and Urs Schreiber

The noncommutative calculus on other shapes do not share the same natural continuum limit.

Posted by: Eric on June 16, 2009 5:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

David wrote:

What can be said about the relationship between the approaches to weak n-categories according to the different shapes one opts for: globular, cubical, simplicial, opetopic, etc.?

Ugh - that’s too big of a question!

Is it that one wants them all to be developed, and shown to be equivalent in some sense, since each finds some area of application in which they are most convenient?

Yes, exactly.

Right now it seems that for each different category of ‘shapes’ which one might use to develop $n$-categories — globes, simplices, cubes, multisimplices, opetopes, etc. — there is a presheaf category: globular sets, simplicial sets, cubical sets, multisimplicial sets, opetopic sets, etc..

A good question is thus: what properties must a presheaf category have in order to be a suitable foundation for a theory of (weak) $n$-categories?

This was already tackled by Grothendieck back in 1983, in his Pursuing Stacks — see the discussion of ‘modelizers’. But, I don’t know anybody who understands what he did!

More recently, Clemens Berger and Denis-Charles Cisinski have made a lot of progress on this issue in their (not yet available) paper on ‘geometric Reedy categories’. I’ve seen Berger give a couple of excellent talks on this subject.

Warning: finding the class of presheaf categories that allow the development of a theory of weak $n$-categories is not so cut-and-dry as I’m making it sound, because different approaches to $n$-categories make very different use of the ‘shapes’ involved. Berger and Cisinski are investigating just one type of approach, and focusing on $n$-groupoids more than $n$-categories.

But, people believe that every definition of $n$-category should eventually come with a concept of the ‘nerve’ of an $n$-category. This will be a simplicial set, and it should be a simplicial weak $n$-category in the set of Street.

Since simplicial sets are so fundamental in mathematics, the nerve seems like a natural ‘hub’ to go through when translating between different definitions of weak $n$-category. It’s a bit like when you fly in the US using Delta Airlines: every flight seems to go through Atlanta.

But, getting this to work seems to be taking a lot of effort on the part of some very smart people — it may take a decade or two.

Posted by: John Baez on December 20, 2006 11:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

Simon wrote:

Massey’s book “Singular Homology Theory” does cubical homology theory.

Yes. Amusingly, this is the book they used in my first course on homology as an undergrad.

One reason that cubical is good is that the product of two cubes is a cube, so, if I remember correctly, you don’t have to worry about Eilenberg–Zilber type of things.

Right. For those who don’t know what ‘Eilenberg–Zilber types of things’ are, let me explain. Suppose you define homology using simplices. Then, to relate the homology groups of a product of spaces to the product of their homology groups, you need to break a product of simplices into a bunch of simplices. This is a combinatorial pain in the butt, which Eilenberg and Zilber figured out how to do.

The problem of breaking the product of two simplices into a bunch of simplices is actually quite fascinating if you let yourself get interested in it. But, kids are usually forced to confront this problem in the middle of trying to do something else: namely, compute the homology of a product of spaces, like $S^n \times S^m$. And then, the Eilenberg–Zilber combinatorics seems like a huge nightmarish digression.

Working with cubes instead of simplices allows us to avoid this, because it’s trivial to break a product of cubes into cubes: the product is a cube. So, Massey wrote a book taking the cubical approach to homology.

As far as I know, this is about the only way in which cubical homology is better than simplicial homology. It’s a great advantage. But, there are more significant advantages to using simplices.

A rather minor one is that you don’t need to mod out by degenerate simplices when doing singular homology — miraculously, you still get the right answer. With cubical homology, you really do need to mod out by degenerate cubes.

But this miracle is sort of misleading: you really should mod out by degenerate simplices, and you need to in fancier situations.

A more significant advantage of simplices is that the category of simplices (including the $-1$-simplex!) is the free monoidal category on a monoid. This is what gives rise to the bar construction, which we use to define the cohomology of groups, Lie algebras, rings, and all other gadgets defined by monads. (If you don’t know about the bar construction, see the stuff around page 18 in my universal algebra notes.)

Luckily, we don’t need to work with just one kind of shape: we can use cubes or simplices, depending on what we’re interested in.

Posted by: John Baez on December 20, 2006 11:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 242)

It turns out most young adults don’t care about the Mars program — so NASA may resort to propaganda in an attempt to change their minds. They don’t seem to have noticed the possibility that the program really is intrinsically boring. Here’s part of a story from CNN which is being discussed by Sean Carroll and others over at Cosmic Variance:

NASA’s vision lost on Web generation

CAPE CANAVERAL, Florida (AP) – Young Americans have high levels of apathy about NASA’s new vision of sending astronauts back to the moon by 2017 and eventually on to Mars, recent surveys show.

Concerned about this lack of interest, NASA’s image-makers are taking a hard look at how to win over the young generation – media-saturated teens and 20-somethings growing up on YouTube and Google and largely indifferent to manned space flight.

“If you’re going to do a space exploration program that lasts 40 years, if you just do the math, those are the guys that are going to carry the tax burden,” said Mary Lynne Dittmar, president of a Houston company that surveyed young people about the space program.

The 2004 and 2006 surveys by Dittmar Associates Inc. revealed high levels of indifference among 18- to 25-year-olds toward manned trips to the moon and Mars.

The space shuttle program is slated to end in 2010 after construction of the international space station is completed with 13 more shuttle flights. The recent 13-day mission by Discovery’s seven astronauts was part of that long-running construction job.

When the shuttles are retired they will be replaced by the Orion spacecraft, which NASA hopes takes humans back to the moon and then on to Mars.

Even though the Dittmar surveys offer a bleak view, NASA Administrator Michael Griffin believes ventures to the moon and Mars will excite young people more than the current shuttle trips to low-Earth orbit.

“If we make it clear that the focus of the United States space program for the foreseeable future will be out there, will be beyond what we do now, I think you won’t have any problem at all reacquiring the interest of young people,” Griffin said in a recent interview.

At an October workshop attended by 80 NASA message spinners, young adults were right up there with Congress as the top two priorities for NASA’s strategic communications efforts.

Tactics encouraged by the workshop included new forms of communication, such as podcasts and YouTube; enlisting support from celebrities, like actors [from obsolete TV shows such as] David Duchovny (“X-Files”) and Patrick Stewart (“Star Trek: The Next Generation”); forming partnerships with youth-oriented media such as MTV or sports events such as the Olympics and NASCAR; and developing brand placement in the movie industry.

Outside groups have offered ideas too, such as making it a priority to shape the right message about the next-generation Orion missions.

And NASA should take a hint from Hollywood, some suggested.

“The American public engages with issues through people, personalities, celebrities, whatever,” said George Whitesides, executive director of the National Space Society, a space advocacy group. “When you don’t have that kind of personality, or face, or faces associated with your issue, it’s a little bit harder for the public to connect.”

Posted by: John Baez on December 29, 2006 10:03 PM | Permalink | Reply to this
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