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August 22, 2007

Gerbes in The Guardian

Posted by John Baez

In his post on Future Gazing, David Corfield invites us to predict the future of nn-categories over the next year — the second year of the n-Café. I’m seeing lots of signs that nn-categories are catching on — but it could be happening even faster than I expected. Here’s one article I would not have thought to see in an important British newspaper:

Thanks go to Eugenia Cheng for pointing this one out!

Alas, the author of this article is the editor of the bimonthly magazine Annals of Improbable Research and organiser of the Ig Nobel Prize — not a promising sign. And, while he limits his mockery of gerbes to a bit of polite ribbing, he makes no effort to explain them. He mainly explains how hard they are to explain. The article begins thus:

What is a gerbe? A gerbe is a mathematical object. It happens to be pretty obscure. Many obscure concepts are easy to understand. One just needs (A) a little patience, and (B) a reminder that most ideas are built upon other ideas. So it is, perhaps, with the gerbe.

To grasp a new concept, just (C) find a concept upon which it is built, and then (D) grasp that earlier concept. To grasp the earlier concept, just (E) find an appropriate earlier concept, and then (F) grasp it. And so on.

A few years ago, Nigel Hitchin used this technique to explain the concept of a gerbe. He wrote a two-page essay called What Is a Gerbe?, which he published in the Notices of the American Mathematical Society.

Nigel Hitchin is Savilian professor of geometry at Oxford University. Several mathematical concepts bear his name. These include the Hitchin integrable system, the Hitchin-Thorpe inequality, Hitchin’s projectively flat connection over Teichmuller space, Hitchin’s self-duality equations, and the Atiyah-Hitchin monopole metric. Hitchin knows his maths.

In What Is a Gerbe?, he begins by explaining that, to understand the concept of a gerbe, one ought first to understand a simpler concept called an “equivalence class of holomorphic line bundles”. To understand equivalence classes of holomorphic line bundles, Hitchin explains, one needs to know the concept called “transition functions relative to open sets of a covering”. And so on.

Mentally gobbling backwards through a few other, increasingly simpler, mathematical concepts, Hitchen soon comes to the end of page one of his essay. Then it’s on to page two. Eventually, with hardly any digressions, he reaches the essay’s conclusion.

The reader is left with a paralytic grasp on the concept of a gerbe - and, perhaps, also with a burning curiosity to see how this simple concept, the gerbe, can be used as a building block to produce new concepts.

I’m not sure what a “paralytic grasp” is — I imagine someone with their arms locked firmly around the concept, unable to be wrested free, but I somehow don’t think that’s what Abraham meant.

By the way, you don’t really need to understand holomorphic line bundles to understand gerbes!

If the Guardian asked me to explain a gerbe, I’d cheat a little and explain a gerbe with connection, which is actually easier. I’d say it’s a well-behaved recipe that’ll tell you a time of day — by giving a position of a clock hand — if you specify a way of sticking a balloon in a given space. The rules for what counts as “well-behaved” would take a few pictures to explain, but the first rule is that as you move the balloon around in a smooth sort of way, the time of day changes in a smooth sort of way.

Of course, this simple explanation would then open the concept up to further questions, like: how could such a recipe actually be good for anything?

That’s actually a great question, which I’ve answered elsewhere.

Posted at August 22, 2007 12:49 PM UTC

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7 Comments & 3 Trackbacks

Re: Gerbes in The Guardian

Yes “paralytic grasp” is odd here. I could imagine being held by the paralytic grasp of something which prevented me from acting, but submitting a gerbe to this treatment seems odd.

Perhaps he just means the weak grasp of one who is paralysed.

Posted by: David Corfield on August 22, 2007 1:49 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

how could such a recipe actually be good for anything?

Maybe the following slight variation of your description is just as close to the math, quite faithful to the physical application and still close to everyone’s everyday experience:

Suppose you are a painter. Your job is to go to the newly built conference building and

- 1) paint all the doorknobs

- 2) paint all the handrails

-3) paint all the walls.

But it tuns out that the architect used lots of different materials all over the place. You find that you need different amounts of paint to color these items at different places.

So you sit down and first make a table which lists

- on the left all the doorknobs

- on the right for each doorknob the amount of color needed to paint it (a small amount, right, but suppose there are many many doorknob).

Maybe you recall from your highschool days that such a table is also sometimes called a function: it maps doorknobs to the amount of color coloring them.

What they don’t tell you in high school these days is that

- a function is also called a 0-bundle with connection

- a function is also called a (-1)-gerbe with connection.

So you already know what a (1)(-1)-gerbe is! It’s just a very strange name for an assignment of milliliters to doorknobs.

Next, you try to sit down and make a table that has

- on the left all the handrails

- on the right the amount of color needed for them.

But now you run into trouble: the material of the handrails turns out to change every few meters. (It’s very modern architecture ). So instead of making a table which maps entire handrails to milliliters, you make a long, long table which maps

- each meter of handrail

- to the amount of color needed for it.

That takes a while. After pages of pages have been filled, this table is finished.

It doesn’t tell you immediately how much color is needed for painting a given handrail. But using the table it is easy to determine this amount:

for every piece of handrail, you chop it up (mentally) into 1-meter parts, look up the color needed for each of them seperatey and add all this up.

It happens that people have invented a funny name for this procedure: this is called

- a bundle with connection

or

- a 0-gerbe with connection.

These are just words. They mean exactly: a procedure for determining how to color handrails, and how to do it piecewise.

At this point, the attentive painter may alreay be able to go all the way up to inventing the concept of gerbes and even 2-gerbes himselves.

For all other painters out there, here is what a gerbe with connection would be:

next the walls need to be painted. They are made of even more variations of material than the handraisl! Grudgingly, you pull out pen and paper and make a huge list which

- has on the left an entry for each square meter of wall in the building

- and next to it on the right the estimated amount of color needed for that,

Cretaing that takes while. When done, you have a procedure that allows to compute the amount of color needed for an arbitrary wall:

for each square meter of it, look up the corresponding number of milliliters in your table, add that all up.

This procedure is what is called

- a 2-bundle with connection

or else

- a gerbe with connection.

(Or maybe a 2-functor with values in Σ\Sigma \mathbb{R}, or maybe a Cheeger-Simons differential 3-character, or maybe…)

Posted by: Urs Schreiber on August 22, 2007 2:18 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

See, John, by making mathematics seem accessible and fun, you’re ruining everything.

The idea is supposed to be that math is ‘cool’, in a way – at least we use a bunch of cool-sounding and mysterious words – but please let’s keep it at arm’s length, because you have to be a genius (or weirdo) or something to actually understand it. It’s hard to find the idea in mass culture that anyone does math because it’s fun.

Recently, the actress Danica McKellar (who played the love interest of Fred Savage’s character in The Wonder Years) has been on TV promoting her new book, “Math Doesn’t Suck”. As it turns out, she’s co-authored a scientific paper or two (I think in statistical mechanics) as an undergraduate at UCLA. As she explained, she wrote the book for young teenage girls, to promote the idea that math and science are actually interesting and fun, and not just for geeks.

Anyway, I was watching this segment on her (as Person of the Week on ABC World News Tonight) and her book, and toward the end she was asked what was the title of this scientific paper she wrote. It was some long title with the word ‘percolation’ in it – I couldn’t tell you exactly because she was drowned out by Charles Gibson’s chuckling voiceover, “Well, you’ll just have to take our word for it,” like, heh, get a load of this brainy chick. Precisely undermining the point of her book!

Posted by: Todd Trimble on August 22, 2007 2:40 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

Terence Tao, a former teacher, has written about her here.

Posted by: David Corfield on August 22, 2007 2:50 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

Todd wrote:

It was some long title with the word ‘percolation’ in it – I couldn’t tell you exactly because she was drowned out by Charles Gibson’s chuckling voiceover, “Well, you’ll just have to take our word for it,” like, heh, get a load of this brainy chick. Precisely undermining the point of her book!

Yeah, I hate that sort of crap.

You’re probably talking about her paper Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on 2\mathbb{Z}^2. This proves that for certain models of magnetic materials, there’s more than one state of thermal equilibrium if and only if there’s ‘percolation’.

To understand the idea of ‘more than one state of thermal equilibrium’, imagine an ordinary chunk of magnetic material at room temperature. The magnetic north pole can point in lots of different directions. Each one of these possibilities is a different state of thermal equilibrium. But when we heat the stuff up sufficiently, it demagnetizes — so at high temperatures, there’s just one state of thermal equilibium.

To understand ‘percolation’, imagine a simplified 2d model of a magnet: a square lattice where each point in the lattice has a little arrow attached to it that can point up or down. There’s a certain amount of randomness involved, thanks to the temperature: the wiggling of atoms makes the arrows flip up and down. Under certain circumstances there will be enough arrows pointing up that at any moment, if you take any up-pointing arrow, there’s a nonzero chance that you can hop in little steps up, down, left or right from it to any other up-pointing arrow — while only landing on up-pointing arrows, throughout your whole path of hops! Then we say there’s ‘percolation’ of up-pointing arrows.

Why? Because when you’re making coffee in a percolator, it only works if the water can find some path all the way through the tightly packed coffee grounds. Here we’re trying to find a path of up-pointing arrows that goes all from one up-pointing arrow to another. It’s analogous!

Anyway, I’m talking about a certain model of 2d magnets called the ‘Potts model’. McKellar and her coauthors studied a fancier model called the ‘Ashkin-Teller model’, where the spins can point in more different directions. They showed there was more than one state of thermal equilibrium in precisely the conditions where there was percolation.

Terry Tao has given a somewhat more detailed description of this result. To someone who knows a little physics, the introduction of her paper is also quite clear.

If ABC news had hired me as their science advisor, and gotten someone to create the right computer graphics, they could have given a thumbnail sketch of the ideas in less than a minute. Picture of magnet; zoom in to show a lattice of atoms with little spin arrows pointing up or down, etc. — all with a nice voiceover. Don’t bother describing the theorem; just give a sense of what it’s about.

But, presumably part of her being ‘Person of the Week’ was her ability to do incomprehensible mathematics while still remaining a ‘cute chick’.

She’s definitely trying to promote an image of herself as “the cute chick who can do math”. But, she actually explains math on her website. She doesn’t just mystify it.

By the way, her coauthor Lincoln Chayes was the teaching assistant for my undergrad quantum mechanics class. His nickname “Link Chayes” fit his punk persona quite well at the time. He and his then-wife Jennifer Chayes were the most glamorous physics grad students at Princeton, wearing leather and doing rigorous stat mech with Aizenman. It turns out Peter Woit cherishes memories of going out with them a to punk rock club in Trenton. It’s a small world.

Now she’s a co-manager of the Microsoft Theory Group. When I met her in 2005 at the Symposium on Discrete Algorithms she was just as friendly and irrepressible as ever. The only difference was that I was no longer too terrified to say hi. With a smile she explained how she was Michael Freedman’s boss.

I have no idea what happened to Lincoln Chayes.

Posted by: John Baez on August 23, 2007 12:08 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

John, you missed out the best bit – the bit where my paper gets cited. Okay, they don’t mention my name and it is sort of in the sense of “Here’s a few weird sounding titles of papers,” but, hey, it’s not everyday I get cited in a national newspaper.

Actually, I’m sure they only chose that paper because they read Urs bigging it up in the nn-Café.

Posted by: Simon Willerton on August 22, 2007 3:24 PM | Permalink | Reply to this

Re: Gerbes in The Guardian

Perhaps the saddest thing about that silly article is that gerbes are MUCH more general than those that Nigel Hitchin works with. If I was to attempt the difficult task of describing them I would probably go for the route via sheaves and bundles rather than that mentioned in the article. To get a plug in: if you do not know of the Bangor website entry you might be amused at trying to build a fibre bundle in wood. (Later it was done in bronze.) Now I challenge anyone to build a gerbe out of cardboard!!!

Posted by: Tim Porter on August 23, 2007 12:01 AM | Permalink | Reply to this
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