## Thursday, January 31, 2013

### International Congress of Mathematicians-Registration is now open

Check the link below if you want to register for next year's ICM in Seul.

International Congress of Mathematicians

## Friday, January 25, 2013

### Springer Verlag is one fucked-up company

I apologize for the profanity in the title but, as Mark Twain so eloquently put it, "there are some instances when profanity provides relief denied even to prayer" .

What is prompting this post?  Things have  been piling up.   I have published a Morse theory book with Springer in 2007 (or thereabout) and the experience was great. In particular,  I worked with  a most professional  editor from whom I learned quite a bit.

The book did well, and Springer offered to    publish a second edition. I jumped at the opportunity. I could  fix errors in the first edition, and add things  that crossed my mind after the publication  of  the first edition. I joyously  worked  on the second edition, sent the manuscript,    kept to the agreed deadlines.  This is when the story goes  dark.

They sent the manuscript for copy-editing in India to a crew  who was obviously   not trained to edit science   books in general, math books  in particular.

Leaving aside the minor fact that they  did not respect the deadlines they themselves imposed,  what came back to me  was a complete disaster.

I could see that  two different people  worked  on the manuscript, but these two people read different grammar and ponctuation books.     One of them was not  even  in great control of English  because he/she    disagreed with  several  of my linguistic constructions which were OK-ed by Oxford and Webster's Online Dictionaries.  Springer    must at least provide these guys with Internet acces to these dictionaries.

The real tragedy  was that these two  untrained   individuals changed the manuscript, without  asking for my consent.  They changed things to the point that  the  mathematical meaning was  severely affected.  They rewrote definitions in what they believed  was proper English,   in the process creating new and meaningless Mathematics. They changed the fonts I was using with   some other idiotic fonts  which did not distinguish very much between  Latin $v$ and the Greek "nu", $\nu$.  I myself could not make  any sense of my own goddamn  proofs.

I was   extremely upset  and  I made sure  that both NY and Europe    headquarters      were aware of  this intellectual rape I was subjected  to by their  perpetual   search for  profit. Ain't publishing books with them anymore.   They ain't professional anymore. They're selling a load a crap    wrapped  in corporate bullshit. (There are still many  professionals working for the company, but they are smothered by the idiots running this joint.)

Yesterday, I received a letter  from a debt collection  agency hired by Springer for a  $50 bill for a mysterious charge which they did not bother to detail. This is adding insult upon insult stacked on injury. The Springer-US site has not information whom to contact about a possible erroneous bill. This is one fucked-up company and it's a pity because it has a great tradition and reputation which it's gradually converting into a pile of shit, all in the name of profit. I'm running out of insults and I think I have already devoted too much of my limited time to this genuinely fucked up company, a shameful anorexic shadow of the former self. Update, Jan 28, 2013: Springer has finally replied to my inquiries regarding the$50 bill.  It  was a screw-up with my credit card    and I sent them a check.

## Wednesday, January 16, 2013

### ArXiv Overlay Journal?

A very interesting new initiative is underway.  More at the post below from Tim Gowers.

Why I’ve also joined the good guys « Gowers's Weblog

## Wednesday, January 2, 2013

### Gauge theory and the variational bicomplex

Hi! Former Notre Dame math student here, posting at Liviu's suggestion to expand on a conversation we were having on Facebook.

Suppose you have a fiber bundle $E \rightarrow M$. Interpret $M$ as "spacetime;" then sections of $E$ are "fields." (Particle dynamics can be recovered by taking $M = \mathbb{R}$.) To set up a classical dynamics on these fields, one writes down a Lagrangian $L$ and associated action functional $S = \int_M L$, then obtains field equations by requiring $\delta S = 0$. When I first read the derivation of these Euler-Lagrange equations in a physics book, I felt like a trick had been played. It wasn't clear to me what the Lagrangian really "was," in a formal mathematical sense, and the formula $\frac{d}{dt}(\delta q) = \delta \dot{q}$ seemed a bit magic.

As usual, the nlab came to my rescue and told me about the "variational bicomplex." (http://ncatlab.org/nlab/show/variational+bicomplex). This is a doubly-graded complex of differential forms on the infinite jet bundle $j_{\infty}(E)$. In particular, any differential form on a finite jet bundle $j_k(E)$ gives you an element of the variational bicomplex via pullback. And both fields and Lagrangians look like forms on finite jet bundles of $E$. A field is a $0$-form on the $0$-jet bundle. A Lagrangian is a bit more complicated- since the action functional is the integral of $L$ over $M$, $L$ must be an $n$-form on $M$... but since its values depend on the 1-jet of the field you're at, it's more like an $n$-form on the $1$-jet bundle.

Write $D$ for the exterior derivative on forms on $j_{\infty}(E)$ (or, for that matter, on any finite jet bundle). We'd like to be able to split $D$ into a sum $d + \delta$, where $d$ is the derivative "along M" and $\delta$ is the derivative along the fiber. What this requires is a splitting of the tangent space at any $\infty$-jet $\varphi$ into horizontal directions and vertical directions. The vertical directions are already there, since we have a fiber bundle, so we just need the horizontal ones. Local coordinates on $j_{\infty}(E)$ are $\{x_1, \ldots, x_n, q_1, \ldots, q_k, \partial_i q_j, \partial_i \partial_j q_k, \ldots \}$.

Trickily, the vectors $\frac{\partial}{\partial x_i}$ are NOT an appropriate choice of horizontal vectors, even if the bundle $E$ happens to be trivial! (As is always the case when $M = \mathbb{R}$.) This is precisely because we want an equation like $\frac{d}{dt}(\delta q) = \delta \dot{q}$. In other words, if we're at the jet $\varphi$, then when we go out from $\varphi$ in a horizontal direction (say the $x_1$ direction), the coordinates $q_1, \ldots, q_k$ of $\varphi$ should change in a manner specified by the coordinates $\partial_1 q_1, \ldots, \partial_1 q_k$ of $\varphi$.

But how should the coordinates $\partial_i q_j$ of $\varphi$ change themselves? Now we need to look at the $2$-jet component of $\varphi$, and it's the same all the way up. Now we see why the full $\infty$-jet bundle was needed- we'd be stuck if we cut ourselves off at a finite jet bundle.

Given this horizontal/vertical splitting of tangent spaces to the $\infty$-jet bundle, we're within our rights to talk about a bicomplex of differential forms on $j_{\infty}(E)$. Sections of $E$ (i.e. fields) yield $(0,0)$-forms, and Lagrangians yield $(n,0)$ forms. We may now happily take a variational derivative of $L$: it's just $\delta L$, the derivative in the vertical direction. This is an $(n,1)$-form, and the usual Euler-Lagrange argument beefs up to show that any $(n,1)$-form splits uniquely as $E + d\Theta$, where $\Theta$ is an $(n-1,1)$-form and $E$ is a "source form," i.e. an $(n,1)$-form such that, when contracted with a vertical vector (represented by a path of germs $\varphi_s$), the result only depends on the values of $\varphi_s$ at the spacetime point in question and not on the higher jet components. So $\delta(L) = E(L) + d\Theta$; the field equations are $E(L) = 0$.

Lots of other nice stuff falls out of this framework: i.e. an infinitesimal symmetry of the system is a vertical vector field $v$ such that $\iota_v \delta L = d \sigma$ for some $(n-1,0)$-form $\sigma$. (Such terms $d \sigma$ affect the action only by a boundary contribution, which can be assumed to be zero in your favorite way). Then you can immediately consider the $(n-1,0)$-form $\sigma - \iota_v \Theta$; being an $(n-1)$-form on spacetime, it represents a Noether current. To see that it's conserved, compute $d(\sigma - \iota_v \Theta) = \iota_v \delta L - \iota_v d\Theta = \iota_v E(L)$. But, at a solution to the field equations, $E(L) = 0$, so Noether's theorem is just a bit of playing around with differential forms.

....................

So, what if you want to study gauge theories? Suppose you have a Lie group $G$ and a $G$-principal bundle $P \rightarrow M$. Then "fields" should be $G$-connections on $P$. These aren't naturally sections of a bundle-- rather, they're an affine space for sections of the bundle $\Omega^1(M; \mathfrak{g})$ where $\mathfrak{g}$ denotes the bundle associated to $P$ via the adjoint representation of $G$. So "variations" in a gauge field look just like the variations above, where $E = \Omega^1(M; \mathfrak{g})$. But somehow this seems unnatural, and I'd like a more convincing way of saying this stuff in a gauge-theory setting.

Furthermore, the principal bundle $P$ shouldn't need to be fixed. Fields should be "bundle plus connection" rather than just "connection on a fixed bundle." Apparently this is where differential cocycles come in. A differential cocycle is supposed to (roughly?) capture the notion of a differential form AND an integer cocycle representing the same real cohomology class. The form gives us the connection, and the integer cohomology class gives us the bundle. (?) Unfortunately, I don't know much about these beasts. What I'd like to know is:

(1) is there a way to set up a variational bicomplex for gauge theory, where the "sections of $E$" are replaced by differential cocycles?

(2) when $G$ is trivial, you don't recover the non-gauge theory. Is there a more general framework which subsumes both gauge fields and non-gauge fields?

(0) what do I need to know about differential cohomology, cocycles, etc., to understand these things? The paper with the right definition, I think, is "Quadratic functions in geometry, topology, and M-theory" (Hopkins, Singer), but it's a bit formidable.