## Tuesday, September 3, 2013

### Academy Fight Song |Thomas Frank | The Baffler

This    is an interesting take of what's going on now in higher education.

Academy Fight Song | Thomas Frank | The Baffler

## Friday, August 9, 2013

### The Fulton-MacPherson compactification of a configuration space


Suppose that $M$ is a real analytic manifold of dimension $m$. Fix a finite set $L$ of labels. For any subset $S\subset L$ we define the following objects.

•  The manifold $M^S$ consisting of  maps $S\to M$. We will indicate a point in $M^S$ as a collection $x_S:=(x_s)_{s\in S}$,  $x_s\in M$, $\forall s\in S$.
•  The configuration space $M(S)\subset$ consisting of injective maps $S\to M$.
• The  thin diagonal  $\Delta_S\subset M^S$ consisting of the constant maps $S\to M$.

The space of configurations $M(L)$ is an open subset of $M^L$. We want to construct   a certain  completion  $M[L]$ of $M(L)$ as a    manifold with corners.  This completion is known as the  Fulton-MacPherson compactification of $M$.  The completion $M[L]$ is compact when $M$ is compact.  We follow closely the approach   of Axelrod and  Singer, Chern-Simons perturbation theory.II,  J. Diff. Geom., 39(1994), 173-213.

We begin with some simple observations. Observe  that if $S\subset S'$, then we  have a natural projection $\pi_S: M^{S'}\to M^S$ which associates to a map $S'\to S$ its restriction to $S$

$$M^{S'}\ni x_{S'}\mapsto x_{S}\in M^S.$$

$\newcommand{\hra}{\hookrightarrow}$  We set

$$\Delta_S^L=\pi_S^{-1}(\Delta_S)\subset M^L.$$

More explicitly, $\Delta_S^L$ consists of the maps $L\to M$ which are constant on $S$. Observe that

$$M^L\setminus M(L)=\bigcup_{|S|\geq 2} \Delta_S^L.$$

For $x\in M$ and $S\subset M$ we denote by $x^S$ the constant map $S\to\lbrace x\rbrace$ viewed as an element in $M^S$.


$$(u_S)_{s\in S}\in (T_xM)^S\sim (v_S)_{s\in S}\in (T_xM)^S\Llra (u_{s_0}-u_{s_1})=v_{s_0}-v_{s_1},\;\;\forall s_0,s_1\in S.$$

We identify $\eN(x_S)$ with the subspace of $Z_S(x)\subset (T_xM)^L$ consisting of vectors $v_L=(v_\ell)_{\ell\in L}$, $v_\ell\in T_xM$ such that

$$v_\ell=0,\;\;\forall \ell\in L\setminus S,\;\; \sum_{s\in S} v_s=0. \tag{1}\label{1}$$

We have a natural $\newcommand{\bsP}{\boldsymbol{P}}$  projector

$$\bsP_S: (T_x M)^L\to Z_S(x)$$

defined as follows. For a vector $\vec{v}=(v_\ell)_{\ell\in L} \in (T_xM)^L$, we denote by $\newcommand{\bb}{\boldsymbol{b}}$ $\bb_S(\vec{v})\in T_xM$  the barycenter of its  $S$-component

$$\bb_S(\vec{v}):=\frac{1}{|S|}\sum_{s\in S} v_s\in T_x M,$$

and we set

$$\bsP_S(\vec{v}) =(\bar{v}_s)_{s\in S},\;\;\bar{v}_s:=v_s-\bb_S(\vec{v}),\;\;\forall s\in S,\;\;v_\ell=0.$$

Note that  for $u_S\in (T_xM)^S$ we have  $u_S\sim \bsP_S u_S$.

We denote by $\Bl(S,M)$ the radial blowup of $M^S$ along $\Delta_S$.  This is  manifold with boundary whose interior  is naturally identified with $M_*^S:=M^S\setminus \Delta_S$. $\newcommand{\bsS}{\boldsymbol{S}}$  Its  boundary is $\bsS(\eN)$, the "unit" sphere  bundle bundle of $\eN$.    Equivalently  we identify the fiber of $\bsS(\eN)$ at $x_s$ with the quotient

$$\bsS(\eN(x^S))=\bigl(\; Z_S(x)\setminus 0\; \bigr)/\propto,$$

where

$$u_S\propto v_S \Llra \exists c>0:\;\; v_S=c u_S.$$

Following Fulton and MacPherson we will refer to the elements  in $Z_S(x)$ as $S$-screens at $x$.   Up to  a a positive rescaling, an $S$-screen  at $x^S\in \Delta_S$ is a collection of points  $u_S\in (T_xM)^S\setminus 0^S$  with barycenter at the origin.  If we fix a metric $g$ on $M$, then the fiber $\bsS\bigl(\;\eN(x_s)\;\bigr)$ that can be identified with the collection $v_L\in (T_xM)^L$ satisfying (\ref{1}) and

$$\max_{s\in S}|v_s|=1. \tag{3}\label{3}$$

There is a natural   smooth surjection (blow-down map)

$$\beta_S:\Bl(S,M)\to M^S$$

whose restriction to the interior of $\Bl(S,M)$ $\DeclareMathOperator{\int}{\boldsymbol{int}}$ induces a diffeomorphism to $M^S_*$    We denote by $\beta_S^{-1}$ the inverse

$$\beta_S^{-1}: M_*^S\to \int \Bl(S,M).$$

For $x_S\in M_*^S$ we set $\newcommand{\ve}{{\varepsilon}}$

$$\hat{x}_S:=\beta_S^{-1}(x_S).$$

If

$$[0,\ve)\ni t\mapsto x_S(t) M^S,\;\; x_S(0)=x_0^S\in\Delta_S$$

is a real analytic   path such that $x_S(t)\in M_*^S$ for $t>0$, then the limit $\lim_{t\searrow 0}\hat{S}(t)$ can be described as follows.
• Fix  local (real analytic) coordinate near $x_0$ so that the points $x_{s}(t)$ can be  identified with points in a neighborhood of $0\in\bR^m$.
• For $t>0$ denote by $\bb(t)$ the barycenter of the collection $(x_s(t))_{s\in S}\subset\bR^m$, $$\bb(t)=\frac{1}{|S|}\sum_{s\in S} x_s(t).$$
• For $t>0$  and $s\in S$  define  $\bar{x}_s(t) =x_{s}(t)-\bb(t)$,  $m(t)=\max_s|\bar{x}_s(t)|$.
Then $\lim_{t\searrow 0}\hat{S}(t)$ can be identified with the vector

$$\lim_{t\searrow 0}\frac{1}{m(t)} \bigl(\;\bar{x}_s(t)_{s_\in S}\;\bigr)\in \eN(x_0^S).$$

We have a natural map $\newcommand{\eX}{\mathscr{X}}$

$$\gamma: M(L)\to \eX(M,L):=M^L\times\prod_{|S|\geq 2} \Bl(S,M),\;\; M(L)\ni x_L\mapsto \gamma(x_L):=\Bigl(\; x_L;\;\;(\hat{x}_S)_{|S|\geq 2}\;\Bigr)\in \eX(M,L).$$

The Fulton-MacPherson compactification of $M(V)$ is the closure of $\gamma\bigl(\;M(L)\;\bigr)$ in $\eX(M,L)$.

We want to give a more  explicit description of this  closure.  Observe first that $\eX(M,L)$ and thus any point in the closure of $\gamma(M[L])$ can be approached from within $\gamma(M[L])$ along a real analytic path.  Suppose  that $(0,\ve)\ni t \mapsto x_L(t)$  is a real analytic path such that $\gamma\bigl(\; x_L(t)\;\bigr)$ approaches a  point $\gamma^0\in \eX(M,L)$. The limit point is a collection $( x^*_L, (y(S))_{|S|\geq 2})\in \eX(M,L)$.

To the point $x^*_L\in M^L$ we associate an equivalence relation on $L$

$$\ell_0\sim_0 \ell_1 \Llra x^*_{\ell_0}=x^*_{\ell_1}.$$

Denote by $\newcommand{\eC}{\mathscr{C}}$ $\eC_0\subset 2^S$ the collection of equivalence classes  of $\sim_0$ of cardinality $\geq 2$.

The  subsets $S$ of $L$ of cardinality $\geq 2$  are of two types.


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.