jump to navigation

Sobolev spaces on manifolds February 22, 2011

Posted by David Speyer in Uncategorized.
14 comments

I just spent the last few days working through the proof of the Hodge theorem in Wells Differential Analysis on Complex Manifolds. There were a few things which confused me right at the start, and which could have been easily cleared up if I had just understood which constructions were canonical and which were not. The point of this blog entry is to record that data. I’ll also add a little extra background, so people other than me have a chance of understanding it. Since I am very much not an analyst, there is a high risk of errors here; I would appreciate anyone who points them out to me. Also, to keep life simple, s will be an integer and p will be 2.

Let M be a smooth n-dimensional Riemmannian manifold and E a complex vector bundle equipped with a positive definite Hermitian form.

Things that are true, important, and canonical

For every integer s, there is a topological vector space W^s(M). There are inclusions W^{s+1}(M) \hookrightarrow W^s(M). These are compact maps, meaning that if v_k is a bounded sequence in W^s then v_k has a convergent subsequence in W^{s+1}. The smooth functions, C^{\infty}(M), have compatible inclusions into all of the W^s‘s, with dense image.

For s \geq 0, a sequence f_i of smooth functions is Cauchy in W^s if, for any differential operator D, of order \leq s, the sequence D f_i is Cauchy in the L^2 topology. So W^0 \cong L^2 and, more generally, W^s can be thought of all the completion of C^{\infty} in this variant of the L^2 topology.

We have containments C^{s-\lfloor n/2 \rfloor -1}(M) \supseteq W^s(M) \supseteq C^s(M), compatible with the injections W^{s+1} \hookrightarrow W^s and C^{k+1} \hookrightarrow C^k. In other words, if we have a sequence in C^{\infty}(M) which is W^s-Cauchy, then it approaches a limit in the C^{s-\lfloor n/2 \rfloor -1}-topology and, if we have a sequence in C^{\infty} approaching a limit in the C^s-topology, then it is W^s-Cauchy.

There is a sesquilinear pairing W^s \times W^{-s} \to \mathbb{C} with respect to which W^{s+1} \hookrightarrow W^s and W^{-s} \hookrightarrow W^{-s-1} are adjoint. This exhibits W^{-s} as the conjugate-linear-dual of W^s. (And, I think, vice versa, although Wells doesn’t say this.) When s=0, this is the standard pairing on L^2(M). The fact that the dual of the smaller space is the larger space is strange. My current intuition is that, although W^{s+1} \hookrightarrow W^{s} is an injection, it schrunches W^{s+1} down, so that it is easier for sequences to converge in the image. Thus, many linear functions which are continuous on W^{s+1} are no longer continuous as functions on W^{s+1} with the topology restricted from W^s. I still find this hard to think about, though.

An order m differential operator L will extends to a continuous map W^s to W^{s-m}. The above pairing will induce an adjoint map L^*: W^{s} \to W^{s-m}. In the cases I care about, L^* is also a differential operator, but I think this is in general not true.

W^s has the structure of a normed vector space, which is canonical in the sense that the norm is well defined up to a constant distortion factor.

The following is true, but should be ignored

The topology on W^s(M) arises from a Hilbert space structure. Nothing is ever adjoint with respect to this inner product. Orthogonality in this inner product is not a canonical property. Since it is a Hilbert space, W^s is isomorphic to its own dual and, hence, W^s \cong W^{-s}; this isomorphism is never used. It would have been better had I not known this structure existed.

Below the fold, for the curious, I will repeat Wells’ definitions in the order I would have given them.
(more…)

A Peter-Weyl “counter-example” February 18, 2011

Posted by David Speyer in Uncategorized.
3 comments

Let K be a compact Lie group. The Peter-Weyl theorem gives a basis for functions on K. In particular, it tells us that the characters are an orthonormal basis for class functions on K.

Let’s look at K=SU(2). Topologically, K is a three sphere, and the conjugacy classes are latitudinal two spheres. We’ll label the conjugacy classes by the line segment [0, \pi], where \theta labels the conjugacy class of matrices with eigenvalues e^{i \theta} and e^{- i \theta}. The conjugacy class \theta is a sphere of radius proportional to \sin \theta, and hence area proportional to \sin^2 \theta.

The characters of SU(2) are indexed by positive integers, with

\displaystyle{\chi_n(\theta) = e^{- (n-1) i \theta} + e^{- (n-3) i \theta} + \cdots + e^{(n-1) i \theta} = \frac{\sin (n \theta)}{\sin \theta} }.

So, if F is a class function on K, then its inner products with the characters are given by the integrals

\displaystyle{ c_n = \frac{2}{\pi} \int_0^{\pi} F(\theta) \chi_n(\theta) \sin^2(\theta) d \theta }.

Here \sin^2 \theta is the area of the conjugacy class \theta and 2/\pi turns out to be the correct normalization factor.

So, we should expect that

\displaystyle{ F(\theta) = \sum c_n \chi_n(\theta) }.

All of this is pretty standard. So, what would you expect happens if you take F to be 1 on [0, \pi/2] and -1 on [\pi/2, \pi]? Seriously, see if you can guess what peculiar behavior these sums show.

(more…)

If You Like Mathematical Physics…. February 11, 2011

Posted by A.J. Tolland in Uncategorized.
4 comments

I visited the University of Pennsylvania on Tuesday, and while I was there, Ron Donagi told me about an upcoming conference at UPenn, called “String-Math 2011″.   It’s a week-long meeting, June 6-11, with a bunch of exciting people on the visitor list.   And it’s the first of a series of such conferences.  If you want to know more, the conference webpage is

http://www.math.upenn.edu/StringMath2011/

The early registration deadline is April 2nd.

In other news, like my co-bloggers, I’m teaching a class this semester:  Quantum Field Theory for Mathematicians, which aims to explain the basic ideas of quantum field theory through the study of mathematically well-defined examples.  In particular, we’re looking to see how Wilson’s “effective field theory” philosophy motivates the rigorous constructions of the path integral measures.  I’ve been putting my lecture notes online; you’re welcome to check them out if you’re interested.  (There have only been 3 lectures so far; we’re still on preliminary material.  However, you can see from the syllabus what I’m hoping to cover.)

Follow

Get every new post delivered to your Inbox.

Join 90 other followers