Lipschitz thoughts, and the spectre of Rectifiability

In the last few days I've accumulated a good share of stir-craziness. Like a laptop running Windows XP, I feel as if my brain is on "Stand By" mode.

To return to my usual work habit, all I must do is

1. flip up the screen,
   
2. tap a key,
   
3. type the magic words (my passphrase)

and after a few process cycles, everything will run smoothly, good as new. However, it's not quite time to work yet, and I'm still in this mental holding pattern of "Stand By" mode.

If now is the time to think about mathematics again, then it's time to think about my thesis problem and to read Hirsch's Differential Topology for clues about the diffeotopy extension problem.

It means looking towards which direction of research is next, and what I must prepare or explore.

It also means addressing the matter of my prelim from 20 December: what exactly happened and why. I'm not ready for that.

So it's not time to think about "work mathematics" quite yet. Let me stay on vacation and think of "play mathematics" instead.

---

What didn't happen during my prelim were notions of rectifiability.

It was the one topic which caused me the most unease, though the framework is elegant. In Mattila's exposition, the essential message is that there are 7-8 equivalent notions of rectifiability in Euclidean space. But as you'd guess, the proofs are somewhat technical.

Time ran out and neither member of my committee asked about any of that, which surprised me to no end. So I have no idea how well (or unwell) I understand this notion. I've only my own guesses, which are unreliable and sketchy, at best.

But I do want to know, if only because it seems like a topic worth investigating and generalising to the context of metric spaces. Recently L. Ambrosio and B. Kirchheim have written a few articles on this general context (one in Acta) ..

.. and as per my usual tyronism, it amazes me that one can say anything about this and metrics spaces: they seem to me too "floppy" and general and too abstract for me.

Let me explain myself. Perhaps I am too stuck on my visual intuition, and I find it hard to imagine such and such a metric space with such and such a measure. Conformal modulus is still a new tool to me, and possible pathologies of curves and their images don't come easily to my mind.

I've warmed up enough to my advisor's suggestions, and now Lipschitz manifolds seem to me rather interesting and somewhat elusive objects.

In the Euclidean case, rectifiability of sets in Rⁿ is inexorably related to Lipschitz functions and parametrizations.

From what I've been told, if you enforce a strong enough condition of rectifiability [1] on a given set, it will give rise to a bi-Lipschitz parametrization, i.e. a Lipschitz manifold structure.

Now that's interesting, if only because images come easily to mind. It is like some geological means of detecting a topological character.

I envision walking across a field, and picking up a rock. With only the flat of my palm I study its exterior and see how flat it gets at this spot or that. If it's almost flat, in some measure sense, then guess what?

It arises from a rough sort of manifold. Huh.

[1] It's something related to tangent plane approximations, called Reifenberg's condition.