The n-Category Café

Now the interview Colin McLarty had with me is available.

Very interesting! I like the way that it’s visibly a conversation between two people who respect each other and are interested in listening as well as talking. In that respect it’s not much like a traditional interview.

I also like Colin’s example of Bollywood movie characters as datatypes.

Thanks, Tom - that’s the one. I couldn’t find my copy of IT and didn’t want to risk misquoting.

To get away from this particular example, for a moment - if there is, as Spinal Tap have taught us, a fine line between stupid and clever, perhaps there is also a fine line between “exaggerating for effect” and “talking out of one’s ****” ?

It seems to come down to this: people who know what they’re doing (Rota? Zeilberger? Gelfand?) exaggerate for effect; people who don’t know what they’re doing, talk out of their ****s. ;) Either that, or this is one of those irregularly declined verbs …

Now I’m curious. Is it really true that you can’t make changes of variables with distributions? What does this even mean?

Let’s see. If we’re talking about integrating functions on measure spaces, I take “change of variables” to mean the following. Let $\alpha\colon X \to Y$ be a map of measurable spaces. Then any measure $\mu$ on $X$ gives rise to a (“pushforward”) measure $\alpha_* \mu$ on $Y$, such that

$$\int_X (g \circ \alpha) d\mu = \int_Y g\, d(\alpha_* \mu)$$

for any $g\colon Y \to \mathbf{R}$ such that $g\circ\alpha$ is $\mu$-integrable.

So, what would change of variables for distributions mean? I suppose it would say that if $X$ and $Y$ were suitable “spaces” (open subsets of $\mathbf{R}^n$?) and $\alpha\colon X \to Y$ a map of a suitable sort ($C^\infty$ and proper to be on the safe side?) then any distribution $u$ on $X$ somehow gives rise to a “pushforward” distribution $\alpha_* u$ on $Y$, such that

$$\langle g\circ\alpha, u \rangle = \langle g, \alpha_* u \rangle$$

for all suitable functions $g$ on $Y$.

Is that plausible? Ignoring all analytical details, I’d imagine that $\alpha\colon X \to Y$ induces a linear map $Test(Y) \to Test(X)$ of spaces of test functions, by composition. The dual of this linear map would be something like $Dist(X) \to Dist(Y)$, since the space $Dist(X)$ of distributions on $X$ is a subspace of the dual of $Test(X)$. And that’s what we want. (If you interpret “$Test(X)$” as meaning “$C(X)$” then pushforward measures can be described like this, I guess.)

So I suppose something must go wrong with the seminorm estimates. Can an expert enlighten me?