Thursday, November 08, 2012

a day in the life: impromptu talks.

i think i'm developing a reputation for responsibility, punctuality, and good cheer ..

.. which, i fear, could very well lead to disaster!

the last thing i need is one more goal to juggle and meet,
and the potential fallout from if/when i drop the ball with something important.

maybe i should start breaking a few promises and failing to show up for a meeting or two .. you know, small things to keep my colleagues on their toes?

on a related note, i've been having a lot of meetings lately .. but more on that later.

so yesterday i addressed the department .. or at least a large chunk of it. of course, i didn't intend on this at all and had i known, i'd have most likely refused.

it all began when
a colleague asked me, last week, to talk informally to a few potential mathematics majors and tell them a thing or two about our research group .. which sounded reasonable enough at the time:
a cup of coffee,
sitting around the departmental lounge,
getting a read on young faces and asking what they wanted to hear.

you know: a conversation.
what i didn't expect was that every other research group would also speak their piece, that most of the presenters were professors and permanent faculty, that they were bringing postdocs and students along (grad and undergrad), and some of the presentations some had slides prepared ..

.. fancy diagrams,
photo rosters of research group members,
bullet points of research goals .. you name it;

so i was completely unprepared for something of this scale. i was just going to talk, maybe pick up a piece of chalk if things became desperate. i was expecting to address students who had been at university for a year or two and who mightn't have an idea of what mathematics is or what the career entails.

thinking about it, if this were really for the students, then is this really the scale at which we should be operating? bullet points and lecture formats? i mean, when i was a student ..

so i sighed inwardly, introduced myself when it was my turn, and told them a ghost story .. or a mathematical version of one, anyway, that would have caused me no end of shock and curiosity [0]. it sounded something like this .. i work in the nοnlinear ΡDE group and it is true that we study ΡDE's. our approach is generally analytical in nature with lots of estimates, but for those interested, there are applications, such as the fluid flow through porous media and perhaps even image processing through minimization methods. [1]

it's not immediately obvious, but even on smooth domains the structure of these equations obey its own strange, intrinsic geometry .. simply the way that solutions scale, or not [2]. analogously, our research interests also include these geometric considerations and geometry problems in their own right.

one of these topics is geometric measure theory. it is, roughly speaking, smooth geometry applied to rather rough sets. to put things concretely, most of you will know what i mean when i say "surface area."

if i hand you a sphere,
then you'll easily remember the rule.

if i give you a smooth object and its parametrization,
then you can probably compute it with the right formula.

the notion breaks down, however, when you consider more general, rougher sets .. like fractals, for example.

let's consider a self-similar construction like this:

the limiting object has zero area. however, if the first square has unit edge lengths, then each approximating shape has length 4 .. and so does the limit [3]. this is rather strange, when you think about it: we've constructed a set which contains no line segments or even curves [4] yet based on our computation, its length should be 4.

so put one way, my research group studies degeneracies of this kind. it is, of course, a very strange example; i still find it strange to this day, as might some of the senior faculty too. that's the compelling thing about mathematics; once you form a definition, you have to allow anything that satisfies its hypotheses, including degenerate examples like this one.

in mathematics we often take two routes: explore the generality further and/or restrict the definition as to eliminate these cases as possible examples. this case is no different. the notion of length here is what's called hausdorff measure, and it has the flexibility that even very rough sets have some notion of length.

if we choose to be more restrictive, however, then an equally interesting thing happens. if you start off with the limiting set and study its heat diffusion, then this gives us a sense of how the set "spreads" under evolution of time. if we allow ourselves short times and measure the spread, then it also gives a notion of surface area [5]; roughly speaking, this is like saying that the area of a sphere is the limit of rescaled volumes of thin annuli .. and in this case, this fractal will have zero "heatwise" surface area [6].

this isn't quite what i meant by the intrinsic geometry of certain ΡDE's, but it just goes to show you that these topics are rather related
at least, that's what i had in mind. in the end, the grey parts of the text i never said, due to time constraints .. which is a shame. i'm still kicking myself for not following through and telling a coherent, full-circle story.

as for the audience's reaction, i couldn't really tell. finns are hard to read.

[0] now that i think it, the moment i knew that metrιc spaces were in my future was when my undergraduate mentor showed me the heisenberg group, treated as a sub-riemannian manifοld. to this day, i occasionally feel the urge to drop everything and work on the isoperimetric problem for that space, hoping to crack it open.

[1] the porous medium equation and the mumford-shah functional, respectively.

[2] that is, parabolic scaling; this is a big issue for harnack-type inequalities.

[3] saying that is slightly misleading. it could mean that length is a continuous operation on compacta (with respect to the hausdorff metric) which is certainly not true in general!

[4] i.e. a so-called purely unrectifiable set of dimension 1.

[5] there's a theorem due to de giorgi that perimeter can be characterised by the heat kernel, as associated to the linear heat equation.

[6] in other words, perimeter charges only the "reduced/measure-theoretic" part of the boundary of a set, where the condition requires that a point sees positive density of the set as well as of the complement. since the fractal has no area, then it has no reduced boundary.

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