Tag Archives: fieldwork

Field(s)notes: Mathematical Box Lunch on the Devil’s Staircase

20 August 2014

Joint lunchbreak with T in the freezing August sun. We sat on the steps of the Warwick Manufacturing Group eating our packed lunches and discussing Sard’s theorem (analysis) which says that for any properly smooth function on the Real line, the set of critical values (on the graph, those are the points at which the line is flat, parallel to the x-axis) has a measure zero (or something to that effect). That means that – although the number of critical values may easily be countably infinite, and even uncountably infinite – it is essentially zero when compared to the whole of the line.  Apparently, once you have too many critical values – imagine driving along a road and stopping everywhere – then the graph of the function is no longer smooth because it becomes broken down into too many infinitesimal segments. I imagine the line becoming very wiggly on a tiny scale – something like the 1.5-dimensional lines of fractals, though I’m not sure this is correct. A better example is the Devil’s Staircase which is very much non-smooth (so nothing prevents it, in theory, to have a large number of critical values).

Figure 1 The Devil's staircase (Source: https://www.math.hmc.edu/funfacts/figures/30003.3.1.gif)

Figure 1 The Devil’s staircase (Source: https://www.math.hmc.edu/funfacts/figures/30003.3.1.gif)

Sard’s theorem brought us to the question whether a flat function with one insanely tall and thin spike in the middle can be called a smooth function… and apparently it can’t, and is instead called a distribution. (http://en.wikipedia.org/wiki/Dirac_delta_function ) The cool thing about the distribution (aka Dirac delta function) is that it is essentially a function over functions, that is, it’s a function-like thing into which you can feed other functions and get results; it is hugely useful in physics. The annoying detail is that, while you can add distributions together, multiply a distribution with a number or even differentiate it, you can’t multiply it by other distributions (or by itself). So δ^δ or other useful operations such as powers or exponentiation make no sense.

"Dirac function approximation" by Oleg Alexandrov - self-made with MATLAB. Licensed under Public domain via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Dirac_function_approximation.gif#mediaviewer/File:Dirac_function_approximation.gif

“Dirac function approximation” by Oleg Alexandrov – self-made with MATLAB. Licensed under Public domain via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Dirac_function_approximation.gif#mediaviewer/File:Dirac_function_approximation.gif

This then brought me to an attempt to grasp the meaning of Martin Hairer’s contribution which brought him (and the Warwick Maths department) one of this year’s four Fields Medals. However, I won’t be summarising it here… I am far from being able to understand the essence of his 180-page manuscript which “must have been downloaded into his brain by a more intelligent alien race” (www.simonsfoundation.org/quanta/20140812-in-mathematical-noise-one-who-heard-music ) and lays out a neat theory of hitherto unknown regularity structures underpinning the previously unruly stochastic partial differential equations… In my defence, a friend who is a lecturer in computer science, said that of course he himself could’t understand the actual contribution either.

While T was drawing the naughty spiky delta nought function on the blackboard, Ian Stewart wandered past, unpacking some mail and recycling the package. I’ve learnt not to be star-struck when sitting around in the Warwick Maths department, mainly thanks to the fact that everyone is very friendly, but still, you don’t get Ian Stewarts wandering past you all the time (you do, if you work at Warwick Maths :p). In the book exchange box under the staircase in the middle of the Common Room, there is a German translation of Ian Stewart’s “Equations that changed the world”. I briefly considered nicking it and then got distracted by hugs at the blackboard (which, as I learnt yesterday from Christopher Zeeman’s essay http://www2.warwick.ac.uk/fac/sci/maths/general/institute/histories-small.pdf , are actually greenglass).

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Friday 13, 3:14

It’s Friday 13th, it is raining and I have no umbrella, an interview I was hoping to do today fell through, and I couldn’t go to the annual von Mises lecture today because I hadn’t registered in advance. (so confusing, this year it is on 13 June 2014, and last year it was on 14 June 2013)

On the plus side, it’s 3.14 πμ and I’m having coffee (or is that co-ϕ?) in the π-Building of the Free University of Berlin. How cool is that.

Das π-Gebäude der FU-Berlin.

Das π-Gebäude der FU-Berlin.

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Proofs, interviews and elephants (field diary notes)

(field diary excerpt, 2013)

This morning I went to a Foundations lecture and a Differential equations lecture. Then I did a brief but very important interview and even got hold of some data about the UK academic labour market. Yay! Successful day, and it’s not even lunchtime. But I was hungry, so lunch has already happened and now I’m onto my second coffee of the day. A sociologist is very much a device for making hypotheses about the world out of coffee, just like a mathematician is a device for turning coffee into theorems.

Important and timely interview gone well, new data at hand, and a steaming cup of coffee: maybe that’s the way a mathematician feels when s/he manages to complete a small step of an elephant proof? Are any mathematicians or students reading this? How does completing a step from a proof make you feel?

Now I’m thinking of an invisible elephant standing patiently in the Mathematics common room, waiting to be drawn out of the air. That’s very much how my project looks like in my imagination. Stuff to be found out, it’s already there, and yet I have infinite possibilities for drawing my elephant well or badly.

Oops. I’ve just committed a cardinal sin by mentioning the word “infinite” – and I don’t have an infinity licence yet!


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