Tag Archives: martin

The mathematics of global warming

“The climate fact no one will admit: 2 °C warming is inevitable”

https://www.newscientist.com/article/dn28430-the-climate-fact-no-one-will-admit-2-c-warming-is-inevitable/

 

Here are some recent mathematicians who work on related questions:

The work of mathematician Guillaume Jouvet (Free University Berlin) and glaciologist Martin Funk of ETH in Zürich is somewhat spooky but awesome. They use computer model of glaciers and data gathered from the slow under-ice motion of the bodies of alpinists who died in a glacier over a number of years. https://plus.maths.org/content/maths-solves-frozen-mystery

And Noemi Petra’s new work is on inferring base slip conditions for Antarctic glacial flows

http://www.youtube.com/watch?v=W_GZb1ej7vk

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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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Bulgarian Solitaire

Today’s new random mathematical discovery: Bulgarian solitaire. It isn’t what your Bulgarian aunt might think it is. It is actually a card game introduced by the famous polymath and mathematician Martin Gardner (1914- 2010). If you don’t know who Martin Gardner was, read his obit.  This year would have been his 100th birthday.

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A computer passed a Turing test!

Yesterday (which was a public holiday here in Germany) I was so excited when I caught a glimpse of a Guardian article which claimed that a computer had passed a Turing test for the first time. Today (which is no longer a holiday, so I should be reading more serious stuff, but this piece news was too exciting to let pass by unnoticed) I read a better journalistic summary in the Vice magazine of what actually happened. Of course, the Turing test is itself is not stable unobjective because the measure of success is a computer being able to fool a jury of human interrogators.  Turing wrote the following in his 1950 paper (quoted in Vice, you can read Turing’s original paper here):

We now ask the question, “What will happen when a machine takes the part of [the test subject] in this game?” Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, “Can machines think?”

As Vice author Martin Robbins point out, “the key words here are “reliably” and “often.” Turing didn’t ask whether a machine could ever, on a single occasion, convince a human judge that it too is human; he asked whether a machine could do so reliably. “

So, the jury is still out… still bloody exciting, though!

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