Tuesday, August 19, 2014

Timothy Gowers on Field Medal winners

Official accounts here. Here is the one on Manjul Bhargava. He has so far written also about Avila and Hairer. I like the way Gowers writes. Even for a mathematician of his calibre, the view of other areas is impressionastic. From the one on Hairer:
"...Hairer studied stochastic PDEs. As I understand it, an important class of stochastic PDEs is conventional PDEs with a noise term added, which is often some kind of Brownian motion term.
Unfortunately, Brownian motion can’t be differentiated, but that isn’t by itself a huge problem because it can be differentiated if you allow yourself to work with distributions. However, while distributions are great for many purposes, there are certain things you can’t do with them — notably multiply them together.
Hairer looked at a stochastic PDE that modelled a physical situation that gives rise to a complicated fractal boundary between two regions. I think the phrase “interface dynamics” may have been one of the buzz phrases here. The naive approach to this stochastic PDE led quickly to the need to multiply two distributions together, so it didn’t work. So Hairer added a “mollifier” — that is, he smoothed the noise slightly. Associated with this mollifier was a parameter \epsilon: the smaller \epsilon was, the less smoothing took place. So he then solved the smoothed system, let \epsilon tend to zero, showed that the smoothed solutions tended to a limit, and defined that limit to be the solution of the original equation.
The way I’ve described it, that sounds like a fairly obvious thing to do, so what was so good about it?"
One of the comments "I am not an expert on this, but I’ve heard that the renormalization procedure using mollifiers results in some limit which is actually not a solution of the original equation (otherwise, it sounds too easy). It is a solution of some modified equation. Then you repeat this procedure and miraculously the process stabilizes after a finite number of steps (five, six?), and that is when you get the solution of the original equation. The fact that the process terminates in finitely many steps is a miracle that has something to do (philosophically, or technically?) with wavelets, since something of this sort happens in wavelets. "

Mathematicians do not seem to think that diferently from others.

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