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Stefan Steinerberger : Vibration and the local structure of elliptic partial differential equations (May 2, 2016 4:25 PM)

If you put sand on a metal plate and start inducing vibrations with a violin bow, the sand jumps around and arranges itself in the most beautiful patterns - this used to be a circus trick in the late 18th century: Napoleon was a big fan and put a prize on giving the best mathematical explanation. Today we know that the sand moves to lines where a certain Laplacian eigenfunction vanishes but these remain mysterious. I will show pictures of sand and demonstrate a new approach: the key ingredient is to make the elliptic equation parabolic and then work with two different interpretations of the heat equation at the same time. If time allows, I will sketch another application of this philosophy to localization phenomena for Schroedinger operators.

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