Seminar Analysis: Stefan Steinerberger (University of Bonn)

A Geometric Uncertainty Principle with an Application to Pleijel's estimate

It is obvious that there is no tiling of the Euclidean plane with unit disks (any three disks have a gap in the middle): we prove a quantitative version of this statement. This simple insight has applications in spectral geometry: it tells us something about the topological structure of the vibration profile of a (possibly oddly-shaped) drum and allows us to recover an improved version of Pleijel's estimate (which was also recently done by Bourgain).

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