Conference Global Portuguese Mathematicians 

14/07/2017, 17:30 — 18:30 — Amphitheatre Pa1, Mathematics Building
Daniel Abreu, Austrian Academy of Sciences

The music of the electrons: time-frequency analysis meets statistical mechanics.

At a first glance it may be hard to tell what musical scores and electron density wave functions have in common, but mathematicians can perceive both as the same object. Such a seemingly bizarre assertion can be better understood in the formulation of uncertainty principles (UPs). UPs tell us that it is impossible to simultaneously measure the position and momentum of an electron and, in time-frequency analysis, that it is impossible to hear an instantaneous frequency. To overcome such obstructions, electron density wave functions are written as functions of position and momentum, while music signals are written as functions of time and frequency (as in a musical score). In this talk we will see how, in mathematics:

1. UPs boil down to the fact that a nonzero function and its Fourier transform cannot both be supported in arbitrary small sets (a simple consequence of Cauchy-Schwarz inequality).
2. The model behind electron density wave functions and time-frequency representations is essentially the same.
3. The identification of the two models has recently led to several new results in the analysis of the Landau equation with a magnetic field and the statistics of multi-layer electron distributions.