In the classical realm of time-frequency analysis, a classical object of interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal $f(x),$ modified by a certain 'window function', in order to make joint time-frequency analysis of functions more feasible.

Since the pioneering work of Daubechies, time-frequency localisation operators have been of extreme importance in that analysis. These are defined through $V^* 1_{\Omega} V f = P_{\Omega} f,$ where $V$ denotes the short-time Fourier transform with some fixed window. These operators seek to measure how much a function concentrates in the time-frequency plane, and thus the study of their eigenvalues and eigenfunctions is intimately connected to the previous questions.

In this talk, we will explore the case of a Gaussian window function $\varphi(x) = e^{-\pi x^2}$, and the operators thus obtained. We will discuss some classical and recent results on domains of maximal time-frequency concentration, their eigenvalues, and inverse problems associated with such properties. During this investigation, we shall see that many of these problems possess some rather unexpected connections with overdetermined elliptic boundary value problems and free boundary problems in general. This is based on recent joint work with Paolo Tilli.