@Article{Turner2020,
author = "Turner, John Wm.",
title = "Echoes and glimpses of a distant drum",
journal = "Opuscula Math.",
year = "2020",
volume = "40",
number = "6",
pages = "737--750",
doi = "https://doi.org/10.7494/OpMath.2020.40.6.737",
url = "https://doi.org/10.7494/OpMath.2020.40.6.737",
abstract = "To what extent does the spectrum of the Laplacian operator on a domain \(D\) with prescribed boundary conditions determine its shape? This paper first retraces the history of this problem, then Kac's approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for \(D\) in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method.",
keywords = "Kac drum problem, inverse methods, diffusion process",
issn = "1232-9274"
}