In 1966, M. Kac published his seminal article ''Can one hear the shape of a drum?'', which deals with the so-called inverse spectral problem: to determine geometric information from spectral data.

In this talk, we want to discuss the inverse spectral problem for polygonal domains in the hyperbolic plane and Dirichlet eigenvalues. We will present spectral invariants, i.e. geometric properties of polygons which can be deduced from the spectrum of the Dirichlet Laplacian. They are obtained by calculating the asymptotic expansion of the heat trace associated with the Dirichlet Laplacian. All coefficients of that expansion, the so-called heat invariants, can be computed explicitly and it turns out that the heat invariants provide interesting information about the polygon.

The results are part of my dissertation, which is published here: https://edoc.hu-berlin.de/handle/18452/19142

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