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Can you hear the shape of a drum and deformational spectral rigidity of planar domains? Kaloshin, Vadim

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M. Kac popularized the question {\em Can you hear the shape of a drum?} Mathematically, consider a bounded planar domain $\Omega$ and the associated Dirichlet problem $\Delta u + \lambda^2 u = 0$ with $u|_{\partial \Omega}$ = 0. The set of $\lambda$s such that this equation has a solution, denoted $\mathcal{L}(\Omega)$ is called the Laplace spectrum of $\Omega.$ Does Laplace spectrum determine $\Omega$? In general, the answer is negative. Consider the billiard problem inside ?. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any generic axis symmetric planar domain with is dynamically spectrally rigid, i.e. can't be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. This is joint works with J. De Simoi, A. Figalli, and J. De Simoi, Q. Wei.

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