Title:  Eigenvalues of the Laplacian on triangles and more
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Abstract:  Sinusoidal eigenfunctions of the second derivative operator represent waveforms on a vibrating string. They generalize in two dimensions to eigenfunctions of the Laplacian, which describe waves on a vibrating membrane or drum. Quantum mechanically, these eigenfunctions represent wavefunctions of particles living in an infinite potential well. The corresponding eigenvalues represent frequencies of vibration, or quantum energy levels.
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How much do we know about these eigenvalues? Not as much as one might suppose! I will describe old results (by Polya), new results (with Siudeja), and open problems, about dependence of the eigenvalues on geometric properties of the membrane such as area and moment of inertia. Especially, we will try to solve max/min problems on the deceptively simple class of triangular drums. One of these problems leads to a Parseval identity for a three vectors in two dimensions, known in harmonic analysis as the "Mercedes-Benz" tight frame.
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Spectral zeta functions and heat traces make guest appearances.