The average of the first k eigenvalues of a self-adjoint o perator\, as a function of k\, has implications for inverse problems. Wh at information does it contain about an operator\, or about the shape of a domain or graph on which the operator is defined? I'll describe some re cently developed tools for approaching inverse problems through averages o f eigenvalues\, with selected applications to PDEs on domains and manifold s and to graphs. This work is joint with J. Stubbe of EPFL and\, in pa rt\, A. El Soufi and S. Ilias of the University of Tours\, and J. Dever of Georgia Tech.

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