Juan Pablo Rossetti works on geometry and spectral questions. With Peter G. Doyle, he proved that closed hyperbolic surfaces with the same Laplacian spectrum have matching closed geodesics, separately counting orientation-preserving and orientation-reversing ones at each length.[ref]
The orientation distinction matters on nonorientable surfaces: different types of closed geodesic contribute differently to a trace formula. Their result explains exactly which geometric data can be recovered from the spectrum, extending the orientable setting of Huber's theorem.[ref]
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