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For closed orientable hyperbolic surfaces of curvature $-1$, the spectrum of the Laplace-Beltrami operator, including multiplicities, determines the unmarked length spectrum of closed geodesics, and conversely. Count either oriented geodesics throughout or unoriented ones throughout. Primitive lengths with multiplicities determine the lengths of all iterates, and these determine the primitive lengths recursively.^[ref]

The Selberg trace formula connects the two lists: the spectral side sums functions of eigenvalues, while the geometric side sums explicit contributions from closed-geodesic lengths. Equality of either complete list forces equality of the other. The theorem concerns these spectral data; distinct surfaces can still have equal spectra.

For nonorientable hyperbolic surfaces, the trace contributions distinguish orientation-preserving and orientation-reversing geodesics. Their separate counts, rather than an unspecified undifferentiated convention, belong in the extended statement.^[ref]

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]