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Alexander Strohmaier studies global analysis, spectral geometry and quantum field theory in curved spacetime. His Hannover research page connects spectra of geometric differential operators with classical dynamics.^[ref]

With Emily B. Dryden, he extended Huber's theorem to compact orientable hyperbolic orbisurfaces: the Laplacian spectrum determines the lengths of closed geodesics and the number of singular points of each order, with a converse. The result uses a Selberg trace formula to relate spectral and geometric terms.^[ref]

Emily B. Dryden studies spectral geometry: the relationship between geometric spaces and spectra of differential operators. Her Bucknell profile identifies this research specialization.^[ref]

With Alexander Strohmaier, she proved an orbisurface version of Huber's theorem. For compact orientable hyperbolic orbisurfaces, the Laplacian spectrum determines both closed-geodesic lengths and singular-point orders with multiplicities; the corresponding geometric data also determine the spectrum. Thus spectral information records some singular geometry as well as ordinary closed paths.^[ref]

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]

Let $X=\Gamma \backslash {\mathbb{H}}^2$ be a closed orientable hyperbolic surface, where $\Gamma \le PS{L}_2(\mathbb{R})$ is a torsion-free discrete subgroup. Write ${\lambda }_j$ for the nonnegative eigenvalues of its Laplace-Beltrami operator, with multiplicities. A heat version of the Selberg trace formula is, for $t>0$,The outer geometric sum is over primitive nonidentity conjugacy classes in $\Gamma$, equivalently oriented primitive closed geodesics; a class and its inverse are both counted. The inner sum records their iterates. The first term is the identity contribution, proportional to hyperbolic area. This formula follows by taking the Gaussian test function $h(r)=e^{-t(r^2+1/4)}$ in the usual spectral-parameter formula, whose Fourier transform is the displayed Gaussian factor.^[ref]

The left side is the heat trace. Thus the same quantity can be computed from vibration frequencies or from closed-geodesic lengths and hyperbolic area. The Huber spectral equivalence theorem extracts the mutual determination of the two spectra. Quotients with cusps or cone points require additional terms, so the displayed version uses the closed smooth orientable setting.