The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$, $n=2,3$. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

Let $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

We answer Mark Kac’s famous question, “Can one hear the shape of a drum?” in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the trace of the heat kernel are not sufficient to determine the above results.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon.

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.