This article concerns the study of a new invariant bilinear form $\mathfrak{B}$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathfrak{B}$ using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathfrak{B}$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin–Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of $\mathfrak{B}$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathfrak{B}$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of pseudocompactly supported automorphic forms. We give a formula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant term operators which suggests that $L^{-1}$ is an analogue of the Aubert–Zelevinsky involution.

For smooth bounded domains in ${\mathbb{R}}^n$, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and we prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of the eigenvalues; this is achieved by working with an appropriate norm for boundary functions, which includes a spectral weight, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery-type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter $E>0$ to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function $u$ solving the Helmholtz equation $(\Delta -E)u=0$. This inclusion bound improves over previously known bounds by a factor of $E^{5/6}$, analogously to a recently improved inclusion bound in the Dirichlet case due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the $42000$th) from nine to fourteen digits, with negligible extra numerical effort.

We give rather simple answers to two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set, and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibers by global blowings-up of the target. These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. We show that the second question has a negative answer, in general, and that the first problem nevertheless has a positive solution.