We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M,g)$ from a restriction ${\Lambda }_{\mathcal{S},\mathcal{R}}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\mathcal{S}$ and $\mathcal{R}$ are open sets in $\partial M$ and the restriction ${\Lambda }_{\mathcal{S},\mathcal{R}}$ corresponds to the case where the Dirichlet data is supported on ${\mathbb{R}}_{+}\times \mathcal{S}$ and the Neumann data is measured on ${\mathbb{R}}_{+}\times \mathcal{R}$. In the novel case where $\bar{\mathcal{S}}\cap \bar{\mathcal{R}}=\varnothing$, we show that ${\Lambda }_{\mathcal{S},\mathcal{R}}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\mathcal{S}$. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,

$${\lambda }_j\le C\Vert {\partial }_{\nu }{\varphi }_j{\Vert }_{L^2\left(\mathcal{S}\right)}^2,j=1,2,\dots ,$$

where ${\lambda }_j$ are the Dirichlet eigenvalues and where $({\varphi }_j{)}_{j=1}^{\infty }$ is an orthonormal basis of the corresponding eigenfunctions.

We modify the definition of the families of $A$ and $B$ stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces have the same dimension as the ones defined in our previous coauthored work with Mavlyutov, but they admit natural flat connections with respect to the appropriate parameters. This solves a longstanding question of relating the Gelfand–Kapranov–Zelevinsky (GKZ) hypergeometric system to stringy cohomology. We construct products on these spaces by vertex algebra techniques. In the process, we fix a minor gap in our coauthored work with Mavlyutov, and we prove a statement on intersection cohomology of dual cones that may be of independent interest.

We prove the universality of the $\beta$-ensembles with convex analytic potentials and for any $\beta >0$; that is, we show that the spacing distributions of log-gases at any inverse temperature $\beta$ coincide with those of the Gaussian $\beta$-ensembles.

Let $\Upsilon$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\Upsilon$ of length at most $L$, choose one, say, ${\gamma }_L$, at random, and let $N\left({\gamma }_L\right)$ be the number of its self-intersections. It is known that there is a positive constant $\kappa$ depending on the metric such that $N\left({\gamma }_L\right)/L^2\to \kappa$ in probability as $L\to \infty$. The main results of this article concern the size of typical fluctuations of $N\left({\gamma }_L\right)$ about $\kappa L^2$. It is proved that if the metric has constant curvature $-1$, then typical fluctuations are of order $L$; in particular, as $L\to \infty$ the random variables $\left(N\right({\gamma }_L)-\kappa L^2)/L$ converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of $N\left({\gamma }_L\right)$ are of order $L^{3/2}$; in particular, the random variables $\left(N\right({\gamma }_L)-\kappa L^2)/L^{3/2}$ converge in distribution to a Gaussian distribution with positive variance. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.