We establish estimates in BMO and Campanato-John-Nirenberg spaces BMO$\sb \psi$ for the second derivatives of solutions to the fully nonlinear elliptic equation $F(D\sp 2u,x)=f(x)$.

Let $G$ be a fixed graph with largest (adjacency matrix) eigenvalue $\lambda\sb 0$ and with its universal cover having spectral radius $rho$. We show that a random cover of large degree over $G$ has its "new" eigenvalues bounded in absolute value by roughly $\sqrt{\lambda\sb 0\rho}$.

This gives a positive result about finite quotients of certain trees having "small" eigenvalues, provided we ignore the "old" eigenvalues. This positive result contrasts with the negative result of A. Lubotzky and T. Nagnibeda which showed that there is a tree all of whose finite quotients are not "Ramanujan" in the sense of Lubotzky, R. Phillips, and P. Sarnakand of Y. Greenberg.

Our main result is a "relative version" of the Broder-Shamir bound on eigenvalues of random regular graphs. Some of their combinatorial techniques are replaced by spectral techniques on the universal cover of $G$. For the choice of $G$ that specializes our main theorem to the Broder-Shamir setting, our result slightly improves theirs.

Let $\overline {\Pr}$ denote the ideal spanned by the characters of projective modules in the Grothendieck ring of the category $\overline {\mathscr{C}\sb f}$ of finite dimensional modules over the small quantum group $U\sp {\rm fin}\sb q(\mathfrak {g})$. We show that $\overline {\Pr}$ admits a description completely parallel to that of the Verlinde algebra of the fusion category (see [AP]), with the character of the Steinberg module playing the role of the identity.

We construct a determinant of the Laplacian for infinite-area surfaces that are hyperbolic near $\infty$ and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order 2 with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near $\infty$ case, the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order 2 with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.

We show that the moduli stack $\mathscr {M}\sb h$ of canonically polarized complex manifolds with Hilbert polynomial $h$ is Brody hyperbolic. Hence if $M\sb h$ denotes the corresponding coarse moduli scheme, and if $U \to M\sb h$ is a quasi-finite morphism, induced by a family, then there are no nonconstant holomorphic maps $\mathbb {C}\to U$.

We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.