Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold $Y°$ that admits a nonconstant map to the moduli stack, we employ extension properties of logarithmic pluriforms to establish a strong relationship between the moduli map and the minimal model program of $Y°$: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much-refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial or more generally semiample canonical bundle.

Let $R$ be an o-minimal expansion of the real field, and let $L_{\mathrm{nest}}(R)$ be the language consisting of all nested Rolle leaves over $R$. We call a set nested sub-Pfaffian over $R$ if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over $R$. Assuming that $R$ admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over $R$ is again a nested sub-Pfaffian set over $R$. As a corollary, we obtain that if $R$ admits analytic cell decomposition, then the Pfaffian closure $P(R)$ of $R$ is obtained by adding to $R$ all nested Rolle leaves over $R$, a one-stage process, and that $P(R)$ is model complete in the language $L_{\mathrm{nest}}(R)$.

We generalize two theorems of Royden about the Teichmüller metric to any complete, finite covolume, mapping class group invariant Finsler (e.g., Riemannian) metric on Teichmüller space. In addition to giving a new proof of Royden's results, the theorems apply to many other metrics on Teichmüller space.

In 1987 Serre conjectured that any mod $\ell$ $2$-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalization of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where $\ell$ is unramified. The hard work is in formulating an analogue of the weight part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod $\ell$ Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod $\ell$ local-global principle for the group $D^{*}$, where $D$ is a quaternion algebra over a totally real field, split above $\ell$ and at $0$ or $1$ infinite places, and we show how it implies the conjecture.

We give a new proof of Hardy uncertainty principle, up to the endpoint case, which is only based on calculus. The method allows us to extend Hardy uncertainty principle to Schrödinger equations with nonconstant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.

In this article we study the local behavior of a solution to the Lamé system with Lipschitz coefficients in dimension $n\ge 2$. Our main result is the bound on the vanishing order of a nontrivial solution, which immediately implies the strong unique continuation property (SUCP). We solve the open problem of the SUCP for the Lamé system with Lipschitz coefficients in any dimension.