In this paper we study positive solutions to the CR Yamabe equation on groups of Heisenberg type. For the subclass of groups of Iwasawa type, we characterize those solutions that have partial symmetry, that is, those that are invariant with respect to the action of the orthogonal group in the first layer of the Lie algebra.

We compute the unramified cohomology of quadrics of dimension 4 in degree 4 over an arbitrary field of characteristic different from 2. We find that it is related to classical invariants of a more elementary nature, such as the group of spinor norms and the projective special orthogonal group modulo Manin's R-equivalence.

We study two boundaries for the Teichmüller space of a surface Teich(S) due to L. Bers and W. Thurston. Each point in Bers's boundary is a hyperbolic 3-manifold with an associated geodesic lamination on S, its end-invariant, while each point in Thurston's is a measured geodesic lamination, up to scale. When dim_ℂ(Teich(S))>1, we show that the end-invariant is not a continuous map to Thurston's boundary modulo forgetting the measure with the quotient topology. We recover continuity by allowing as limits maximal measurable sublaminations of Hausdorff limits and enlargements thereof.

In previous work, the authors computed archimedian heights of hermitian line bundles on families of polarized, n-dimensional abelian varieties. In this paper, a detailed analysis of the results obtained in the setting of abelian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian fibrations and hence extend to a quite general situation. Specifically, we let f : X→Y be any family of smooth, projective, n-dimensional complex varieties over some base, and consider a line bundle on X equipped with a smooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chern class. Under mild additional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n+1)-fold star product of Green's currents associated to the sections, integrated along the fibers of f, is the log-norm of a global section of M. Furthermore, it is proven that in certain general settings the global section of M can be explicitly expressed in terms of point evaluations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces when M is a moduli space of degree-2 polarizations. In this setting, the global section constructed via Green's currents is equal to a power of the Φ-function first studied by R. Borcherds. Additional examples and problems are presented.

We prove that every element of the mapping class group Γ_g has linear growth (confirming a conjecture of N. Ivanov) and that Γ_g is not boundedly generated. We also provide restrictions on linear representations of Γ_g and its finite index subgroups.

Characteristic numbers of compact hyper-Kähler manifolds are expressed in graph-theoretical form, considering them as a special case of the curvature invariants introduced by L. Rozansky and E. Witten. The appropriate graphs are generated by "wheels," and the recently proved Wheeling theorem is used to give a formula for the $\mathscr{L}$²-norm of the curvature of an irreducible hyper-Kähler manifold in terms of the volume and Pontryagin numbers. The formula involves the multiplicative sequence that is the square root of the Â-polynomial.

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.