It is known that a family of meromorphic functions is normal if each function in the family shares a 3-element set with its derivative. In this paper we consider value distribution and normality problems with regard to 2-element shared sets. First we construct an example, by use of the Weierstrass doubly periodic functions, to show that a 3-element shared set can not be reduced to a 2-element shared set in general. We obtain a new criterion of normal families and new Picard-type theorems. The proofs make use of some results in complex dynamics. More examples are constructed to show that our assumptions are necessary.

The Oda's Strong Factorization Conjecture states that a proper birational map between smooth toric varieties can be decomposed as a sequence of smooth toric blowups followed by a sequence of smooth toric blowdowns. This article describes an algorithm that conjecturally constructs such a decomposition. Several reductions and simplifications of the algorithm are presented and some special cases of the conjecture are proved.

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions in the standard construction of the KP hierarchy and its solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of conformally immersed tori in the 4-sphere via quaternionic holomorphic geometry. The Sato-Segal-Wilson construction of KP solutions is adapted to this setting and the connection with quaternionic holomorphic curves is made. We then compare three different notions of "spectral curve": the QKP spectral curve; the Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in the 4-sphere.

Let $K$ be a nonarchimedean local field, let $h$ be a positive integer, and denote by $D$ the central division algebra of invariant $1/h$ over $K$. The modular towers of Lubin-Tate and Drinfeld provide period rings leading to an equivalence between a category of certain $\mathrm{GL}_h(K)$-equivariant vector bundles on Drinfeld's upper half space of dimension $h-1$ and a category of certain $D^*$-equivariant vector bundles on the $(h-1)$-dimensional projective space.

In this paper, the authors give the $L^p$ boundedness of a class of the Carleson type maximal operators with rough kernel, which improves some known results.

We give a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group.

É. Cartan proved that conformally flat hypersurfaces in $S^{n+1}$ for $n>3$ have at most two distinct principal curvatures and locally envelop a one-parameter family of $(n-1)$-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in $S^4$ is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in $S^4$ and their loop group symmetries. We also generalise these results to conformally flat $n$-immersions in $(2n-2)$-spheres with flat and non-degenerate normal bundle.