We investigate a class of isolated hypersurface singularities, the so-called purely elliptic singularities, of complex algebraic varieties of dimension greater than or equal to two. We show that, for hypersurface purely elliptic singularities defined by nondegenerate polynomials, Calabi-Yau varieties arising among the irreducible components of the essential divisors are concretely associated with the defining equations of these singularities, and that the birational class of the Calabi-Yau varieties does not depend on the irreducible components.

Characterizations of invertible and Fredholm composition operators are obtained for Bergman spaces on connected domains in the complex plane. In addition, the isomorphism between Toeplitz algebras and their K-theory are discussed.

In this paper, we investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set. We shall establish the connections between the Fatou components and the singularities of the inverse function and, accordingly, give sufficient conditions for the non-existence of wandering domains or Baker domains, and for the Julia set to be the Riemann sphere. Through the discussion of permutability of such functions, we shall construct several transcendental meromorphic functions which have Baker domains and wandering domains with special properties; for example, wandering and Baker domains with a critical value on the boundary and a wandering domain with the boundary being a Jordan curve (some such examples for entire functions were exhibited in other papers) and those of non-finite type which have no wandering domains.

We consider a quasiconformal automorphism of a Riemann surface, which fixes the homotopy class of a simple closed geodesic. Under certain conditions on the injectivity radius of the surface and bounds on the dilatation of the map, the automorphism induces a periodic element of the Teichmüller modular group. We may also estimate the order of the period.

Generalizing the notion of a numerically flat vector bundle over a Kähler manifold $M$, we define a numerically flat principal $G$-bundle over $M$, where $G$ is a semisimple complex algebraic group. It is proved that a principal $G$-bundle $E_G$ is numerically flat if and only if $\text{ad}(E_G)$ is numerically flat. Numerically flat bundles are also characterized using the notion of semistability.

Let $k$ denote the algebraic closure of the finite field, $\mathbb F_p,$ let $\mathcal O$ denote the Witt vectors of $k$ and let $K$ denote the fraction field of this ring. In the first part of this paper we construct an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional $k$-schemes and we apply this to semisimple groups. In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, $p^{-r}\mathcal O^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.

This paper contains several results relating $Q$ spaces in several real variables with their dyadic counterparts, which are analogues of theorems for BMO and for $Q$ spaces on the circle. In addition, it gives an atomic (or quasi-orthogonal) decomposition for these $Q$ spaces in terms of the same type of atoms used to decompose BMO.