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 o...

We consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.

In response to a question of Y.-T. Siu, we show that for any algebraic variety V of codimension 2 in C, there is a neighborhood U of V and an injective holomorphic map Φ : C → C \ U . That is, there is a Fatou-Bieberbach domain (a proper subdomain in C biholomorphic to C) in the complement of some neighborhood of V . In particular, Φ is a dominating map. In case n = 1, V is empty, so the result...

A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that ηi : Xi → Xi is a continuous proper map on a locally compact Hausdorff space Xi, for i = 1, 2. We show that the dynamical systems (X1, η1) and (X2, η2) are conjugate if and only if some topological conjugacy algebra of (X1, η1) is is...

In response to a question of Y.-T. Siu, we show that for any algebraic variety V of codimension 2 in C, there is a neighborhood U of V and an injective holomorphic map Φ : C → C \ U . That is, there is a Fatou-Bieberbach domain (a proper subdomain in C biholomorphic to C) in the complement of some neighborhood of V . In particular, Φ is a dominating map. In case n = 1, V is empty, so the result...

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in [16], [8], [6], [7]. We then apply the equation to show that the generalized Chen’s conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-p...

We have identified a MAP kinase kinase (DdMEK1) that is required for proper aggregation in Dictyostelium. Null mutations produce extremely small aggregate sizes, resulting in the formation of slugs and terminal fruiting bodies that are significantly smaller than those of wild-type cells. Time-lapse video microscopy and in vitro assays indicate that the cells are able to produce cAMP waves that ...

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension. In this paper we are interested in the following question. Question. Classify all smooth affine surfaces X/C which admit a proper morphism f : X → X with degree f > 1. In [5] and [18], a classification of smooth projective surfaces with a self-map of degree > 1 has been given. Th...

Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufcient conditions for the proper part of P being homotopy equivalent to the suspension of the proper part of Q. An application of this lemma is a uniied proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result by ZIEGLER) and single step inclusion order (whi...