Volume 2005, Issue 586

A hypothesis is introduced under which a compact complex analytic space, X , viewed as a structure in the language of analytic sets, is essentially saturated. It is shown that this condition is met exactly when the irreducible components of the restricted Douady spaces of all the cartesian powers of X   are compact. Some implications of saturation on Kähler-type spaces, which by a theorem of Fujiki meet the above condition, are discussed. In particular, one obatins a model-theoretic proof of the fact that relative algebraic reductions exist in the class of Kähler-type spaces.

In this paper we solve the inverse problem in differential Galois theory over K  ( t  ) where K   is an algebraically closed field of characteristic zero.

Let  be a smooth algebraic group, defined over an algebraically closed field k of characteristic p ≧ 3. We determine the structure of the representation-finite and tame blocks of the algebras of distributions associated to the Frobenius kernels of .

The classical mean curvature flow of hypersurfaces with boundary satisfying a Neumann condition on an arbitrary, fixed, smooth hypersurface in Euclidean space is examined. In particular, the problem of singularity formation on the free-boundary and the classification of the limiting behaviour thereof is focused on. A monotonicity formula is developed and used to show that any smooth blow up centred about a boundary point is self-similar, with smoothness of the blow up being shown to necessarily follow in the case of Type I singularities. This leads to a classification of boundary singularities for mean convex evolving hypersurfaces.

We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G | P . In the case of SL flags, we find all relations between our generators, showing that they essentially come from . In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.