Given a logarithmic $1$-form on the snc locus of log canonical surface pair $(X, D)$ over perfect field characteristic $p \ge 7$, we show that it extends with at worst poles to any resolution singularities. We also prove analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences tamely dlt pairs are established. gi...

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of...

These theories present the verified enumeration of tame plane graphs as defined by Thomas C. Hales in his revised proof of the Kepler Conjecture. Compared with his original proof, the notion of tameness has become simpler, there are many more tame graphs, but much of the earlier verification [1] carries over. For more details see http:// code.google.com/p/flyspeck/ and the forthcoming book “Den...

In 2003, Shestakov-Umirbaev solved Nagata’s conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the “generalized Shestakov-Umirbaev inequality”, which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we deduce that no tame automorphism of a polyn...

We show that if a closed irreducible three-manifold with hyperbolic fundamental group contains a surface subgroup satisfying a certain geometric regularity assumption, then the surface is either quasiconvex or a virtual fiber. In the latter case, the manifold is hyperbolic. The regularity condition ensures that we may find algebraic bounds on the surface group which are analogous to the diamete...

Let H be a Krull monoid with infinite cyclic class group G and let GP ⊂ G denote the set of classes containing prime divisors. We study under which conditions on GP some of the main finiteness properties of factorization theory—such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the exist...