Model theory of special subvarieties and Schanuel-type conjectures

We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure)

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

Center vortex model for the infrared sector of SU(4) Yang-Mills theory: String tensions and deconfinement transition

A random vortex world-surface model for the infrared sector of SU(4) Yang-Mills theory is constructed, focusing on the confinement properties and the behavior at the deconfinement phase transition. Although the corresponding data from lattice Yang-Mills theory can be reproduced, the model requires a more complex action and considerably more tuning than the SU(2) and SU(3) cases studied previously. Its predictive capabilities are accordingly reduced. This behavior has a definite physical origin, which is elucidated in detail in the present work. As the number of colors is raised in Yang-Mills theory, the corresponding infrared effective vortex description cannot indefinitely continue to rely on dynamics determined purely by vortex world-surface characteristics; additional color structures present on the vortices begin to play a role. As evidenced by the modeling effort reported here, definite signatures of this behavior appear in the case of four colors.Comment: 24 pages, 7 figures containing 8 ps file

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.Comment: 16 page

Pseudo-dualizing complexes and pseudo-derived categories

The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing "in between" the conventional derived category and the coderived or the contraderived category. The constructions of these triangulated categories use appropriate versions of the Auslander and Bass classes of modules. The constructions of derived functors providing the triangulated equivalence are based on a generalization of a technique developed in our previous paper arXiv:1503.05523.Comment: LaTeX 2e with pb-diagram, xy-pic, and tikz-cd, 60 pages, 14+3 commutative diagrams; v.4: sections 10-12 added, new subsections 0.8 and 0.10 inserted in the Introduction; v.5: subsection 0.2 shortened, basic definitions added in section 1, explanations added in the proof of Theorem 4.2, several references added; v.7: misprints corrected, references updated -- this is intended as the final versio