The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

External definability and groups in NIP theories

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{ext}, of a model M. In the light of these results we continue the study of the "definable topological dynamics" of groups in NIP theories. In particular we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were corrected. Journal of the London Mathematical Society, accepte

On function field Mordell-Lang and Manin-Mumford

We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reorganized.Comment: 21 page