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This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups, are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra.
Table of contents:
Preface; 1. Introduction; 2. Finite covers; 3. Definable subgroups of algebraic groups over pseudo-finite fields; 4. Groups in pseudo-finite fields; 5. The groups of automorphisms of the field of complex numbers leaving fixed the algebraic numbers are simple; 6. The automorphism groups of the field of complex numbers are complete; 7. The algebra of an age; 8. Elimination of inverses in groups; 9. Model-theoretic properties of polycyclic-by-finite groups; 10. Non-standard free groups; 11. Finitely generated subgroups of the free Z(t)-group on two generators; 12. Rings of definable scalars and biendomorphism rings; 13. Recent results on simple first-order theories.
Surveys recent interactions between model theory and algebra, notably group theory. Topics include automorphism groups of algebraically closed fields, aspects of model theory of various classes of groups, and model theory of pseudo-finite fields and of modules. Contains the first comprehensive survey of finite covers.
Surveys recent interactions between model theory and other branches of mathematics, notably group theory.
Table of contents:
Preface; 1. Introduction; 2. Finite covers; 3. Definable subgroups of algebraic groups over pseudo-finite fields; 4. Groups in pseudo-finite fields; 5. The groups of automorphisms of the field of complex numbers leaving fixed the algebraic numbers are simple; 6. The automorphism groups of the field of complex numbers are complete; 7. The algebra of an age; 8. Elimination of inverses in groups; 9. Model-theoretic properties of polycyclic-by-finite groups; 10. Non-standard free groups; 11. Finitely generated subgroups of the free Z(t)-group on two generators; 12. Rings of definable scalars and biendomorphism rings; 13. Recent results on simple first-order theories.
Surveys recent interactions between model theory and algebra, notably group theory. Topics include automorphism groups of algebraically closed fields, aspects of model theory of various classes of groups, and model theory of pseudo-finite fields and of modules. Contains the first comprehensive survey of finite covers.
Surveys recent interactions between model theory and other branches of mathematics, notably group theory.