Model theory, a major branch of mathematical logic, plays a key role connecting logic and other areas of mathematics such as algebra, geometry, analysis, and combinatorics. Simplicity theory, a subject of model theory, studies a class of mathematical structures, called simple. The class includes all stable structures (vector spaces, modules, algebraically closed fields, differentially closed fields, and so on), and also important unstable structures such as the random graph, smoothly approximated structures, pseudo-finite fields, ACFA and more. Simplicity theory supplies the uniform model theoretic points of views to such structures in addition to their own mathematical analyses. This book starts with an introduction to the fundamental notions of dividing and forking, and covers up to the hyperdefinable group configuration theorem for simple theories. It collects up-to-date knowledge on simplicity theory and it will be useful to logicians, mathematicians and graduate students working on model theory.
Byunghan Kim is Professor of Mathematics at Yonsei University, Seoul, Korea. Logic and Foundations sectional invited speaker at the International Congress of Mathematicians 2014, Seoul.
Introduction ; 1. Dividing, Forking, and Simplicity ; 2. Lascar Strong Types and Type-Amalgamation ; 3. Hyperimaginaries and canonical bases ; 4. Elimination of hyperimaginaries ; 5. Constructing simple structures ; 6. Groups ; 7. A geometry of forking ; 8. Generalized amalgamation and the group