Introduction

The main purpose of writing this book is to convey to the general mathematical audience the notion of a Zariski geometry with the whole spectrum of geometric ideas arising in model-theoretic context. The idea of a Zariski geometry is intrinsically linked with algebraic geometry, as are many other model-theoretic geometric ideas. However, there are also very strong links with combinatorial geometries, such as matroids (pre-geometries) and abstract incidence systems. Model theory developed a very general unifying point of view based on the model-theoretic geometric analysis of mathematical structures as diverse as compact complex manifolds and general algebraic varieties, differential fields, difference fields, algebraic groups, and others. In all of these, Zariski geometries have been detected and have proved crucial for the corresponding theory and applications. In more recent works, this author has established a robust connection to non-commutative algebraic geometry.

Model theory has always been interested in studying the relationship between a mathematical structure, such as the field of complex numbers (ℂ, +, ·), and its description in a formal language, such as the finitary language suggested by D. Hilbert: the first-order language. The best possible relationship is when a structure M is the unique, up-to-isomorphism model of the description Th(M): the theory of M. Unfortunately, for a first-order language, this is the case only when M is finite because in the first order, it is impossible to fix an infinite cardinality of (the universe of) M.