Point-Counting and the Zilber-Pink Conjecture : Cambridge University Press : 9781009170321 : 1009170325 : 09 Jun 2022 : Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only