Explore the foundational concepts of o-minimality and the Pila-Wilkie Theorem in this comprehensive graduate-level lecture. Delve into key definitions, references, and examples as Gareth Jones from the University of Manchester guides you through the intricacies of this mathematical field. Examine the Monotheistic Theorem, analyze observations and partitions, and engage with practical exercises to reinforce your understanding. Investigate the domain of o-minimality, distinguish between good and bad points, and visualize concepts through illustrations. Conclude with a thorough review of essential definitions, providing a solid foundation for further study in this advanced mathematical topic.