Explore the fundamental concepts and initial examples of positive geometries in this comprehensive lecture from Harvard CMSA's Summer School in Total Positivity and Quantum Field Theory. Delve into the mathematical foundations that connect algebraic geometry with quantum field theory applications, examining how positive geometries provide a geometric framework for understanding scattering amplitudes and related physical phenomena. Learn through detailed mathematical exposition and concrete examples that illustrate the key definitions and properties of these geometric structures. Gain insights into how positive geometries serve as a bridge between pure mathematics and theoretical physics, with particular emphasis on their role in modern approaches to quantum field theory calculations.