Embark on a comprehensive journey through the foundations of differential geometry in this inaugural lecture by Professor Claudio Arezzo. Delve into the fundamental concepts that form the bedrock of this mathematical discipline, exploring the intricate relationships between geometry and calculus. Gain insights into the study of curves, surfaces, and manifolds, and discover how these concepts apply to various fields of mathematics and physics. Throughout this 90-minute session, engage with rigorous mathematical proofs, intuitive explanations, and illustrative examples that illuminate the core principles of differential geometry. Prepare to expand your mathematical horizons and develop a solid groundwork for further exploration in this captivating field.