Introduction to geometric quantization

This thesis explores the interplay between symplectic geometry and geometric quantization, a mathematical framework that seeks to transition from classical mechanics to quantum mechanics. Starting with the foundations of smooth manifolds, tensors, and symplectic structures, the geometric prerequisites for quantization are established, to be able to then move on to introduce the process of quantization, highlighting its challenges, focusing on the possibility of having different quantizations. As a preliminary step, the concept of prequantization is explored, followed by the optimization provided by polarization. By making use of examples and counterexamples, we are able to analyze the limits of quantization methods and to demonstrate the urge for more general structures.