Harmonic maps and their geometric structure: applications to quantum mechanics, general relativity, and Yang-Mills theory

Motivated by recent developments in theoretical physics, this thesis presents a self-contained analysis of harmonic maps and explores their modern applications in quantum mechanics, general relativity, and Yang-Mills theory. The aim is to introduce the geometric and analytical foundations necessary to understand the structure of harmonic maps and to provide a physical motivation for their study. In particular, the combination of Riemannian geometry and the calculus of variations offers a rigorous mathematical framework to model gravitational nonlinearities and to couple Schrödinger-type equations with an extended harmonic map theory. The project begins by laying out the necessary geometric background and analytical tools, then moves on to a broader exploration of harmonic maps and their key properties. Finally, it highlights applications in theoretical physics, along with current challenges and potential directions for future research.