This course aims to provide graduate students with a solid foundation in the advanced mathematical methods widely used in modern physics research and to develop their ability to apply these techniques to problems encountered in quantum mechanics, electromagnetism, statistical physics, condensed matter physics, and particle physics. The course focuses on the theoretical foundations and practical applications of mathematical tools used in the formulation and solution of physical problems. It aims to equip students with the skills required to model complex physical systems, develop analytical and semi-analytical solution methods, interpret mathematical results in a physical context, and utilise advanced mathematical techniques effectively in independent scientific research.

Fundamental concepts of vector and tensor analysis, orthogonal coordinate systems, linear algebra and eigenvalue problems, complex variables and complex analysis, ordinary and partial differential equations, Fourier series and integral transforms, Laplace transforms, special functions (Legendre, Bessel, Hermite, and Laguerre functions), Green’s functions, Sturm–Liouville theory, calculus of variations, integral equations, and an introduction to group theory. Mathematical modeling of physical systems and applications of these methods to problems in classical mechanics, electromagnetism, quantum mechanics, statistical physics, and condensed matter physics.