Module #1 Introduction to Mathematical Physics Overview of the importance of mathematical methods in theoretical physics, historical context, and course objectives
Module #2 Vector Calculus Review of vector algebra, vector differentiation, and integration, with emphasis on physical applications
Module #3 Differential Equations Linear and nonlinear ODEs and PDEs, with techniques for solving and interpreting physical systems
Module #4 Linear Algebra Vector spaces, linear transformations, eigenvalues, and eigenvectors, with applications to quantum mechanics
Module #5 Group Theory Introduction to abstract groups, group representations, and their role in particle physics
Module #6 Tensor Analysis Introduction to tensors, tensor algebra, and covariance, with applications to general relativity
Module #7 Complex Analysis Complex functions, Cauchy-Riemann equations, and residue theory, with applications to quantum field theory
Module #8 Special Functions Gamma functions, Legendre functions, and other special functions used in theoretical physics
Module #9 Fourier Analysis Fourier series and transforms, with applications to signal processing and quantum mechanics
Module #10 Laplace Transforms Definition and properties of Laplace transforms, with applications to differential equations and circuit analysis
Module #11 Operators and Eigenvalue Problems Linear operators, eigenvalues, and eigenvectors, with applications to quantum mechanics and electromagnetism
Module #12 Greens Functions Definition and properties of Greens functions, with applications to potential theory and scattering problems
Module #13 Path Integrals Introduction to path integrals, Feynman diagrams, and their role in quantum field theory
Module #14 Riemann Surfaces and Topology Introduction to Riemann surfaces, genus, and topological invariants, with applications to string theory
Module #15 Symmetries and Conservation Laws Noethers theorem, conservation laws, and symmetries in classical and quantum mechanics
Module #16 Lie Groups and Lie Algebras Introduction to Lie groups and Lie algebras, with applications to particle physics and quantum field theory
Module #17 Functional Analysis Introduction to Hilbert spaces, Banach spaces, and operator theory, with applications to quantum mechanics
Module #18 Numerical Methods Introduction to numerical methods for solving partial differential equations and eigenvalue problems
Module #19 Renormalization and Regularization Introduction to renormalization and regularization techniques in quantum field theory
Module #20 Computational Physics Introduction to computational physics, including numerical methods, programming languages, and scientific computing
Module #21 Applications to Quantum Mechanics Applications of mathematical methods to quantum mechanics, including Schrödinger equation and spin systems
Module #22 Applications to Quantum Field Theory Applications of mathematical methods to quantum field theory, including Feynman diagrams and renormalization group
Module #23 Applications to General Relativity Applications of mathematical methods to general relativity, including curvature tensors and geodesics
Module #24 Applications to Condensed Matter Physics Applications of mathematical methods to condensed matter physics, including topological insulators and superconductors
Module #25 Course Wrap-Up & Conclusion Planning next steps in Mathematical Methods in Theoretical Physics career
