Mathematical Physics

  1. Advanced Topics in Mathematical Physics
    1. Statistical Mechanics
      1. Measure Theory and Probability
        1. Probability Spaces
          1. Random Variables
            1. Expectation and Variance
            2. Ergodic Theory
              1. Measure-Preserving Transformations
                1. Ergodic Hypothesis
                  1. Mixing Properties
                  2. Thermodynamic Formalism
                    1. Gibbs Measures
                      1. Partition Functions
                        1. Phase Transitions
                        2. Lattice Models
                          1. Ising Model
                            1. Potts Model
                              1. Exact Solutions and Approximations
                              2. Critical Phenomena
                                1. Scaling Theory
                                  1. Renormalization Group
                                    1. Critical Exponents
                                  2. Topology in Physics
                                    1. Algebraic Topology
                                      1. Fundamental Group
                                        1. Homology Groups
                                          1. Cohomology Groups
                                          2. Homotopy Theory
                                            1. Homotopy Groups
                                              1. Fiber Bundles
                                                1. Characteristic Classes
                                                2. Topological Defects
                                                  1. Solitons
                                                    1. Vortices
                                                      1. Monopoles
                                                      2. Topological Phases of Matter
                                                        1. Topological Insulators
                                                          1. Quantum Hall Effect
                                                            1. Chern Numbers
                                                            2. Topological Quantum Field Theory
                                                              1. Axioms of TQFT
                                                                1. Examples and Applications
                                                              2. Geometry in Modern Physics
                                                                1. Fiber Bundles
                                                                  1. Principal Bundles
                                                                    1. Associated Bundles
                                                                      1. Connection Forms
                                                                      2. Gauge Theory
                                                                        1. Yang-Mills Theory
                                                                          1. Gauge Transformations
                                                                            1. Curvature and Field Strength
                                                                            2. Characteristic Classes
                                                                              1. Chern Classes
                                                                                1. Pontryagin Classes
                                                                                  1. Applications to Physics
                                                                                  2. Index Theory
                                                                                    1. Atiyah-Singer Index Theorem
                                                                                      1. Applications to Quantum Field Theory
                                                                                    2. String Theory Mathematics
                                                                                      1. Conformal Field Theory
                                                                                        1. Conformal Transformations
                                                                                          1. Virasoro Algebra
                                                                                            1. Primary Fields
                                                                                            2. Complex Geometry
                                                                                              1. Kähler Manifolds
                                                                                                1. Calabi-Yau Manifolds
                                                                                                  1. Mirror Symmetry
                                                                                                  2. Supersymmetry
                                                                                                    1. Superspace
                                                                                                      1. Superfields
                                                                                                        1. Supersymmetric Quantum Mechanics
                                                                                                        2. Modular Forms
                                                                                                          1. Elliptic Functions
                                                                                                            1. Modular Transformations
                                                                                                              1. Applications to String Theory

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                                                                                                          6. Functional Analysis and Advanced Quantum Theory

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                                                                                                          1. Foundations in Mathematical Methods