Ⅰ. Review of Fundamental Notions of Analysis A. Set Theory, Definitions B. Algebraic Structures, Definitions C. Topology D. Integration E. Key Theorems in Linear Functional Analysis Problems and Exercises Problem 1: Clifford algebra; Spin(4) Exercise 2: Product topology Problem 3: Strong and weak topologies in Lz Exercise 4: Holder spaces See Problem VI 4: Application to the Schrtdinger equationⅡ. Differential Calculus on Banach Spaces A. Foundations B. Calculus of Variations C. Implicit Function Theorem. Inverse Function Theorem D. Differential Equations Problems and Exercises Problem 1: Banach spaces, first variation, linearized equation Problem 2: Taylor expansion of the action; Jacobi fields; the Feynman-Green function; the Van Vleck matrix: conjugate points; caustics Problem 3: Euler-Lagrange equation: the small disturbance equation: the soap bubble problem: Jacobi fieldsⅢ. Differentiable Manifolds, Finite Dimensional Case A. Definitions B. Vector Fields; Tensor Fields C. Groups of Transformations D. Lie Groups Problems and Exercises Problem 1: Change of coordinates on a fiber bundle, configuration space, phase space Problem 2: Lie algebras of Lie groups Problem 3: The strain tensor Problem 4: Exponential map; Taylor expansion; adjoint map; left and right differentials; Haar measure Problem 5: The group manifolds of SO(3) and SU(2) Problem 6: The 2-sphereⅣ. Integration on Manifolds A. Exterior Differential Forms B. Integration C. Exterior Differential Systems Problems and Exercises Problem 1: Compound matrices Problem 2: Poincare lemma, Maxwell equations, wormholes Problem 3: Integral manifolds Problem 4: First order partial differential equations, Hamilton-Jacobi equations, lagrangian manifolds Problem 5: First order partial differential equations, catastrophes Problem 6: Darboux theorem Problem 7: Time dependent hamiltonians See Problem Ⅵ 11 paragraph c: Electromagnetic shock wavesⅤ. Riemannian Manifolds. Kahlerian Manifolds A. The Riemannian Structure B. Linear Connections C. Geodesics D. Almost Complex and Kahlerian Manifolds Problems and Exercises Problem I: Maxwell equation; gravitational radiation Problem 2: The Schwarzschild solution Problem 3: Geodetic motion; equation of geodetic deviation; exponentiation; conjugate points Problem 4: Causal structures; conformal spaces; Weyl tensorⅤbis. Connections on a Principal Fibre Bundle A. Connections on a Principal Fibre Bundle B. Holonomy C. Characteristic Classes and Invariant Curvature Integrals Problems and Exercises Problem 1: The geometry of gauge fields Problem 2: Charge quantization Monopoles Problem 3: Instanton solution of eucfidean SU(2) Yang-Mills theory (connection on a non-trivial SU(2) bundle over S4) Problem 4: Spin structure, spinors, spin connectionsⅥ. Distributions A. Test Functions B. Distributions C. Sobolev Spaces and Partial Differential Equations Problems and Exercises Problem 1: Bounded distributions Problem 2: Laplacian of a discontinuous function Exercise 3: Regularized functions Problem 4: Application to the Schrodinger equation Exercise 5: Convolution and linear continuous responses Problem 6: Fourier transforms of exp (-x2) and exp (ix2) Problem 7: Fourier transforms of Heaviside functions and Pv(I/x) Problem 8: Dirac bitensors Problem 9: Legendre condition Problem 10: Hyperbolic equations; characteristics Problem 11: Electromagnetic shock waves Problem 12: Elementary solution of the wave equation Problem 13: Elementary kernels of the harmonic oscillatorⅦ. Differentiable Manifolds, Infinite Dimensional Case A. Infinite-Dimensional Manifolds B. Theory of Degree; Leray-Schauder Theory C. Morse Theory D. Cylindrical Measures, Wiener Integral Problems and Exercises Problem A: The Klein-Gordon equation Problem B: Application of the Leray-Schauder theorem Problem C1: The Reeb theorem Problem C2: The method of stationary phase Problem D1: A metric on the space of paths with fixed end points Problem D2: Measures invariant under translation Problem D3: Cylindrical σ-field of C([a, b]) Problem D4: Generalized Wiener integral of a cylindrical functionReferencesSymbolsIndex
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