Contents › Part 2

Part 2 — Mathematical Methods

1. Logic, Sets, and Maps
   • Logic
   • Sets
   • Maps
   • Equivalence classes
   • The infinite
   • The crisis of foundations
   • The analytical perspective
   • Formal systems and incompleteness
   • Computability and decidability
   • Can mathematics be built without contradictions?
2. Algebraic Structures
   • Binary operations and algebraic structures
   • Simple algebraic structures
   • Composite algebraic structures
   • Group theory: further definitions
   • Basic groups
   • Functions on groups
   • The symmetric group
   • Decompositions of a group
   • Direct products
3. Linear Algebra and Representation Theory
   • Preliminary formulas from elementary algebra
   • Vector spaces
   • Inner product, norm, and metric
   • Linear transformations
   • The dual space
   • Eigenvectors, eigenvalues, and decompositions
   • Algebras
   • Representations of groups
   • Representations of finite groups
   • Properties of irreducible representations
   • Representations of algebras
4. Topological and Metric Spaces
   • Topological spaces
   • Metric spaces
   • The topology of $\R^{N}$
5. Real Analysis
   • The real numbers
   • Sequences
   • Real functions
   • Limits
   • Continuity
   • The derivative
   • The Riemann integral
   • The fundamental theorem of calculus
   • Series
   • Multivariable calculus
   • Vector analysis
   • Sets of measure zero and Hausdorff dimension
   • Functional calculus
   • The Dirac delta
6. Complex Analysis
   • The complex field
   • Holomorphic functions
   • Contour integrals
   • Cauchy's integral theorem
   • The Cauchy integral formula
   • Laurent series and singularities
   • The residue theorem
7. Ordinary Differential Equations and Sturm–Liouville Theory
   • Ordinary differential equations
   • Linear systems with constant coefficients
   • Linear systems with periodic coefficients
   • Orthonormal systems of functions
   • The Sturm–Liouville operator
   • The semiclassical approximation
   • The one-dimensional Helmholtz equation
   • The Bessel equation
   • The Legendre equation and spherical harmonics
   • The Hermite equation
   • The Laguerre equation
   • Elliptic integrals and the Jacobi elliptic functions
   • The Laplace transform
   • Summary of the classical families
8. Partial Differential Equations
   • Definitions
   • From partial to ordinary differential equations
   • Green's functions
   • The canonical equations and their fundamental solutions
   • Maximum principles
   • Weak solutions
9. Probability and Statistics
   • Discrete statistics
   • Continuous statistics
   • Characteristic functions and the central limit theorem
   • Estimation
   • Hypothesis testing
   • The look-elsewhere effect
10. Hilbert Spaces
    • Definitions
    • Orthonormal bases and separability
    • Bounded operators and duality
    • Self-adjointness and the spectral theorem
    • Unbounded operators and domains
    • Direct sums, tensor products and rigged spaces
    • The canonical commutation relations in Weyl form
11. Differentiable Manifolds, Tensors, and Curvature
    • Index conventions and the symmetry ladder
    • Tensors under orthogonal transformations
    • The theory of curves and surfaces in $\R^{3}$
    • Differentiable manifolds
    • Tensor analysis on manifolds
    • Differential forms: $k$-forms and $k$-vectors
    • The metric tensor
    • Vielbein and local frames
    • Flows, the Lie derivative, and Killing vectors
    • Connections, parallel transport, and torsion
    • Curvature
    • The Cartan formalism
    • Maximally symmetric spaces and the symmetry ladder
12. Lie Groups, Lie Algebras, and Fibre Bundles
    • Lie groups
    • Representations of Lie groups
    • The Lie algebra associated with a Lie group
    • Central extensions
    • Contraction of Lie algebras
    • Fibre bundles
    • Invariant polynomials
    • The Chern–Weil theorem and transgression forms
13. Expansions of Lie Algebras
    • The tensor product construction
    • Ideals, the radical, and the Killing form
    • The Takiff tower
    • Gradings and resonant subalgebras
    • Contraction as a degeneration of $A$
    • Central charges as the top level of an expansion
    • The Bacry–Lévy-Leblond cube as a lattice of algebras
    • Invariant forms and why the extension is needed
    • Relation to the published expansion methods
    • The construction in $3{+}1$ dimensions
    • Positive functionals and the Gram form
14. Calculus of Variations
    • The classical problems
    • Functionals and the first variation
    • Constrained variation
    • The second variation
    • Direct methods and existence
    • Symmetry and conservation: Noether's theorems
    • The variational principles of physics
15. Fourier Analysis and Integral Transforms
    • Fourier series
    • The Fourier transform
    • Convolution and correlation
    • The Laplace transform
    • Sampling and the discrete transform
    • Other integral transforms