class plan

  • 1.17 Introduction and course overview. Complex numbers. Interpretation as a commutative algebra (of matrices). Algebraic closeness.
  • 1.19 Linear spaces. Bases. Exchanges. Subspaces, factorspaces. Finite dimensional invariant subspaces and differential equations.
  • 1.24 Linear operators. Matrices. Base changes.
  • 1.26 Determinants; definitions and properties. Remarkable determinants. Cauchy-Binet formula.
  • 2.2 Matrix polynomials; matrix functions. Matrix/operator exponentials; Rotations as exponentials of skew-symmetric matrices. SO(3).
  • 2.7 Normal forms. Mappings between different spaces.
  • 2.9 Self-mappings, Jordan normal form.
  • 2.14 Quadratic forms. Mappings to conjugate spaces. Normal forms.
  • 2.16 Applications: Statistics; variance, correlations. Orthogonality, orthogonalization.
  • 2.23 midterm 1
  • 2.28 Hermitian spaces. Self-adjoint operators and their spectra. normal operators
  • 3.2 Applications: electric networks, Applications: random walks.
  • 3.7 Finite groups of symmetries.
  • 3.9 Representations. Schur lemma. Representations of finite abelian groups.
  • 3.14 Vector calculus. Classical identities in 3D; scalar, vector products.
  • 3.16 Line integrals and generalizations. Exterior forms. Differential forms. (See Arnold, Ch. 7)
  • 3.21,23 no class (spring break)
  • 3.28 Stokes and relatives.
  • 3.30 Div, grad, rot.
  • 4.4 Complex analysis. Analytic functions. Cauchy-Riemann.
  • 4.6 Laurent series. Integrals via residues.
  • 4.11 Applications: logarithmic residues, Rouche theorem. Ideal fluids. Fractional-linear mappings and hyperbolic plane.
  • 4.13 Elements of Fourier transform. Parseval theorem. Inverse transform, duality.
  • 4.18 Elements of asymptotic analysis. Laplace method. Steepest descent.
  • 4.20 Tauber theorems. Asymptotics for coefficients of generating functions; Darboux transfer.
  • 4.21 midterm 2 (take home).
  • 4.25 Partial differential equations. First order equations; examples (eikonal).
  • 4.27 Cauchy problem. Quasilinear PDEs. Wave equation: d’Alembert solution.
  • 5.2 Wave equation via Fourier series. Dirichlet problem and Poisson kernel.