MATH487, Fall 2018: class plan

8.27

Introduction and course overview

Systems of linear equations.
Gauss elimination

8.29

Basic properties of matrices.
Regular matrices; \(LU\) decomposition of regular matrices.

8.31

Notion of (matrix) group. Examples: groups of lower, upper triangular matrices. Matrix inverses.
Row swaps and permutation matrices. Group of permutation matrices.
Non-singular matrices and row-echelon normal form.

9.5

Determinants: definition, existence and properties.
Matrix is singular iff its determinant vanishes.

9.7

Determinants and \(LU\) decomposition.
Example: tridiagonal matrices.
Kramer’s rule.

9.10

Remarkable determinants: VanderMonde, Cauchy.
Applications: Lagrange interpolation formula.

9.12

Minors, adjoint matrices.
Cauchy-Binet formula.

9.14

Matrix tree theorem.

9.17

General linear spaces.
Linear operators.

9.19

Bases. Bases changes.

9.21

Normal forms: mappings between different spaces.

9.24

Endomorphisms: classification problem.

9.26

Eigenvectors. Characteristic polynomial.

9.28

Jordan normal forms.

10.1

Cayley-Hamilton.

10.3

Functions of matrices.
Matrix exponentials.

10.5

Midterm 1

10.8

Euclidean and Hermitean spaces.

10.10

Self-adjoint operators and their spectra.

10.12

Orthogonalization. Orthogonal polynomials.

10.15

Rayleigh min-max theorem

10.17

Spectra interlacing.

10.19

Exterior forms.

10.22

Exterior differential forms. Line integrals

10.24

Integrals over higher-dimensional chains.
Exterior differentiation
Stokes and relatives.

10.26

Classical identities in 3D; scalar, vector products.
Div, grad, rot.

10. 29

Applications: winding numbers.

Complex analysis. Complex analytic functions.

10. 31

Analytic functions. Cauchy-Riemann.
Integrals along a curve.

11. 2

Laurent series. Domains of convergence.

11. 5

Integrals via residues.

11. 7

Midterm 2

11.9

Integrals involving multivalued functions.

11.12

Fresnel integral. Logarithmic residues. Rouche theorem.

11.14.

Elements of Fourier transform.
Parseval theorem.

11.16.

Inverse transform.
Duality (smoothness-localization). “Uncertainty principle”.

Week of 11.19

no classes

Week of 11.26

Elements of asymptotic analysis.
Laplace method. Steepest descent.
Tauber theorems.
Asymptotics for coefficients of generating functions; Darboux transfer.

Week of 12.10

Partial differential equations.
First order equations; examples (eikonal).
Cauchy problem. Quasilinear PDEs.

Week of 12. 3

Wave equation: d’Alembert solution.
Wave equation via Fourier series.
Elliptic problems.

Harmonic functions.
Dirichlet, Neumann.

Additional material

Rotation matrices in 3D spaces, SO(3).
Roll, pitch, yaw.
Commutators, Jacobi identity.Vector calculus.

Metric spaces: completeness, examples.
Contractions and their fixpoints.

Hyperbolic spaces.
Applications: search and navigation.

Applications: Ideal fluids.
Fractional-linear mappings.

 


Course recap.