# 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 andÂ $$LU$$ decomposition.
Example: tridiagonal matrices.
Kramer’s rule.

#### 9.10

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

#### 9.12

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.

Cayley-Hamilton.

#### 10.3

Functions of matrices.
Matrix exponentials.

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.

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.

#### 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.

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”.

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: dâ€™Alembert solution.
Wave equation via Fourier series.
Elliptic problems.

Harmonic functions.
Dirichlet, Neumann.

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.