### 1. Introduction to ultracold atoms

Bose-Einstein condensation.

Optical lattices.

Chosen experiments.

### 2. Quantum dynamics in ultracold atoms

Collapse and revival of a
wave function.

Time-resolved control of super-exchange interaction.

### 3. Dynamics after a sudden change of parameters: quantum quenches

Experiments in
ultracold gases.

Short time dynamics.

Long time dynamics.

### 4. Numerical methods for quantum dynamics I: 'Static' density matrix renormalization group method (DMRG)

Schmidt decomposition.

Algorithm.

Convergence analysis.

### 5. Numerical methods for quantum dynamics II

Numerical methods for time-evolution.

Trotter-Suzuki algorithm.

Runge-Kutta algorithm, and more specialized algorithms.

### 6. Numerical methods for quantum dynamics III: Generalization of the DMRG to time-dependent (finite temperature) situations

Adaptation of the effective Hilbert space.

Algorithm.

Convergence analysis.