Review of Ordinary Differential Equations: analytical methods, stability – Fourier series, orthogonal functions, Fourier integrals, Fourier transform – Partial Differential Equations: first-order PDEs, method of characteristics, linear advection equation, Burgers’ equation, shock formation, Rankine-Hogoniot jump condition; classification, canonical forms; Laplace equation, min-max principle, cylindrical coordinates; heat equation, method of separation of variables, similarity transformation method; wave equation, d’Alembert solution – Calculus of Variations: standard variational problems, E

Flows on the line: Introduction; Fixed points and stability; Population growth; Linear Stability Analysis; Existence and Uniqueness; Impossibility of oscillations; Potentials

Bifurcations: Saddle-node bifurcation; Transcritical bifurcation; Laser threshold; Pitchfork bifurcation; Overdamped bead on a rotating hoop; Imperfect bifurcations and catastrophes; Insect outbreak

Flows on a circle: Examples and Definitions; Uniform Oscillator; Nonuniform Oscillator; Overdamped Pendulum; Fireflies; Superconducting Josephson junctions

Quantum bits and quantum gates: quantum bits, basic computations with 1-qubit quantum gates, Pauli matrices or I, X, Y, Z-gates, Hadamard matrix gate or H-gate, quantum gates with multiple qubit inputs and outputs, quantum circuits, non cloning theorem.

Quantum measurements: quantum measurements and types, quantum measurements in the orthonormal basis,  Projective  or  von-Neumann  measurements,   POVM   measurements,   quantum   measurements on joint states.

Sky coordinates and motions: Earth Rotation – Sky coordinates – seasons – phases of the Moon – the Moon's orbit and eclipses – timekeeping (sidereal vs synodic period).

Planetary motions – Kepler's Laws – Gravity. Light & Energy – Telescopes – Optics – Detectors.

Planets: Formation of Solar System – planet types – planet atmospheres – extrasolar planets.

Stars: Measuring stellar characteristics (temperature, distance, luminosity, mass, size) – HR diagram – stellar structure (equilibrium, nuclear reactions, energy transport) – stellar evolution.