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.

Bonding in Condensed matter Physics: Forces and energy of interatomic bonding, Primary bonds: Covalent bonds, Ionic bonds, Metallic bonds etc. Secondary bonds: Van der Waals bonds, Hydrogen bonds etc.

Crystal structure: Bravais lattice, primitive vectors, primitive unit cell, conventional unit cell, Wigner-Seitz cell; Symmetry operations and classification of 2- and 3-dimensional Bravais lattices; Common crystal structures; Reciprocal lattice and Brillouin zone; Bragg-Laue formulation of X-ray diffraction by a crystal.

Preliminary concepts, probability theory, introduction to central limit theorem, random walk problem, quasi-static process,thermal and mechanical interactions, laws of thermodynamics, thermodynamic potentials. Statistical description of a system of particles,microstates, concept of ensembles, basic postulates, phase space, Liouville's theorem, microcanonical, canonical and grand canonical ensembles. Partition functions. Chemical potential,free energy and connection with thermodynamic variables. Equivalence of ensembles. Ideal gas,Gibbs paradox, M-B gas velocity distribution.

Probability Theory: Elementary concepts on probability – axiomatic definition of probability – conditional probability – Bayes’ theorem – random variables – standard discrete and continuous distributions – moments of random variables – moment generating functions – multivariate random variables – joint distributions of random variables – conditional and marginal distributions – conditional expectation – distributions of functions of random variables – t and χ2 distributions – Schwartz and Chebyshev inequalities – weak law of large numbers for finite variance case – central limit theorem for