Second quantization: Fock space representation, creation and annihilation operators for bosons and fermions, representation of many-body operators. Green's functions at zero temperature: Interaction representation, Wick's theorem, Feynman diagrams. Finite temperatures: Matsubara functions, retarded and advanced Green's functions. Linear response, Kubo formula. Interacting fermions: Fermi liquid theory, Hubbard model, Heisenberg model. Electron-Phonon interaction, BCS theory of superconductivity

Errors and uncertainties in computations: Types of errors, error in functions, errors in algorithms. Matrix computing and scientific libraries.

Zero-finding and matching: Newton's rule for finding roots. Quantum eigenvalues, particle in a box. Fields due to moving charges.

Integration: Trapezoid rule, Simpson's rule, Gaussian quadrature, multi-dimensional integrals. Monte-Carlo integrations.

Phase Transitions and Critical Phenomena
General Introduction: Origin of phase transition, thermodynamic instabilites, Maxwell construction. Classification of phase transitions: first order and second order. Phase transitions in different systems (e.g. liquid-gas and paramagnet-ferromagnetic transition), order parameter, critical exponents, concept of long-range order.

Lattice models: Ising Model, exact solution in one dimension, high-temperature and low-temperature expansions. Phase transitions in X-Y and Heisenberg Models.