Algebra I
Vector spaces — bases and dimension — linear transformations — the matrix of a linear transformation — linear operators and eigen values — the characteristic polynomial — orthogonal matrices and rotations — diagonalization.
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Analysis I
Axioms of the real number system without construction, applications of the least-upper-bound- property, Archimedean principle, existence of n^{th} roots of positive real numbers, a^{x} for a > 0 and x > 0.
Convergence of sequences, monotonic sequences, subsequences, Heine-Borel theorem, lim sup and lim inf Cauchy sequences, completeness of R. Infinite series, absolute convergence, comparison test, root test, ratio test, conditional convergence, complex numbers, power series, radius of convergence of power series.
Continuous functions on intervals of R, intermediate value theorem, boundedness of continuous functions on closed and bounded intervals.
Differentiation, mean value theorem, Taylor's theorem, application of Taylor's theorem to maxima and minima, L'Hôpital rules to calculate limits.
Construction of e^{z} using power series, proof of the periodicity of sin and cos.
Riemann Integration: Riemann integrals, Riemann integrablity of continuous functions, fundamental theorem of calculus.
Recommended Texts (Analysis I, II, III and Calculus)
Introduction to Programming
The course will be based on the programming language Haskell.
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English
Poetry: Selections from Indian and English poets.
Prose: Three Short Stories and one Novel.
Drama:
Interactive Communication:
Self-Expression
Effective Language
Algebra II
The group of motions of the plane — finite and discrete groups of motions — abstract symmetry — group operations on cosets — permutation representations — the operations of a group on itself — class equation of icosahedral group — the Sylow theorems — free groups — generators and relations. Bilinear forms — symmetric and hermitian forms — spectral theorem — conics and quadrics — skew-symmetric forms.
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Analysis II
Topology of R^{n} : Euclidean, l_{1} and l_{∞} norms on R^{n} and the equivalence of convergence of sequences in R^{n}, open and closed sets, sequential compactness, continuous functions defined on subsets of R^{n}, boundedness of continuous functions defined on compact subsets. Characterisation of open sets in R^{n}.
Differentiation : Review of inner product spaces and linear maps on inner product spaces, Derivative as a linear map, Chain rule, Matrix representation and partial derivatives, Comparison of real and complex derivatives, sufficient condition for differentiablity, equality of mixed partial derivatives. Taylor's formula and its application to maxima and minima, Inverse function theorem, Implicit function theorem, Tangent space of level sets and gradient, Lagrange multiplier method, diagonalization of symmetric operators.
Recommended Texts (Analysis I, II, III and Calculus)
Advanced Programming
Imperative programming using Python and C.
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Algebra III
Rings, ideals, homomorphisms, quotient rings, fraction fields, maximal ideals, factorization, UFD, PID, Gauss Lemma, fields, field extensions, finite fields, function fields, algebraically closed fields.
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Analysis III
Construction of the real number system: assuming N and induction, construct Z, Q and the Cauchy construction of R.
Metric spaces: examples of metric spaces, convergence of sequences in metric spaces.
Continuous functions , Open sets, closed sets, Connected sets, Completeness, Completions, Various formulations of Compactness, Consequences of compactness, Baire category theorem and some applications. Uniform convergence, stability of Uniform convergence i.e. stability under continuity etc., Dini's theorem.
Stone-Weierstrass theorem: A brief introduction to convolution of compactly supported functions, approximate identity, Weierstrass theorem, Fourier transform for 2π-periodic functions, injectiveness of the Fourier transform, Riemann-Lebesgue lemma, Contraction mapping theorem, Arzela-Ascoli theorem and their applications to ODEs.
Recommended Texts (Analysis I, II, III and Calculus)
Calculus
Improper integrals, Multiple integrals, Fubini's theorem atleast for continuous functions, Change of co-ordinates formula, several examples (Polar and spherical co-ordinates etc.) Line integrals, gradients, path-independence of line integrals; Green's Theorem in the plane. Integrals as iterated integrals and primitive mappings; partitions of unity; change of variables; differential forms; simplexes and chains; Stokes' Theorem; closed and exact forms; vector analysis.
Recommended Texts (Analysis I, II, III and Calculus)
Algebra IV
Modules, generators and relations, structure theorem for Abelian groups/modules of Euclidean domains/PIDs. Applications to linear operators. Galois theory: separable and normal field extensions, fundamental theorem of Galois theory.
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Complex Analysis
Complex numbers and geometric representation — analytic functions — power series — exponential and logarithmic functions — conformality — Mobius transformations — complex integration — Cauchy's theorem — Cauchy's integral formula — singularities — Taylor's theorem — The maximum principle — The residue theorem and applications.
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Topology
Point set topology — connectedness — compactness — Tychonoff's theorem — Stone Czech compactification — covering spaces — fundamental groups
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Probability Theory
discrete probability — simple random variables — the law of large numbers — Binomial, Poisson and normal distributions — central limit theorem — random variables — expectations and moments — Markov chains.
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Classical Mechanics I
Space and Time — Newton's Laws — Conservation Laws — Harmonic, Damped, Forced, and Kicked Oscillators — Rocket Motion — Collision Problems — Projectiles — Central Forces — Inverse Square Law — Rutherford Scattering — Centrifugal and Coriolis Forces — Potential Theory.
Principle of Least Action — Constraints and Generalised Coordinates — La- grange's Equations — Noether's Theorem and Symmetries — Applications — Hamilton's Equations — Small Oscillations — Stability — Normal Modes.
Lorentz Transformations — Space-Time Diagrams — Length Contraction, Time Dilation — Kinematics and Dynamics of a Particle — Composition of Velocities - Proper Time — Equations of Motion in Absolute Form and Relative Form.
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Classical Mechanics II
Review of Hamilton's theory — Liouville's theorem — Poincare Recurrence Theorem — Poisson's Brackets — Canonical Transformations — Action-Angle Variables — Adiabatic Invariants — Hamilton-Jacobi Theory.
Phase Space and Phase Portraits — First and Second Order Systems — Predator-Prey Problems — Limit Cycles — Sensitivity to Initial Conditions and Predictability — Integrability — Some Hamiltonian Systems which Exhibit Chaos — Near Integrable Systems.
General Mathematical Formulation of Kinematics and Dynamics of Continuum Systems — Eulerian and Lagrangian Descriptions — Rigid Body Dynamics: Angular Velocity — The Inertia Tensor — Angular Momentum — The Equations of Motion — Eulerian Angles — Euler's Equations — Elasticity: The Strain Tensor — The Stress Tensor — Hooke's Law — Homogeneous and Temperature-dependent Deformations — Elastic Waves — Thermal Conduction and Viscosity — Fluid Dynamics: Conservation Laws — Ideal Fluids — Viscous Fluids — Basics of Turbulence — Thermal Conduction and Diffusion in Fluids.
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Electrodynamics
Gradient, Divergence, Curl — Theorems of Gauss, Green, and Stokes. Electrostatics: Charges, Fields, Potentials, Capacitance — Magnetostatics: Currents, Fields, Potentials, Inductance — Electromagnetic Induction: Faraday's Law — Displacement Current.
Currents and Conductors: Uniqueness Theorems — Method of Images — Ohms' Law — Microscopic Theory of Conduction — Hall Effect.
Electric and Magnetic Fields in Matter: Polarization — Displacement — Magnetization — Boundary Conditions at a Surface of Discountinuity.
Conservation Laws: Conservation of of Energy — Poynting's Theorem — Con- servation of Momentum and Angular Momentum — Maxwell's Stress Tensor.
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Optics
Waves: Plane waves — Spherical Waves — Harmonic Waves — Phase Velocity — Wavepackets — Group Velocity — Plane Electromagnetic Waves: Linear, Circular, and Elliptic Polarizations — Stokes Parameters, Polarisers.
Eikonal Approximation — Ray and Matrix Optics — Fermat's Principle — Optical Imaging — Aberrations: Chromatic, Spherical, Coma, Astigmatism, Distortion — Optical Instruments.
Wave Optics: Reflection and Refraction — Interference and Interferometers — Multiple-beam Interference — Coherent and Incoherent Light — Elementary Theory of Diffraction: Kirchoff theory — Fraunhofer and Fresnel Diffraction — Elementary Dispersion Theory — Elementary Scattering Theory.
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Thermal Physics
The laws of Thermodynamics — Thermodynamic Potentials — Applications of Thermodynamics — Equation of State — Description of Phase Transitions — Surface Effects in Condensation — Van der Waals Equation of State — Osmotic Pressure.
Probability — General Definitions — One Random Variable — Some Important Probability Distributions — Many Random Variables.
Binary Collisions — Boltzmann Transport Equation — Boltzmann's H Theorem — Maxwell-Boltzmann Distribution — Most Probable Distribution — Trans- port Phenomena — Mean Free Path — Conservation Laws — The Zeroth and First Order Approximations — Viscosity — The Navier-Stokes Equation, Examples in Hydrodynamics.
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Statistical Mechanics
General Definitions — The Microcanonical Ensemble — The Ideal Gas — Mixing Entropy and the Gibbs Paradox — The Canonical Ensemble — Examples — The Grand Canonical Ensemble — The Equivalence of the Canonical and the Grand Canonical Ensemble — Interacting Particles — The Cumulant Expansion — The Cluster Expansion — Critical Point Behaviour.
The Postulates of Quantum Statistical Mechanics — Density Matrix — Ensembles in Quantum Statistical Mechanics — Ideal Gases: The Ideal Fermi Gas — The Ideal Bose Gas — Applications — Statistical Mechanical Theory of Phase Transitions: Ising Model.
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Quantum Mechanics I
Experimental Background — The Old Quantum Theory — Uncertainty and Complementarity — Discussion of Measurement — The Schrodinger and Heisen- berg Pictures and Equivalence — Development of the Wave Equation — Interpretation of the Wave Function — Wave Packets in Space and Time — Eigenfunctions and Eigenvalues — Energy and Momentum Eigenfunctions — Expectation Values — Two-level System — One-dimensional Square Well and Barrier Potential — Linear Harmonic Oscillator — The Hydrogen Atom — Collisions in Three Dimensions — Scattering by a Coulomb Field.
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Quantum Mechanics II
Matrix Formulation of Quantum Mechanics: Bra and Ket Formulation — Transformation Theory — Equations of Motion — Symmetry in Quantum Mechanics — Space and Time Displacements — Rotation — Angular Momentum and Unitary Groups — Combination of Angular Momentum States and Tensor Operators — Space Inversion and Time Reversal — Dynamical Symmetry.
Approximation Methods for Bound States: Stationary Perturbation Theory — Variational Method — Dalgarno-Lewis Method – WKB Approximation — Time-dependent Perturbation Theory.
Approximation Methods in Collision Theory: The Scattering Matrix — Sta- tionary Collision Theory — Born Approximation — Distorted Wave Born Approximation — Partial Wave Analysis.
Identical Particles and Spin: Stern-Gerlach Experiment — Pauli Matrices — Boson and Fermion Wavefunctions — Density Operator and Density Matrix.
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Optional Courses
Classical Field Theory
Special Theory of Relativity: Experimental Basis — Lorentz Transformations — Basic Kinematic Results — Addition of Velocities — 4-Velocity — Relativistic Momentum and Energy of a Particle — Covariant formulation of Electrodynamics — Transformation of Electromagnetic Fields — Solution of the Wave Equation Covariant Form.
Classical Radiation Theory: Radiation by Moving Charges — Lienard-Wiechert Potentials — Radiation from Relativistic Acceleration — Larmor Formula as Non-relativistic Limit.
General theory of Relativity: The Principles of Equivalence — Local Lorentz Invariance — General Coordinate Invariance — Geometry of Curved Spacetime.
Tensor Analysis: Parallel Displacement — Christoffel Symbols — Geodesics — Covariant Differentiation — The Curvature Tensor — The Bianchi Relations — The Ricci Tensor — Einstein's Field Equations — The Schwarzschild Solution.
Experimental Tests: The Gravitational Redshift, Deflection of Light by the Sun, Precession of Perihelia of Mercury.
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Advanced Quantum Mechanics
Nonrelativistic Quantum Mechanics: Selection rules for dipole transitions in hydrogen — Spontaneous and stimulated emission, absorption — Lifetime of excited states — Line shape and width — Fine structure of hydrogen — Path Integrals — Aharonov Bohm Effect — Geometric Phase — Coherent States.
Relativistic Quantum Mechanics: Klein Gordon Equation — Dirac Equation — Canonical (second) quantization of a wave field — Quantization of the free Maxwell field.
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Condensed Matter Physics
Non-Equilibrium Statistical Mechanics: Systems Close to Equilibrium — Onsager's Regression Hypothesis and Time Correlation Functions — Application to Chemical Kinetics — Application to Self-Diffusion — Fluctuation-Dissipation Theorem — Response Functions — Absorptions — Friction and Langevin Equation — Fokker-Planck Equations — Master Equations — Quantum Dynamics — Linear Response Theory.
Condensed Matter Physics: Crystal Structure — Wave Diffraction and Reciprocal Lattice — Crystal Binding and Elastic Constants — Crystal Vibrations: Phonons — Thermal Properties — Free Electron Fermi Gas — Energy Bands — Semiconductor Crystals — Fermi Surfaces and Metals — Superfluidity and Superconductivity — Diamagnetism and Paramagnetism — Ferromagnetism and Antiferromagnetism — Magnetic Resonance — Dielectrics and Ferroelectrics — Point Defects — Dislocations.
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Structure of Matter
Atomic Physics: Review of Hydrogen Atom — Pauli Exclusion Principle and Helium Atom — Many-Electron Systems — Dipole transitions (selection rules) — Photoelectric Effect.
Molecular Physics: Born- Oppenheimer Approximation — Hydrogen Molecule — Concept of Valence — Vibrational and Rotational Levels - Selection Rules — Raman effect.
Nuclear Physics: Basic Nuclear Properties (Mass, Radius, Isotopes) — Mass Formula — Deuteron Problem (Scattering Length, Effective Radius) — Shell Model — Liquid Drop Model — Radioactivity — Parity Violation in Beta Decay — Detectors and Accelerators.
Particle Physics: Properties of Elementary Particles — Conservation Laws — Quark Model — Strong Interactions — pi-N Scattering (Including Isospin) — Electromagnetic Interaction- Magnetic Moments of the Baryon Octet — Weak Interactions — Parity and Charge Conjugation Violation — Neutrino Oscillations — CP-Violation.
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