Matrix operations — Groups and subgroups — homomorphisms — equivalence relations and partitions — quotient groups — 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.
Functions of one variable: Real number system — Limits and continuity — differentiation — chain rule — mean value theorems and applications — Taylor expansion — L'Hospital's rule — integration — fundamental theorem of calculus — change of variable and applications — exponential, logarithmic, trigonometric and inverse trigonometric functions — applications of integration.
Introduction to Programming
The course will be based on the programming language Haskell.
Poetry: Selections from Indian and English poets.
Prose: Three Short Stories and one Novel.
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.
Functions of several variables: Limits and continuity — partial derivatives and applications — chain rule — directional derivatives and the gradient vector — Lagrange multiplier — Taylor expansion — implicit and inverse function theorems — double integrals, iterated integrals and applications — multiple integrals — change of variables.
Imperative programming using Python and C.
Principles of Counting — binomial coeffecients — generating functions — partitions — Graph Theory: paths — degree sequences — trees — minimum spanning trees — shortest path — bipartite matching — Tutte's theorem — connectivity — flows — graph colouring
Rings: Formal construction of integers and polynomials — ideals and homomorphisms — quotient rings — integral domains and fraction fields — maximal ideals — factorization of integers and polynomials — UFDs and PIDs — Gauss lemma — algebraic integers — ideal factorization — real quadratic fields.
Modules: Matrices, free modules and bases — generators and relations for modules — structure theorem for Abelian groups.
Finite, countable and uncountable sets — metric spaces — compact sets — perfect sets — connected sets — convergent and Cauchy sequences — power series — absolute convergence — rearrangements — limits — continuity and compactness — connectedness — discontinuities — infinite limits — sequences and series of functions — equicontinuous family of functions and the Stone-Weierstrass theorem.
Vector calculus: Vector fields — line integrals — Green's theorem — curl and divergence — area of a surface — surface integrals — Stokes' theorem.
Differential equations: Homogeneous equations — first order linear equations — second order linear equations — non homogeneous linear equations — applications.
Field Theory: Fields — algebraic and transcendental elements — field extensions — ruler and compass constructions — finite fields — function fields — transcendental extensions.
Galois theory: The main theorem of Galois theory — cubic equations — symmetric functions — primitive elements — quartic equations — cyclotomic extensions — quintic equations.
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.
Point set topology — connectedness — compactness — Tychonoff's theorem — Stone Czech compactification — covering spaces — fundamental groups
Machine Language and Assembly Language Programming — Computer Arithmetic- CPU Internals — Memory subsystems — I/O and Interfacing — Introduction to System Software
Laboratory: Assembly Language Programming — Programming Assignments.
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.
Linear equations — Gaussian elimination — integral solutions — Smith normal form — linear functionals — duality — cones — Simplex method — discrete probability theory — the probabilistic method
Mathematical Theory of Computation
Finite automata — regular languages — pumping lemma — stack automata — context free languages — applications to compilers — Turing machines — universal Turing machines — halting problem — non deterministic Turing machines — complexity classes — P v/s NP
Programming Language Concepts
Propositional and Predicate Logic: syntax, semantics, axiomatic systems, completeness, compactness, model theory
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.
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.