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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Discrete Mathematics
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
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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, Arzelo-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)
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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Big O notation — sorting and searching — algorithm analysis techniques, recurrences — graph algorithms: DFS, BFS, shortest paths, spanning trees — divide and conquer — greedy algorithms — dynamic programming — data structures: heaps, binary search trees, union-finde — advanced topics: LP, network flows
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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
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Programming Language Concepts
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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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Optional Courses
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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Mathematical Logic
Propositional and Predicate Logic: syntax, semantics, axiomatic systems, completeness, compactness, model theory
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Operations Research
Linear equations — Gaussian elimination — integral solutions — Smith normal form — linear functionals — duality — cones — Simplex method — discrete probability theory — the probabilistic method
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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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