• 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.

    Recommended Texts

    1. Artin, M., Algebra, Prentice-Hall.
    2. Jacobson, N., Basic Algebra I, II, Hindustan Publishing Corporation, 1991.
    3. P. R. Halmos, Finite-dimensional Vector Spaces, Springer.
  • Analysis I

  • Axioms of the real number system without construction, applications of the least-upper-bound- property, Archimedean principle, existence of nth roots of positive real numbers, ax 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 ez 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)

    1. R. Goldberg, Methods of Real Analysis.
    2. Bartle and Sherbett, Introduction to Real Analysis.
    3. Rudin, Principles of Mathematical Analysis.
    4. Spivak, Calculus on Manifolds.
    5. Chapter 1 of S. Kumaresan, A Course in Differential Geometry and Lie Groups.
    6. L. Cohen and Ehrlick, Structure of the Real Number System.
    7. T. Apostol, Calculus, vols I and II
  • Introduction to Programming

  • The course will be based on the programming language Haskell.

    • Function definitions: pattern matching, induction
    • Basic data types, tuples, lists
    • Higher order functions
    • Polymorphism
    • Reduction as computation, lazy evaluation
    • Measuring computational complexity
    • Basic algorithms: sorting, backtracking, dynamic programming
    • User-defined datatypes: enumerated, recursive and polymorphic types
    • Input/output

    Recommended Texts

    1. R. Bird and P. Wadler, Introduction to Functional Programming Prentice Hall, 1988.
    2. R. Bird, Introduction to Functional Programming using Haskell, Prentice Hall, 1998.
    3. Paul Hudak, The Haskell school of expression, Cambridge University Press, 2000.
    4. Graham Hutton, Programming in Haskell, Cambridge University Press, 2007.
    5. Bryan O'Sullivan, John Georzen and Don Stewart, Real World Haskell, O'Reilly, 2007.
  • English

  • Poetry: Selections from Indian and English poets.

    • Tagore,
    • Keats,
    • Hopkins.

    Prose: Three Short Stories and one Novel.

    • Sharatchandra,
    • R. K. Narayan,
    • Somerset Maugham,
    • Hemmingway's Old Man & The Sea.

    Drama:

    • Shakespeare's The Tempest (selected portions)

    Interactive Communication:

    • Public Speaking
    • Field-work: interviews with selected individuals

    Self-Expression

    • Paper to be submitted on a subjective topic.

    Effective Language

    • Use of language in different contexts — technical, informal, literary etc.
  • 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.

    Recommended Texts

    1. Artin, M., Algebra, Prentice Hall of India, 1994.
    2. Jacobson, N., Basic AlgebraI, II, Hindustan Publishing Corporation, 1991.
  • Analysis II

  • Topology of Rn : Euclidean, l1 and l∞ norms on Rn and the equivalence of convergence of sequences in Rn, open and closed sets, sequential compactness, continuous functions defined on subsets of Rn, boundedness of continuous functions defined on compact subsets. Characterisation of open sets in Rn.

    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)

    1. R. Goldberg, Methods of Real Analysis.
    2. Bartle and Sherbett, Introduction to Real Analysis.
    3. Rudin, Principles of Mathematical Analysis.
    4. Spivak, Calculus on Manifolds.
    5. Chapter 1 of S. Kumaresan, A Course in Differential Geometry and Lie Groups.
    6. L. Cohen and Ehrlick, Structure of the Real Number System.
    7. T. Apostol, Calculus, vols I and II
  • Advanced Programming

  • Imperative programming using Python and C.

    • Control flow: assignment, conditionals, loops
    • Lists and arrays
    • Basic algorithms: searching, sorting, backtracking, dynamic programming
    • Implementing recursive datatypes
    • Memory management
    • Basics of graphs and graph algorithms: representations of graphs, breadth-first search, depth-first search

    Recommended Texts

    1. Brian W. Kernighan and Dennis M. Ritchie, The C Programming Language, Prentice-Hall, 1999.
    2. Jon Kleinberg and Eva Tardos, Algorithm Design, Pearson/Addison-wesley, 2006.
    3. Mark Lutz, Programming Python, O'Reilly, 2001.
    4. Mark Pilgrim, Dive into Python, available online.
  • 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

    Recommended Texts

    1. Biggs, N. L., Discrete Mathematics, Oxford Science Publications, 1989.
    2. Douglas B. West, Introduction to Graph Theory, Prentice-Hall India, 2001.
  • 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.

    Recommended Texts

    1. M. Artin, Algebra, Prentice Hall of India, 1994.
    2. T. Hungerford, Algebra, Springer.
  • 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)

    1. R. Goldberg, Methods of Real Analysis.
    2. Bartle and Sherbett, Introduction to Real Analysis.
    3. Rudin, Principles of Mathematical Analysis.
    4. Spivak, Calculus on Manifolds.
    5. Chapter 1 of S. Kumaresan, A Course in Differential Geometry and Lie Groups.
    6. L. Cohen and Ehrlick, Structure of the Real Number System.
    7. T. Apostol, Calculus, vols I and II
  • 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)

    1. R. Goldberg, Methods of Real Analysis.
    2. Bartle and Sherbett, Introduction to Real Analysis.
    3. Rudin, Principles of Mathematical Analysis.
    4. Spivak, Calculus on Manifolds.
    5. Chapter 1 of S. Kumaresan, A Course in Differential Geometry and Lie Groups.
    6. L. Cohen and Ehrlick, Structure of the Real Number System.
    7. T. Apostol, Calculus, vols I and II
  • 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.

    Recommended Texts

    1. Ahlfors, L. V. Complex analysis, Tata Mc-Graw Hill.
  • Topology

  • Point set topology — connectedness — compactness — Tychonoff's theorem — Stone Czech compactification — covering spaces — fundamental groups

    Recommended Texts

    1. Munkres, J. Topology, Prentice Hall of India, 1987
  • 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.

    Recommended Texts

    1. W. Feller, W: An Introduction to Probability Theory and its Applications, Vol.1, John Wiley.
    2. G. R. Grimmett and D. R. Stirzaker: Probability and Random Processes, Oxford Science Publications.
    3. K. S. Trivedi: Probability and Statistics with Queuing, Reliability and Computer Science Applications. Prentice-Hall.


  • Design and Analysis of Algorithms

    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

    Recommended Texts

    1. T.H. Cormen, C.E. Leiserson, and R.L. Rivest: Introduction to algorithms, Prentice-Hall (1998).
    2. J. Kleinberg and E. Tardos: Algorithm design, Pearson/Addison-Welsey (2006).
  • 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

    Recommended Texts

    1. Hopcroft, J. E. and Ullman, J. D, Introduction to Automata theory, Languages and Computation Narosa.
    2. D. Kozen: Automata and Computability, Springer.
  • Programming Language Concepts

    • Object oriented-programming
    • Event driven programming
    • Exception handling
    • Concurrent programming
    • Foundations of functional programming: λ-calculus, type checking
    • Logic programming
    • Scripting languages

    Recommended Texts

    1. John C Mitchell, Concepts in Programming Languages, Cambridge University Press, 2003.
    2. Ravi Sethi, Programming Languages, Addison Wesley, 1996.
  • 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.

    Recommended Texts

    1. Mechanics: Berkeley Physics Course, Vol. 1, by C. Kittel, W. D. Knight, M. A. Ruderman, C. A. Helmholz, and B. J. Moyer; Tata-McGraw Hill.
    2. Classical Mechanics, T.W.B. Kibble, F. H. Berkshire, World Scientific.
    3. Principle of Mechanics by J. L. Synge and B. A. Griffith, Nabu Press, 2011.
  • 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.

      Recommended Texts

      1. M. Artin, Algebra, Prentice Hall of India, 1994.
      2. E. Artin, Galois Theory.
      3. S. Lang, Algebra
    • Mathematical Logic

    • Propositional and Predicate Logic: syntax, semantics, axiomatic systems, completeness, compactness, model theory

      Recommended Texts

      1. D. Ebbinghaus, J. Flum and W. Thomas: Mathematical Logic, Springer-Verlag.
      2. Herbert B. Enderton: A Mathematical Introduction to Logic, Academic Press.
    • Operations Research

    • Linear equations — Gaussian elimination — integral solutions — Smith normal form — linear functionals — duality — cones — Simplex method — discrete probability theory — the probabilistic method

      Recommended Texts

      1. Chvatal, V. Linear Programming,
      2. Schrijver, A. The theory of Integer and Linear programming Addison Wesley
      3. Alon, N., Spencer, The Probabilistic Method, John Wiley, 1991
    • 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.

      Recommended Texts

      1. Quantum Physics: Berkeley Physics Course, Vol. 4, by E. H. Wichman; Tata-McGraw Hill.
      2. Quantum Mechanics by L. I. Schiff, McGraw Hill.
      3. Quantum Mechanics by E. Merzbacher, John Wiley.
    • Complex function theory
    • Number Theory
    • Representation theory
    • Differential geometry of curves and surfaces
    • Partial differential equations
    • Algebraic topology
    • Differential topology
    • Introduction to Algebraic geometry
    • Operations Research
    • Complexity Theory
    • Advanced Algorithms