Course
Structure and Syllabus for MSc in Mathematics and
Computing (to be applicable from 2011 batch onwards) 

Semester1 

Course No. 
Course Name 
L 
T 
P 
C 
MA 501 
Discrete
Mathematics 
4 
0 
0 
8 
MA 511 
Computer
Programming 
3 
0 
2 
8 
MA 521 
Modern
Algebra 
3 
1 
0 
8 
MA 522 
Linear
Algebra 
3 
1 
0 
8 
MA 541 
Real
Analysis 
3 
1 
0 
8 


15 
4 
2 
40 
Semester2 

Course No. 
Course Name 
L 
T 
P 
C 
MA 512 
Data
Structures and Algorithms 
3 
0 
0 
6 
MA 513 
Formal
Languages and Automata Theory 
3 
0 
0 
6 
MA 542 
Differential
Equations 
4 
0 
0 
8 
MA 547 
Complex
Analysis 
3 
1 
0 
8 
MA 591 
Optimization
Techniques 
3 
1 
0 
8 
MA 515 
Data
Structures Lab with ObjectOriented Programming* 
0 
1 
2 
4 


16 
3 
2 
40 
Semester3 

Course No. 
Course Name 
L 
T 
P 
C 
MA 514 
Theory
of Computation 
3 
0 
0 
6 
MA 543 
Functional
Analysis 
3 
1 
0 
8 
MA 572 
Numerical
Analysis 
3 
0 
2 
8 
MA 590 
Probability
Theory and Random Processes 
3 
1 
0 
8 
MA XXX 
Elective
– I 
3 
0 
0 
6 
MA 697 
Seminar 
0 
0 
3 
3 


15 
2 
5 
39 






Semester4: 

Course No. 
Course Name 
L 
T 
P 
C 
MA 571 
Numerical
Linear Algebra 
3 
0 
2 
8 
MA 573 
Numerics of Partial
Differential Equations 
3 
0 
2 
8 
MA XXX 
Elective
– II 
3 
0 
0 
6 
MA XXX 
ElectiveIII 
3 
0 
0 
6 
MA 699 
Project 
0 
0 
12 
12 


12 
0 
16 
40 
Total No. of
Credits: 159 
MA
501 Discrete
Mathematics (4008) Set
theory – sets, relations, functions, countability;
Logic – formulae, interpretations, methods of proof, soundness and
completeness in propositional and predicate logic; Number theory – division
algorithm, Euclid's algorithm, fundamental theorem of arithmetic, Chinese
remainder theorem, special numbers like Catalan, Fibonacci, harmonic and Stirling; Combinatorics –
permutations, combinations, partitions,
recurrences, generating functions; Graph Theory – paths,
connectivity, subgraphs, isomorphism, trees,
complete graphs, bipartite graphs, matchings, colourability, planarity, digraphs; Algebraic Structures
– semigroups,
groups, subgroups, homomorphisms, rings, integral
domains, fields, lattices and Boolean algebras. Texts:
2.
R. C. Penner, Discrete
Mathematics: Proof Techniques and Mathematical Structures, World
Scientific, 1999. References: 1.
R. L. Graham, D. E. Knuth, and O. Patashnik,
Concrete Mathematics, 2^{nd} Ed., AddisonWesley, 1994. 2.
K. H. Rosen, Discrete Mathematics & its
Applications, 6^{th} Ed., Tata McGrawHill, 2007. 3.
J. L. Hein, Discrete Structures, Logic, and
Computability, 3^{rd} Ed., Jones and Bartlett, 2010. 4.
N. Deo, Graph Theory,
Prentice Hall of India, 1974. 5.
S. Lipschutz and M. L.
Lipson, Schaum's Outline of Theory and
Problems of Discrete Mathematics, 2^{nd} Ed., Tata McGrawHill,
1999. 6.
J. P.
Tremblay and R. P. Manohar, Discrete Mathematics
with Applications to Computer Science, Tata McGrawHill, 1997. 
MA 511 Computer
Programming
( 3028 ) Introduction
– the von Neumann architecture, machine language, assembly language,
high level programming languages, compiler, interpreter, loader, linker, text
editors, operating systems, flowchart; Basic features of programming (Using
C) – data types, variables, operators, expressions, statements, control
structures, functions; Advance programming features – arrays and
pointers, recursion, records (structures), memory management, files,
input/output, standard library functions, programming tools, testing and
debugging; Fundamental operations on data – insert, delete, search,
traverse and modify; Fundamental data structures – arrays, stacks,
queues, linked lists; Searching and sorting – linear search, binary
search, insertionsort, bubblesort, selectionsort; Introduction to object
oriented programming. Programming
laboratory will be set in consonance with the material covered in lectures.
This will include assignments in a programming language like C and C++ in GNU
Linux environment. Texts: 1.
A. Kelly and I. Pohl, A
Book on C, 4^{th} Ed., Pearson Education, 1999. References: 1.
H. Schildt, C: The
Complete Reference, 4^{th} Ed., Tata Mcgraw
Hill, 2000. 2.
B. Kernighan and D. Ritchie, The
C Programming Language, 2^{nd} Ed., Prentice Hall of India, 1988. 3.
B. Gottfried and J. Chhabra, Programming
With C, Tata Mcgraw Hill, 2005. 
MA 521 Modern
Algebra (3
1 0 8) Prerequisites: Nil
Review
of Groups, Subgroups, Homomorphisms, Group actions,
Sylow theorems, Generators and Relations, Free
groups, Solvable and Nilpotent groups. Rings, Ideals and Quotient rings,
Maximal, Prime and Principal ideals, Euclidean and Polynomial rings, Modules.
Field extensions, Finite fields, Algebraically closed fields. Texts / References:
1.
M. Artin, Algebra,
Prentice Hall India, 1994. 2. I. N. Herstein, Topics in Algebra, 2^{nd}
Ed, Wiley Eastern, 1975. 3. C. Musili, Introduction to Rings
and Modules, 2^{nd} Ed, Narosa, 1992. 4. F. Loonstra, Introduction to Algebra, McGraw Hill,
London, 1969. 
MA 522
Linear Algebra
(3 1 0 8) Prerequisites:
Nil
Systems
of linear equations, Vector spaces, Bases and dimensions, Change of bases and
change of coordinates, Sums and direct sums, Quotient spaces. Linear
transformations, Representation of linear transformations by matrices, The
rank and nullity theorem, Dual spaces, Transposes of linear transformations.
Trace and determinant, Eigenvalues and
eigenvectors, Invariant subspaces, DirectSum decomposition, Cyclic subspaces
and Annihilators, The minimal polynomial, The Jordan canonical form. Inner
product spaces, Orthonormal bases, GramSchmidt
process. Adjoint operators, Normal, Unitary, and
Selfadjoint operators, Spectral theorem for normal
operators. Texts / References:
1.
K. Hoffman and R. Kunze, Linear Algebra,
Prentice Hall India, 1996. 2.
G. Schay, Introduction to Linear Algebra, Narosa, 1997. 3.
G. C. Cullen, Linear Algebra with Applications, 2^{nd} Ed,
Addison Wesley, 1997. 4.
S. Axler, Linear Algebra Done Right, 2^{nd}
ed, UTM, Springer, 1997. 5.
K. Janich, Linear
Algebra, UTM, Springer, 1994 
MA 541 Real Analysis (3 1 0 8) Prerequisites:
Nil
Sequences
and Series of functions, Uniform convergence, Equicontinuity,
Ascoli’s theorem, Weierstrass
approximation theorem. Functions of several variables, Inverse and Implicit
function theorems, Multiple integration, Change of
variables. Lebesgue measure, Lebesgue
integral, Introduction to Lp spaces. Fourier
Series. Texts / References:
1.
W. Rudin, Principles of Mathematical Analysis, 3^{rd}
Ed, McGraw Hill, 1976. 2.
T. M. Apostol, Mathematical Analysis, 2^{nd}
Ed, Narosa, 1985. 3.
H. L. Royden, Real Analysis, 3^{rd}
edition, Prentice Hall India, 1995. 
MA 512 Data
Structures and Algorithms
(3006) Prerequisite:
MA 511 or
equivalent. Asymptotic
notation; Sorting – merge sort, heap sort, priortiy
queue, quick sort, sorting in linear time, order statistics; Data structures
– heap, hash tables, binary search tree, balanced trees (redblack
tree, AVL tree); Algorithm design techniques – divide and conquer,
dynamic programming, greedy algorithm, amortized analysis; Elementary graph
algorithms, minimum spanning tree, shortest path algorithms. Texts: 1. T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 2^{nd}
Ed., PrenticeHall of India,
2007. References: 1.M. T. Goodrich and R. Tamassia, Data Structures and Algorithms in Java,
Wiley, 2006. 2.A.V. Aho
and J. E. Hopcroft, Data Structures and Algorithms,
AddisonWesley, 1983. 3.S. Sahni, Data
Structures, Algorithms and Applications in C++, 2^{nd} Ed.,
Universities Press, 2005. 
MA 513
Formal Languages and Automata Theory (3 0 0 6) Prerequisite: MA 501 or equivalent. Alphabets,
languages, grammars; Finite automata, regular languages, regular expressions;
Contextfree languages, pushdown automata, DCFLs; Context sensitive
languages, linear bounded automata; Turing machines, recursively enumerable
languages; Operations on formal languages and their properties; Chomsky
hierarchy; Decision questions on languages. Texts:
References:

MA 542
Differential Equations
(4 0 0 8) Prerequisites: Nil
Review
of fundamentals of ODEs, Existence and uniqueness theorems, Power series
solutions of ODE, Systems of Linear ODEs, Reduction
of higher order linear ODEs to first order linear systems, Stability of
linear systems, SturmLiouville problems. First
order linear and quasilinear PDEs, The Cauchy problem, Classification of
PDEs, Characteristics, Wellposed problems, Solutions of hyperbolic, parabolic
and elliptic equations, Dirichlet and Neumann
problems, Maximum principles, Greens functions for elliptic, parabolic and
hyperbolic equations. Texts / References:
1.
E. A. Coddington and N. Levinson, Theory of
Ordinary Differential Equations, Tata McGraw Hill, 1990. 2.
P.F. Hsieh and Y. Sibuya, Basic Theory of
Ordinary Differential Equations, UTX, Springer, 1999. 3.
I. N. Sneddon, Elements of Partial Differential
Equations, McGraw Hill, 1957. 4.
F. John, Partial Differential Equations, Springer Verlag,
1982. 5.
W. E. Willams, Partial Differential Equations,
Oxford, 1980. 6.
W.A. Strauss, Partial Differential Equations: An Introduction, John Wiley,
1992. 
MA 547
Complex Analysis
(3 1 0 8) Prerequisites: Nil
Spherical
representation of extended complex plane, Analytic functions, Harmonic
functions, Elementary functions, Branches of multiplevalued functions,
Mappings of elementary functions, Bilinear transformations, Conformal mappings
and Computational aspects. Complex integration, Cauchy’s integral
theorem, Cauchy’s integral formula, Theorems of Morera
and Liouville, MaximumModulus theorem, Power
series, Taylor’s theorem and analytic continuation, Zeros of analytic
functions, Hurwitz theorem, Singularities, Laurent’s theorem, CasoratiWeierstrass theorem, Argument principle,
Theorems of Rouche and GaussLucas, Residue theorem
and its applications to evaluate real integrals and to compute the inverse
Laplace transform, Contour integration of functions with branch points. Texts / References:
1. L. V. Ahlfors, Complex Analysis, 3^{rd} Ed,
McGraw Hill, 1979 2. R.V. Churchill and
J. W. Brown, Complex Variables and Applications, 6^{th} Ed,
McGraw Hill, 1995. 3. J. H. Mathews and
R. W. Howell, Complex Analysis for Mathematics and Engineering, 3^{rd}
Ed, Narosa, 1998. 4. E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis
with Applications to Engineering, Science and Mathematics, 3^{rd} Ed,
Prentice Hall, 2002. 
MA 591
Optimization Techniques (3108) Mathematical foundations and basic
definitions: concepts from linear algebra, geometry, and multivariable
calculus. Linear optimization: formulation and geometrical
ideas of linear programming problems, simplex method, revised simplex method,duality,
sensitivity snalysis, transportation and assignment
problems. Nonlinear
optimization: basic theory, method of Lagrange multipliers, KarushKuhnTucker
theory, convex optimization. Numerical optimization techniques: line search methods, gradient methods, Newton’s
method, conjugate direction methods, quasiNewton methods, projected gradient
methods,
penalty methods. Texts: 1.
N. S. Kambo, Mathematical Programming Techniques,
East West Press, 1997. 2.
E.K.P. Chong and S.H. Zak,
An Introduction to Optimization, 2^{nd} Ed.,
Wiley, 2001 (WSE,
2004). References: 1.
R. Fletcher, Practical
Methods of Optimization, 2^{nd} Ed., John Wiley, 1987. 2.
D. G. Luenberger, Linear and Nonlinear Programming, 2^{nd}
Ed., Kluwer, 2003. 3.
M. S. Bazarra, J.J. Jarvis,
and H.D. Sherali, Linear Programming and Network Flows, 2^{nd} Ed.,1990. [Also available as WSE (2003) edition]. 4.
U. Faigle, W. Kern, and G.
Still, Algorithmic Principles of
Mathematical Programming, Kluwe, 2002. 5.
D.P. Bertsekas, Nonlinear Programming, 2^{nd} Ed.,
Athena Scientific, 1999. 6.
M. S. Bazarra, H.D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and
Algorithms, 2^{nd} Ed., John Wiley, 1993. (WSE, 2004). 
MA 515 Data Structures Lab With
ObjectOriented Programming (0124
) The
tutorials will be based on objectoriented programming concepts such as
classes, objects, methods, interfaces, packages, inheritance, encapsulation,
and polymorphism. Programming laboratory will be set in consonance with the
material covered in MA 512. This will include assignments in a programming language
like C++ in GNU Linux environment. References: 1. T. Budd, An Introduction to
ObjectOriented Programming, AddisonWesley, 2002. 
MA 514 Theory
of Computation (3006) Prerequisite: MA
513 or equivalent
Models of computation – Turing machine, RAM,
µrecursive function, grammars; Undecidability
– Rice's theorem, Post
correspondence problem, logical theories; Complexity
classes – P, NP, coNP, EXP, PSPACE, L, NL, ATIME, BPP, RP, ZPP, IP. Texts: 1.
M. Sipser, Introduction to the Theory of Computation,
Thomson, 2004. 2.
H. R. Lewis and C.
H. Papadimitriou, Elements of the Theory of Computation, PHI, 1981. References:

MA 543
Functional Analysis
(3 1 0 8) Prerequisites:
MA 541 (Real Analysis)
Banach
spaces, Continuity of linear maps, HahnBanach
theorem, Open mapping and closed graph theorems, Uniform boundedness
principle. Duals and Transposes. Compact operators and their spectra. Weak
and Weak* convergence, Reflexivity. Hilbert spaces, Bounded operators on
Hilbert spaces. Adjoint operators, Normal, Unitary,
Selfadjoint operators and their spectra. Spectral
theorem for compact selfadjoint operators. Texts / References:
1.
B. V. Limaye, Functional Analysis, 2^{nd}
Ed, Wiley Eastern, 1996. 2.
E. Kreyszig, Introduction to Functional Analysis
with Applications, John Wiley and Sons, 1978. 
MA572
NUMERICAL ANALYSIS (3028) Prerequisites: Nil Definition
and sources of errors, Propagation of errors, Backward error analysis,
Sensitivity and conditioning, Stability and accuracy, Floatingpoint
arithmetic and rounding errors. Nonlinear equations, Bisection method,
Newton's method and its variants, Fixed point iterations, Convergence
analysis. Newton’s method for nonlinear systems. Finite differences,
Polynomial interpolation, Hermite interpolation, Spline interpolation, Bsplines.
Numerical integration, Trapezoidal and Simpson's rules, NewtonCotes formula,Gaussian quadrature, Richardson Extrapolation. IVP: Taylor series
method, Euler and modified Euler methods, RungeKutta methods, Multistep methods,
PredictorCorrector method. Accuracy and stability, Solution for Stiff
equations. BVP: Finite difference method, Collocation method, Galerkin method. Texts
/ References: 1.
M. T. Heath, Scientific Computing: An Introductory
Survey, McGraw Hill, 2002. 2.
S. D. Conte and Carl de Boor, Elementary Numerical
Analysis  An Algorithmic Approach, 3^{rd}
Edition, McGraw Hill, 1980. 3.
K. E. Atkinson, Introduction to Numerical Analysis,
2nd Edition, John Wiley, 1989. 4.
C. F. Gerald and P. O. Wheatley, Applied Numerical
Analysis, 5th edition, Addison Wesley, 1994. 
MA 590 Probability Theory and
Random Processes (3108) Axiomatic
construction of the theory of probability, independence, conditional
probability, and basic formulae, random variables, probability distributions,
functions of random variables; Standard univariate discrete
and continuous distributions and their properties, mathematical expectations,
moments, moment generating function, characteristic functions; Random
vectors, multivariate distributions, marginal and conditional distributions,
conditional expectations; Modes of convergence of sequences of random variables,
laws of large numbers, central limit theorems. Definition and classification
of random processes, discretetime Markov chains, Poisson process, continuoustime
Markov chains, renewal and semiMarkov processes, stationary processes,
Gaussian process, Brownian motion, filtrations and martingales, stopping
times and optimal stopping. Texts/References: 1. G. R. Grimmett
and D. R. Stirzaker, Probability and Random
Processes, 3rd Edition, Oxford University Press, 2001. 2. P. G. Hoel,
S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal
Book Stall, 2000. 3. W. Feller, An Introduction
to Probability Theory and its Applications, Vol. 1, 3rd Edition, Wiley,
1968 (WSE Edition). 4. K. S. Trivedi,
Probability and Statistics with Reliability, Queuing, and Computer Science
Applications, 2nd Edition, Wiley, 2001 (WSE Edition). 5. A. Papoulis and S. Unnikrishna Pillai, Probabilities,
Random Variables and StochasticProcesses, 4th Edition,
Tata McGrawHill, 2002. 6. S.M. Ross, Stochastic
Processes, 2nd Edition, Wiley, 1996 (WSE Edition). 7. J. Medhi,
Stochastic Processes, 3rd Edition, New Age International, 2009. 
MA571 Numerical Linear Algebra (3028) Prerequisites: MA522 Linear Algebra Fundamentals.
Linear systems, LU decompositions, Gaussian elimination with partial
pivoting, Banded systems, Positive definite systems, Cholesky
decomposition. Vector and matrix norms, Perturbation theory of linear
systems, Condition numbers, Estimating condition numbers, IEEE floating point
arithmetic, Analysis of roundoff errors. GramSchmidt
orthonormal process, Orthogonal matrices,
Householder transformation, Givens rotations, QR factorization, Roundoff error analysis of orthogonal matrices, Stability
of QR factorization. Solution
of linear least squares problems, Normal equations, Singular Value
Decomposition(SVD), Polar decomposition, MoorePenrose inverse, Rank
deficient least squares problems, Sensitivity analysis of leastsquares
problems. Review of eigenvalues and canonical forms
of matrices, Sensitivity of eigenvalues and
eigenvectors, Reduction to Hessenberg and tridiagonal forms, Power and inverse power methods,
Rayleigh quotient iteration, Explicit and implicit QR algorithms for
symmetric and nonsymmetric matrices, Implementation of implicit QR
algorithm. Computing the SVD, Sensitivity analysis of singular values and
singular vectors. Overview of iterative methods: Jacobi, GaussSeidel and
successive overrelaxation methods, Krylov subspace method, The Arnoldi
and the Lanczos iterations. Texts/References: 1. L. N. Trefethen
and David Bau, Numerical Linear Algebra,
SIAM, 1997. 2. D. S. Watkins, Fundamentals
of Matrix Computation, Wiley, 1991. 3. G. H. Golub
and C.F.Van Loan, Matrix Computation, John
Hopkins U. Press, Baltimore, 1996. 4. G. W. Stewart, Introduction
to Matrix Computations, Academic Press, 1973. 5. J.W. Demmel, Applied numerical
linear algebra, SIAM, Philadelphia, 1997. 
MA573 Numerics
of Partial Differential Equations (3028) Prerequisites: Nil Finite
difference methods for Parabolic, Elliptic and Hyperbolic equations. Dirichlet, Neumann and Mixed problems. Sparseness and the
ADI method, Iterative methods for Laplace equation. Backward Euler, CrankNicolson
schemes, Stability and convergence analysis, Lax's
equivalence theorem. Method of characteristics, LaxWendroff
explicit method, CFL conditions, Wendroff implicit
approximation. Introduction to FEM. Texts / References: 1. J. D. Hoffman, Numerical
methods for Engineers and Scientists, McGraw Hill, 1993. 2. G. D. Smith, Numerical
solutions to Partial Differential Equations, Brunel University, Clarendon Press,Oxford, 1985. 3. C. Johnson, Numerical
Solution of Partial Differential Equations by the Finite Element Method, Cambridge
University Press, 1987. 4. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential
Equations, Cambridge University Press, 1996. 5. L. Lapidus
and G. F. Pinder, Numerical Solution of Partial
Differential Equations in Science and Engineering, John Wiley, 1982. 6. H. P. Langtangen,
Computational Partial Differential Equations Springer Verlag,
1999. 