## First Semester |
|||||||

S. No. | Course Code | Course Description | Type | L | T | P | Credit |
---|---|---|---|---|---|---|---|

1. | MTH6011 | Analysis – I | PC | 3 | 1 | 0 | 4 |

2. | MTH6041 | Linear Algebra | PC | 3 | 1 | 0 | 4 |

3. | CSE104 | Computer programming | PC | 3 | 0 | 0 | 3 |

4. | CSE104(L) | Computer programming Lab | PC | 0 | 0 | 3 | 2 |

5. | MTH6021 | Optimization | PC | 3 | 1 | 0 | 4 |

6. | MTH6012 | Algebra | PC | 3 | 1 | 0 | 4 |

Total Credits = 21 |
|||||||

## Second Semester |
|||||||

S. No. | Course Code | Course Description | Type | L | T | P | Credits |

1. | MTH6031 | Probability & Statistics | PC | 3 | 1 | 0 | 4 |

2. | MTH6022 | Analysis-II | PC | 3 | 1 | 0 | 4 |

3. | MTH6032 | Complex Analysis | PC | 3 | 1 | 0 | 4 |

4. | MTH6052 | Numerical Analysis | PC | 3 | 1 | 0 | 4 |

5. | MTH4042 | Ordinary Differential Equations | PC | 3 | 1 | 0 | 4 |

Total Credits = 20 |
|||||||

## Third Semester |
|||||||

S. No. | Course Code | Course Description | Type | L | T | P | Credits |

1. | MTH7011 | Topology | PC | 3 | 1 | 0 | 4 |

2. | MTH7021 | Functional Analysis | PC | 3 | 1 | 0 | 4 |

3. | MTH7031 | Partial Differential Equations | PC | 3 | 1 | 0 | 4 |

4. | -- | Other Elective-I | OE | 3 | 1 | 0 | 4 |

5. | -- | Other Elective-II | OE | 3 | 1 | 0 | 4 |

6. | -- | Project | PC | -- | -- | -- | 4 |

Total Credits = 24 |
|||||||

## Fourth Semester |
|||||||

S. No. | Course Code | Course Description | Type | L | T | P | Credits |

1. | -- | Mathematical Methods | PC | 3 | 1 | 0 | 4 |

2. | -- | Other Elective-III | OE | 3 | 1 | 0 | 4 |

3. | -- | Other Elective-IV | OE | 3 | 1 | 0 | 4 |

4. | -- | Other Elective-V | OE | 3 | 1 | 0 | 4 |

5. | -- | Project | -- | -- | -- | 4 | |

Total Credits = 20 |

Minimum number of credits required to complete the Programme is 85.

## Possible Elective Courses*: |
||||||

S. No. | Course No. | Course Name | L | T | P | Credit |
---|---|---|---|---|---|---|

1. | MTH5041 | Numerical Methods for Partial Differential Equations | 3 | 1 | 1 | 4 |

2. | MTH5022 | Finite Element Methods | 3 | 1 | 1 | 4 |

3. | MTH5051 | Computational Topology | 3 | 1 | 1 | 4 |

4. | MTH5061 | Number Theory | 3 | 1 | 0 | 4 |

5. | MTH5071 | Differential Geometry | 3 | 1 | 0 | 4 |

6. | MTH5032 | Spectral Element Methods | 3 | 1 | 1 | 4 |

7. | MTH5042 | Computational Fluid Dynamics | 3 | 1 | 0 | 4 |

8. | MTH5081 | Fractal Geometry and Applications | 3 | 1 | 0 | 4 |

9. | MTH5091 | Elements of Sobolev Spaces and Applications | 3 | 1 | 0 | 4 |

10. | MTH5052 | Applied Harmonic Analysis | 3 | 1 | 0 | 4 |

11. | MTH5062 | Wavelets and Applications | 3 | 1 | 0 | 4 |

12. | MTH5072 | Parallel Numerical Algorithms | 3 | 0 | 1 | 4 |

13. | MTH5082 | Image Processing and Computer Graphics | 3 | 0 | 1 | 4 |

14. | MTH5092 | Mathematics in Multi-media | 3 | 1 | 0 | 4 |

15. | MTH5102 | Data Mining: A Mathematical Perspective | 3 | 1 | 0 | 4 |

16. | MTH5112 | Introduction to Stochastic Differential Equations | 3 | 1 | 0 | 4 |

17. | MTH5122 | Mechanics | 3 | 1 | 0 | 4 |

18. | MTH5132 | Continuum Mechanics | 3 | 1 | 0 | 4 |

19. | MTH5142 | Cryptography | 3 | 1 | 0 | 4 |

20. | MTH5152 | Graph Theory | 3 | 1 | 0 | -- |

* Some additional courses will be added to this list.

**First Semester**

**Title of the course :**Analysis-I**Course Code:**MTH4011**Prerequisite:**None**Proposed Curriculum :**

Real number system and set theory: Completeness, Archimedean property, Denseness of rationals and irrationals, Countable and uncountable, Cardinality, Zorn’s lemma, Axiom of choice. Metric spaces: Open sets, closed sets, Continuous functions, Completeness, Cantor intersection theorem, Baire category theorem, Compactness, connectedness, Finite intersection property. Functions of several variables: Differentiation, inverse and implicit function theorems. Rlemann-Stieitjes integral: Definition and existence of the integral, Properties of the integral, Differentiation and integration. Sequence and Series of functions: Uniform convergence, Uniform convergence and continuity, Uniform Convergence and integration, Uniform convergence and differentiation. Equicontinuity, Ascoli’s Theorem.

**W. Rudin,**Principles of Mathematical, Analysis, 3rd ed., McGraw-Hill, 1983.**T. Apostol,**Mathematical Analysis, 2nd ed., Narosa Publishers, 2002.

**Title of the course :**Linear Algebra**Course Code:**MTH4021**Prerequisite:**None

Vector spaces over fields, subspaces, bases and dimension. Systems of linear equations, matrices, rank, Gaussian elimination. Linear transformations and projections, representation of linear transformations by matrices, rank-nullity theorem, duality and transpose. Determinants. Eigenvalues and eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem, Elementary canonical forms: diagonalization, triangulation, primary decomposition etc. Secondary decomposition theorem, Rational canonical forms, Jordan canonical forms and some applications. Inner product spaces, Selfadjoint, Unitary and normal operators, Orthogonal projections. Bilinear forms, Symmetric, Skew-symmetric, Positive and semi-positive forms etc.

**K. Hoffman and R. Kunze,**Linear Algebra, Pearson Education (India), 2003, Prentice-Hall of India, 1991.**G. Strang,**Linear Algebra and Its Applications, Thomson Brooks/Cole, 2007.

**Title of the course :**Computer Programmeming and Lab**Course Code :**MTH4031**Prerequisites :**None**Proposed Curriculum :**- Fortran 95: Integer and real operations, logic and complex operations, Control statements, Do statement, arrays subroutines and functions. Introduction to data structures in C/C++ Programmeming Language; Arrays: linear, Multidimensional, Records, Pointers, Stacks, queues, Linked Lists; Singly linked lists, doublyd linked lists, circular linked lists, Application of Linked Lists; Polynomial addition, sparse matrices, Trees: binary trees, redblack trees, Hash tables. Some discussion about data structures in F90/F95 with examples.
**Suggested Readings :**

**Aho, Hopcroft and Ullmann,**Data Structures and Algorithms, Addison Wesley.**M. Weiss,**Data Structures and Algorithms in C++, Addison Wesley.

**Title of the course :**Optimization**Course Code :**MTH4041**Pre-requisites :**Nil**Proposed Curriculum :**- Mathematical foundations and basic definitions: concepts from linear algebra, geometry, and multivariable calculus. Linear optimization: formulation and geometrical ideas of linear Programmeming problems, simplex method, revised simplex method, duality, sensitivity analysis, transportation and assignment problems. Nonlinear optimization: basic theory, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, convex optimization. Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods
**Suggested Readings :**

**D. G. Luenberger,**Linear and Nonlinear Programmeming, 2nd Ed., Kluwer, 2003.**M. C. Joshi,**Optimization: Theory and Practice, Alpha Science International, Ltd; 1st edition, 2004.**S. S. Rao,**Optimization: Theory and applications.**E.K.P. Chong and S.H. Zak,**An Introduction to Optimization, 2nd Ed., Wiley, 2001 (WSE, 2004).

**Title of the course :**Algebra**Course Code :**MTH4012**Pre-requisites :**Elementary set theory**Proposed Curriculum :**- Groups: Binary operation and its properties, Definition of a group, Examples and basic properties, Subgroups, Cyclic groups, Dihedral Groups, Permutation groups, Cayley’s theorems. Coset of a subgroup, Lagrange’s theorem, Order of a group, Normal subgroups, Quotient group, Homomorphisms, Kernel Image of a homomorphism, Isomorphism theorems, Direct product of groups, Group action on a set, Semi-direct product, Sylow’ theorems, Structure of finite abelian groups. Rings: Definition, Examples and basic properties. Zero divisors, Integral domains, Fields. Characteristic of a ring, Quotient field of an integral domain. Subrings, Ideals, Quotient rings, Isomorphism theorems, Ring of polynomials. Prime, Irreducible elements and their properties, UFD, PID and Euclidean domains. Prime ideal, Maximal ideals, Prime avoidance theorem, Chinese remainder theorem. Fields: Field of fractions, Gauss lemma, Fields, field extension, Galois theory.
**Suggested Readings :**

**E. Artin, Algebra,**Prentice-Hall of India.**I.N. Herstein,**Topics in Algebra, Wiley, 2008.**J. Gallian,**Contemporary Abstract Algebra, 4th edition, Narosa, 2009.**J. B. Fraleigh,**A First Course in Abstract Algebra, Pearson, 2003.

**Title of the course :**Probability & Statistics**Course Code :**MTH4012**Pre-requisites :**Elementary Calculus**Proposed Curriculum:**- Probability:- Axiomatic definition, Properties. Conditional probability, Bayes rule and independence of events. Random variables, Distribution function, Probability mass and density functions, Expectation, Moments, Moment generating function, Chebyshev’s inequality. Special distributions: Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson, Uniform, Exponential, Gamma, Normal, Joint distributions, Marginal and conditional distributions, Moments, Independence of random variables, Covariance, Correlation, Functions of random variables, Weak law of large numbers, P. Levy’s central limit theorem (i.i.d. finite variance case), Normal and Poisson approximations to binomial, Statistics:- Introduction: Population, Sample, Parameters. Point Estimation: Method of moments, MLE, Unbiasedness, Consistency, Comparing two estimators (Relative MSE). Confidence interval estimation for mean, difference of means, variance, proportions, Sample size problem, Test of Hypotheses:-N-P Lemma, Examples of MP and UMP tests, p-value, Likelihood ratio test, Tests for means, variance, Two sample problems, Test for proportions, Relation between confidence intervals and tests of hypotheses, Chi-square goodness of fit tests, Contingency tables, SPRT, Regression Problem:- Scatter diagram, Simple linear regression, Least squares estimation, Tests for slope and correlation, Prediction problem, Graphical residual analysis, Q-Q plot to test for normality of residuals, Multiple regression, Analysis of Variance: Completely randomized design and randomized block design, Quality Control: Shewhart control charts and Cusum charts.
**Suggested Readings:**

**W. Feller,**Introduction to Probability Theory and its Applications, Wiley Eastern, New Delhi**V.K. Rohatgi,**An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern, New Delhi.**A. Papoulis & S.U. Pillai.**Probability, Random Variables and Stochastic Processes (4^{th}Ed.), McGraw-Hill Higher Education.**Miller & Freund,**Probability and Statistics for Engineers, 7th Edition, Pearson-Prentice Hall, 2005.

**Title of the course :**Analysis-II**Course Code :**MTH4022**Pre-requisite :**Analysis-I**Proposed Curriculum :**- Lebesgue measure on R
^{n}: Introduction, outer measure, measurable sets, Lebesgue measure, regularity properties, measurable functions, Egoroff’s theorem, Lusin’s theorem. Lebesgue integration: Simple functions, Lebesgue integral of a bounded function over a set of finite measure, bounded convergence theorem, integral of nonnegative functions, Fatou’s Lemma, monotone convergence theorem, the general Lebesgue integral, Lebesgue convergence theorem, change of variable formula. Product measure, Fubini’s theorem, Differentiation and integration: Functions of bounded variation, differentiation of an integral, absolutely continuity, Lp-spaces: The Minkowski’s inequality and Hölder’s inequality, completeness of Lp, denseness results in Lp. Fourier series: Definition of Fourier series, formulation of convergence problems, The L2 theory of Fourier series, convergence of Fourier series. **Suggested Reading :**

**Walter Rudin,**Real and complex analysis, McGraw-Hill, New York, 1966**H.L. Royden,**Real Analysis, 3rd ed., Macmillan, 1988.**P.R. Halmos,**Measure Theory, Graduate Text in Mathematics, Springer-Verlag,

**Title of the course :**Complex Analysis**Course Code :**MTH4032**Pre-requisites :**Analysis-I**Proposed Curriculum :**- Inequalities involving complex numbers, The spherical representation of the complex plane C, Limit, Continuity and differentiability, Necessary and sufficient conditions for differentiability(Cauchy-Riemann equations), Analytic functions, Polynomials, Rational functions , Harmonic conjugates, Elementary functions, Connectedness, Compactness in C, Heine-Borel Property in C, Bolzano-Weierstrass theorem for C, Continuous functions on compact subsets of C, Arcs and closed curves, Analytic functions in regions, Conformal mapping, Linear transformation, The linear group, The cross ratio, Symmetry. Line intergrals, Rectifiable arcs, Line integrals as functions of arcs, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disk, The index of a point with respect to a closed curve, Cauchy’s intergral formula, Higher derivatives, Morera’s theorem, Liouville’s theorem, Power series expansions, The Weierstrass theorem, Taylors Theorem, Laurent’s Theorem, Classification of singularities, Classical theorem of Weierstrass concerning behavior of a function in the neighborhood of an essential singularity, zeros of analytic functions, open mapping theorem, The maximum principle, Schwarz’s lemma, Residue theorem and applications, The argument principle, Rouche’s theorem, Evaluation of definite integrals.
**Suggested Readings :**

**Lars V. Ahlfors,**Complex Analysis, McGraw-Hill International Editions.**J.B. Conway,**Functions of one complex variable, Narosa, New Delhi.**T.W. Gamelin,**Complex Analysis, Springer International Edition, 2001.**R.V. Churchill and J.W. Brown,**Complex Variables and Applications, Wiley.

**Title of the course :**Numerical Analysis**Course Code :**MTH4042**Prerequisites :**Basic linear algebra**Proposed Curriculum :**- This course will emphasize the development of numerical algorithms to provide solutions to common problems formulated in science and engineering. The primary objective of the course is to develop the basic understanding of the construction of numerical algorithms, and more importantly, the applicability and limitations of their appropriate use. The emphasis of the course will be the thorough study of numerical algorithms to understand (1) the guaranteed accuracy that various methods provide, (2) the efficiency and scalability for large scale systems. and (3) issues of stability.
- Topics include the standard algorithms for numerical computation: Numerical methods for systems of linear numerical methods for systems of linear and nonlinear equations, interpolation and approximation, differentiation and integration, and differential equations. Specific topics include basic direct and iterative methods for linear systems; classical root finding methods; Newton’s method and related methods for non-linear systems; fixed-point iteration; polynomial, piecewise polynomial, and spline interpolation methods: least-squares approximation; orthogonal functions and approximation; basic techniques for numerical differentiation; numerical integration, including adaptive quadrature; and methods for initial-value problems for ordinary differential equations. Both theory and practice are examined. Error estimates, rates of convergence, and the consequences of finite precision arithmetic are also discussed. Topics from linear algebra and elementary functional analysis will be introduced as needed. These may include norms and inner products, orthogonality and orthogonalization, operators and projections, and the concept of a function space.
- An important component of numerical analysis is computational implementation of algorithms which were developed in the course in order to observe first-hand the issues of accuracy, computational work effort, and stability. Assignments were included computational experiments in a Programmeming language of the student's choice.
**Suggested Reading :**

**S. D. Conte, C. d. Boor,**Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition, McGraw-Hill International Editions.**Richard L. Burden and J. Douglas Faires,**Numerical Analysis, Brooks Cole.

**Title of the course :**Ordinary Differential Equations**Course Code :**MTH4052**Pre-requisites :**Calculus, Linear algebra**Proposed Curriculum :**- Review of solution methods for first order, Existence and uniqueness of initial value problems: Picard's and Peano's Theorems, Gronwall's inequality, Picard’s theorem for systems, Continuation of solutions and maximal interval of existence, Continuous dependence, General solution of second order and higher equations, Higher order linear equations and linear Systems: fundamental solutions, Wronskian, variation of constants, exponential matrix and asymptotic behaviour of solutions, Power series methods with properties of Legendre polynomials and Bessel functions., oscillations, and Sturm comparison theorems, boundary value problems for second order equations: Green's function, Two dimensional autonomous systems and phase space : analysis and classification of critical points, Asymptotic behavior and stability of linear systems, linearized stability and Lyapunov methods.
**Suggested Reading :**

- E. Kreyszig, Advanced Engineering Mathematics. (8
^{th}Edition) - G.F. Simmons, Differential Equations with Applications and Historical Notes. New York: McGraw-Hill, 1991.
- L. Perko, Differential Equations and Dynamical Systems, Texts in Applied
Mathematics, Vol. 7, 2
^{nd}ed., Springer Verlag, New York, 1998. - V Raghavendra, V Lakshmikantam, S Deo, Text book of ordinary differential equations, Tata McGraw-Hill Education, 2008

**Title of the course :**Topology**Course Code :**MTH5011**Pre-requisites :**Analysis-I**Proposed Curriculum:****Topological Spaces:**open sets, closed sets, neighbourhoods, bases, sub bases, limit points, closures, interiors, continuous functions, homeomorphisms. Examples of topological spaces: subspace topology, product topology, metric topology, order topology. Quotient Topology: Construction of cylinder, cone, Moebius band, torus, etc.**Connectedness and Compactness:**Connected spaces, Connected subspaces of the real line, Components and local connectedness, Compact spaces, Heine-Borel Theorem, Local -compactness. Separation Axioms: Hausdorff spaces, Regularity, Complete Regularity, Normality, Urysohn Lemma, Tychonoff embedding and Urysohn Metrization Theorem, Tietze Extension Theorem. Tychnoff Theorem, One-point Compactification. Complete metric spaces and function spaces, Characterization of compact metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Category Theorem. Applications: space filling curve, nowhere differentiable continuous function.**Suggested Readings :**

**J. R. Munkres,**Topology, 2nd Edition, Pearson Education (India), 2001.**H. L. Royden,**Real Analysis, 3rd edition, Prentice Hall of India, 1995.**G.F. Simmons,**Introduction to Topology and Modern Analysis, McGraw-Hill, New York, 1963.**J. L. Kelley,**General Topology, Van Nostrand, 1955.

**Title of the Course :**Functional Analysis**Course Code :**MTH5021**Prerequisites :**Analysis-1**Proposed Curriculum :**- Fundamentals of normed linear spaces: Normed linear spaces, Riesz lemma, characterization of finite dimensional spaces, Banach spaces. Bounded linear maps on a normed linear space: Examples, linear map on finite dimensional spaces, finite dimensional spaces are isomorphic, operator norm. Hahn-Banach theorems: Geometric and extension forms and their applications. Three main theorems on Banach spaces: Uniform boundedness principle, divergence of Fourier series, closed graph theorem, projection, open mapping theorem, comparable norms. Dual spaces and adjoint of an operator: Duals of classical spaces, weak and weak* convergence, Banach Alaoglu theorem, adjoint of an operator. Hilbert spaces: Inner product spaces, orthonormal set, Gram-Schmidt ortho-normalization, Bessel’s inequality, Orthonormal basis, Separable Hilbert spaces. Projection and Riesz representation theorem: Orthonormal complements, orthogonal projections, projection theorem, Riesz representation theorem. Bounded operators on Hilbert spaces: Adjoint, normal, unitary, self adjoint operators, compact operators, eigen values, eigen vectors, Banach algebras. Spectral theorem: Spectral theorem for compact self adjoint operators, statement of spectral theorem for bounded self adjoint operators.
**Suggested Readings:**

**B. V. Limaye,**Functional Analysis.**K. Yoshida,**Functional Analysis, Springer.**S. Nanda and B. Choudhari,**Functional Analysis With Application, New Age International Ltd.**S. C. Bose,**Introduction to Functional Analysis, Macmillan India Ltd.

**Title of the course :**Partial Differential Equations**Course Code :**MTH5031**Pre-requisite :**ODEs, Functional Analysis**Proposed Curriculum:**- Mathematical models leading equations. First order PDE solutions by characteristics, Cauchy-Kowalewski’s theorem. Higher order equations and characteristics. Classification of second order equations. Riemann’s method and applications. One dimensional wave equation and De’Alembert’s method. Solution of three dimensional wave equation. Method of decent and Duhamel’s principle. Solutions of equations in bounded domains and uniqueness of solutions. BVPs for Laplace’s and Poisson’s equations. Mean value property, Maximum principle and applications. Green’s functions and properties. Existence theorem by Perron’s method. Heat equation, Maximum principle. Uniqueness of solutions via energy method. Uniqueness of solutions of IVPs for heat conduction equation. Green’s function for heat equation, Finite difference method for the existence and computation of solution of heat conduction equation.
**Suggested Readings:**

**E. DiBenedetto,**Partial Differential Equations, Birkhauser, Boston, 1995.**L.C. Evans,**Partial Differrential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, Providence, 1998.**F. John,**Partial Differential Equations, 3rd ed., Narosa Publ. Co., New Delhi,1979.

**Title of the Course :**Mathematical Methods**Course Code :**MTH5041**Pre-requisites :**Analysis**Proposed Curriculum:**- Definition and classification of linear integral equations. Conversion of initial and boundary value problems into integral equations. Conversion of integral equations into differential equations. Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel. Basic concepts of the calculus of variations such as functional, extremum, variations, function spaces. Euler-Lagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations. Invariance of Euler`s equations. Variational problem in parametric form. Fourier Series and Fourier Transforms: Orthogonal set of functions, Fourier series, Fourier sine and cosine series, Half range expansions, Fourier integral Theorem, Fourier Transform, Fourier Cosine Transform, Fourier Sine Transform, Transforms of Derivatives, Fourier transforms of simple Functions, Fourier transforms of Rational Functions, Cosine and Sine Transforms, Inversion Theorem. Laplace Transform: Definition, Transform of some elementary functions, rules of manipulation of Laplace Transform, Transform of Derivatives, relation involving Integrals, the error function, Transform of Bessel functions, Periodic functions, convolution of two functions, Inverse Laplace Transform of simple function, Tauberian Theorems, Solution of Differential Equations Hankel Transform: Elementary properties, Inversion theorem, transform of derivatives of functions, transform of elementary functions.
**Suggested Reading:**

**Jerry, Abdul J.,**Introduction to Integral Equations with applications, Clarkson University Wiley Publishers, 2nd Revised edition edition (11 October 1999)**Elsgolc, L.E.:**Calculus of Variations, Dover Publications Inc. (15 January 2007)**Loknath Debnath,**Integral Transforms and their applications, Chapman and Hall/CRC; 2 edition ,2006.**Donald A. Mc Quarrie:**Mathematical Methods for Scientists & Engineers, University Science Books, Edition: 2008.**Cordumeanu, C. :**Integral Equations and Applications, Cambridge University Press, 1991.**Curant, R. and D. Hilbert:**Methods of Mathematical Physics, Vol I. Interscience Press, 1953.

Ian N. Sneddon , The use of Integral Transforms ,McGraw Hill; Second Printing edition ,1972.

**Title of the course :**Numerical Methods for PDEs**Course Code :**MTH5041**Pre-requisites :**ODE & PDE, Elementary Linear Algebra and Numerical analysis**Proposed Curriculum :**- Basic linear algebra – vector and matrix norms and related theorems, Some physics behind the PDEs, Introduction to finite differences, Parabolic equations in 1-D: Explicit and implicit finite difference schemes, Truncation error and consistency, Stability analysis (matrix method, maximum principle, Fourier analysis, energy method),
- Maximum principle and convergence, Lax equivalence theorem, general boundary conditions, split operator methods, multilevel difference schemes, nonlinear problems, Parabolic equations in 2D: explicit and implicit methods, ADI methods,
- Hyperbolic Equations: Vector systems and characteristics, Shock formation, The entropy condition, domain of dependence, The CFL theorem, Dissipation and dispersion, Upwind scheme, Lax-Wendroff scheme, Lax-Friedrich scheme, Leap-Frog scheme, conservative schemes, Non-linear system, Artificial viscosity, Conservation properties, Mac-Cormack’s scheme, Richmyer method, Conservation laws in two space dimensions.
- Introduction, Solvability, Maximum principle, Dirichlet, Neumann and mixed problems, Basic iterative schemes, Direct factorization methods and successive over-relaxation (S.O.R.), ADI and conjugate gradient methods.
**Suggested Reading:**
.

**K.W. Morton and D.F. Mayers,**Numerical Solution of Partial Differential Equations, Cambridge, 2nd Edition.**Thomas,**Numerical Partial Differential Equations: Finite Difference Methods, Springer, 1995.**LeVeque, Randall J.,**Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM 2007.- Finite Volume Methods for Hyperbolic Problems. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002.

**Title of the Course :**Finite Element Methods**Course Code :**MTH5022**Prerequisites :**Basic linear algebra, elementary numerical Analysis**Proposed Curriculum :**In this course we first study how to develop basic mathematical theory of finite element method, construct finite element function spaces based on triangular or rectangular element domains and piecewise polynomials approximation; Ritz-Galerkin methods and variational crimes; energy and maximum norm estimates; mixed finite element method; applications to diffusion-reaction problems; Laplace, Poisson and heat equations; computer implementations for each methods.**Suggested Reading:**

**S. C. Brenner and L. R. Scott,**The mathematical theory of finite element methods, 3rd ed., Springer, 2008.**Claes Johnson,**Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009.**P.G. Ciarlet,**The Finite Element Method for Elliptic Problems, SIAM: Society for Industrial and Applied Mathematics; 2nd edition, April 2002.**Vidar Thomee,**Galerkin Finite Element Methods for Parabolic Problems; Springer Berlin Heidelberg, 2006.

**Title of the Course :**Computational Topology**Course Code :**MTH5051**Pre-requisites :**Topology, Consent of the instructor**Proposed Curriculum:**- Overview of computational topology, point set topology, surface topology, Simplical & cubical complexes, Group theory, Homotopy, Simplical homology, Filtrations & persistent homology, Computing homology, Data Analysis, Learning & Visualization: Topology of point cloud data, Manifold Learning from point cloud data sets (PCD), Persistence and its Stability in PCDs, Natural Image Classification, Homology computation using harmonic analysis, Networks and Sensing: Coverage problems in sensor networks –I, Coverage problems in sensor networks –II, Landmarks, routing and homology feature size in sensor networks, Robotics and Coordination: Morse theory – continuous, discrete & combinatorial, Navigation in robotics, Configuration spaces – I: Distributed coordination, Configuration spaces – II: Reconfigurable systems,
**Suggested Readings:**

**J. R. Munkres,**Topology, 2nd Edition, Pearson Education (India), 2001.**J. Milnor,**Morse Theory.**Y. Matsumoto,**An introduction to Morse Theory.**A. Zomorodian,**Topology for Computing, Cambridge University Press 2009.

**Title of the Course :**Number Theory**Course Code :**MTH5061**Prerequisites :**Elementary Set Theory**Proposed Curriculum :**- Divisibility, Division Algorithm, Greatest Common Divisor, Euclidean Algorithm, Primes and their distribution, Analytic proof of the prime number theorem, Congruences, Congruences with prime modulus, Representation of integers in baseb, Some Diophantine equations, The Chinese remainder theorem, Sum of integer squares, Fermat's little theorem, Wilson's theorem and applications, Primitive roots, Quadratic reciprocity, Continued fractions and rational approximations, Pell's equation, Number theoretic functions, Euler phi function and MÃ¶bius inversion formula, Applications to primality testing & cryptography, Dirichlet series and Euler products.
**Suggested Readings:**

**D. Burton,**Elementary Number Theory, McGraw Hill.**I. Niven, H.S. Zuckerman,**An Introduction to the Theory of Numbers (5th Edition), Wiley.**Keneth Rosen,**Elementary Number Theory and its Applications(5th Edition), McGraw Hill.**T. Koshy,**Elementary Number Theory with Applications.

**Title of the Course :**Differential Geometry**Course Code :**MTH5071**Pre-requisites :**Analysis-I**Proposed Curriculum:**- Parametrized curves in R3, length of curves, integral formula for smooth curves, regular curves, parametrization by arc length. Osculating plane of a space curve, Frenet frame, Frnet formula, curvatures, invariance under isometry and reparametrization. Discussion of the cases for plane curves, rotation number of a closed curve, osculating circle, ‘Umlaufsatz’.Smooth vector fields on an open subset of R3, gradient vector field of a smooth function, vector field along a smooth curve, integral curve of a vector field. Existence theorem of an integral curve of a vector field through a point, maximal integral curve through a point. Level sets, examples of surfaces in R3. Tangent spaces at a point. Vector fields on surfaces. Existence theorem of integral curve of a smooth vector field on a surface through a point. Existence of a normal vector of a connected surface. Orientation, Gauss map. The notion of geodesic on a surface. The existence and uniqueness of geodesic on a surface through a given point and with a given velocity vector thereof. Covariant derivative of a smooth vector field. Parallel vector field along a curve. Existence and uniqueness theorem of a parallel vector field along a curve with a given initial vector. The Weingarten map of a surface at a point, its self-adjointness property. Normal curvature of a surafce at a point in a given direction. Principal curvatures, first and second fundamental forms, Gauss curvature and mean curvature. Surface area and volume. Surfaces with boundary, local and global stokes theorem. Gauss-Bonnet theorem.
**Suggested Readings:**

**B O’Neill,**Elementary Differential Geometry, Academic Press (1997).**A. Pressley,**Elementary Differential Geometry, Springer (Indian Reprint 2004).**J.A. Thorpe,**Elementary topics in Differential Geometry, Springer (Indian reprint, 2004).**Michael Spivak,**A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1994.

**Title of the course :**Spectral Element Methods**Course Code :**MTH5032**Prerequisites :**Basic/Numerical Linear Algebra, C/FORTAN/C++/MATLAB**Proposed Curriculum :**- Galerkin, Collocation & Tau methods, Spectral approximation, The Fourier system, Continuous & discrete Fourier expansion, Orthogonal polynomials in (-1,1), Fundamentals of spectral methods for PDEs, Temporal discretization, The Galerkin Collocation method, Implicit spectral equations, Initial Value Problems, Boundary Value Problems, Unbounded Domains, Polar Coordinates, Iteration and Preconditioning, Spectral Element Method, Case of non-smooth solutions.
**Suggested Readings:**

**C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang,**Spectral Methods in Fluid Dynamics: Fundamentals in Single Domains by , Springer (2005).**Constantine Pozrikidis,**Introduction to Finite and Spectral Element Methods using MATLAB, Chapman & Hall CRC (2005).**Lloyd N. Trefethen,**Spectral methods in MATLAB.**J. P. Boyd,**Chebyshev and Fourier Spectral Methods.**G. E. Karniadakis and S. J. Sherwin,**Spectral/hp Element Methods for CFD, Oxford University Press.

**Title of the Course :**Computational Fluid Dynamics**Course Code :**MTH5042**Prerequisites :**Elementary Numerical Analysis, C/FORTRAN/MATLAB**Proposed Curriculum:**- Introduction to Computational Fluid Dynamics and Principles of Conservation: Continuity Equation, Navier Stokes Equation, Energy Equation and General Structure of Conservation Equations, Classification of Partial Differential Equations and Physical Behavior, Approximate Solutions of Differential Equations: Error Minimization Principles, Variational Principles and Weighted Residual Approach, Fundamentals of Discretization: Finite Element Method, Finite Difference and Finite Volume Method, Finite Volume Method: Some Conceptual Basics and Illustrations through 1D Steady State Diffusion Problems, Boundary Condition Implementation and Discretization of Unsteady State Problems, Important Consequences of Discretization of Time Dependent Diffusion Type Problems and Stability Analysis : Consistency, Stability and Convergence, LAX Equivalence theorem, Grid independent and time independent study, Stability analysis of parabolic equations (1D unsteady state diffusion problems): FTCS (Forward time central space) scheme, Stability analysis of parabolic equations (1D unsteady state diffusion problems): CTCS scheme (Leap frog scheme), Dufort Frankel scheme, Stability analysis of hyperbolic equations: FTCS, FTFS, FTBS and CTCS Schemes, Finite Volume Discretization of 2D unsteady State Diffusion type Problems, Solution of Systems of Linear Algebraic Equations: Elimination Methods, Iterative Methods, Gradient Search Methods, Discretization of Convection Diffusion Equations: A Finite Volume Approach, Discretization of Navier Stokes Equations: Stream Function Vorticity approach and Primitive variable approach, SIMPLE Algorithm,SIMPLER Algorithm, Unstructured Grid Formulation , Introduction to Turbulence Modeling.
**Suggested Readings:**

**T. J. Chung,**Computational Fluid Dynamics, Cambridge University Press.**H. K. Versteeg and W. Malalasekera,**An Introduction to Computational Fluid Dynamics, Longman Scientific & Technical.**J. H. Ferziger and M. Peric,**Computational Methods for Fluid Dynamics, Springer.**John D. Anderson Jr,**Computational Fluid Dynamics, McGraw Hill Book Company.**J. Blazek,**Computational Fluid Dynamics: Principles and Applications, Elsevier.

**Title of the Course :**Fractal Geometry and Applications**Course Code :**MTH5081**Prerequisites :**Basic Linear Algebra, Basic Courses in Real & Complex Analysis**Proposed Curriculum:**- Introduction: Euclidean Spaces, Metric Spaces and its Properties, Space of Fractals, Contraction Mappings, Affine and Mobius Transformations, Iterated Function System (IFS), Attractors, Random Iteration Algorithm, Collage Theorem. Fractal Interpolation: Address of points of Fractals, Dynamical Systems, Simulation of 2D and 3D Fractal objects, Fractal and Housdroff Dimension, Fractal Interpolation Function (FIF): Construction and Properties, Hidden Variable FIF, Coalescence FIF, Fractal Dimension of FIF, Interpolation Error Bounds. Fractals in dynamical systems: Classification of Periodic Points, Kinds of Bifurcation, Chaotic Sets, Julia Sets, Fatou Sets, Mandelbrot Sets, Simulation and Interpolation. Applications of Fractals: Application of Fractal Interpolation in Image Compression, Fractal Antenna and in Simulation of Highly Irregular Objects in Nature.
**Suggested Readings:**

**Michael Barnsley,**Fractals Everywhere (03rd Ed.), Dover Publication (2012).**K.J. Falconer,**Fractal Geometry Mathematical Foundations and Applications, John Wiley (1997).**B. Mandelbrot,**Fractal Geometry of Nature, Henry Holt and Company (1983).

**Title of the course :**Elements of Sobolev Spaces and Applications**Course Code :**MTH5091**Prerequisites :**Real Analysis, Functional Analysis, ODEs & PDEs**Proposed Curriculum:**- In many problems of mathematical physics and variational calculus it is not sufficient to deal with the classical solutions of differential equations. It is necessary to introduce the notion of weak derivatives and to work in the so called Sobolev spaces. Sobolev spaces are a fundamental tool in the modern study of partial differential equations. The topics covered in this course are: an introduction to the theory of Distributions, Sobolev Spaces, imbedding theorem, Poincare inequality, Fourier transformation, Trace theorem, the results are applied to the study of weak solutions of elliptic boundary value problems.
**Suggested Reading:**

**S. Kesavan,**Topics in Functional Analysis and Applications, New Age International, 1989.**Robert A. Adams, John J. F. Fournier,**Sobolev Spaces, Academic Press, 2003.

**Title of the Course :**Applied harmonic analysis**Course Code :**MTH5052**Pre-requisites :**Consent to teacher**Proposed Curriculum :**- Basic Fourier Analysis - a review Convolutions, Multipliers and Filters, Poisson Summation Formula, Shannon Sampling Discrete Fourier Transform, Fast Fourier Transform, Discrete Wavelets, Continuous Wavelets, Uncertainty Principles, Radar Ambiguity, Phase Retrieval, Radon Transform, Basic Properties, Convolution and Inversion, Computerized Tomography.
**Suggested Readings:**

**G.B. Folland,**A course in Abstract Harmonic Analysis. Studies in Advanced mathematics, CRC press, 1995.**E.M. Stein and R. Shakarchi,**Fourier Analysis: An Introduction, Princeton University Press, Princeton 2003.**F. Natterer,**The Mathematics of Computerized Tomography, SIAM.

**Title of the Course:**Wavelets and Applications**Course Code :**MTH5062**Pre-requisites :**Basic Linear Algebra**Proposed Curriculum:****Basics of Fourier Analysis:**Discrete and Continuous Fourier Transform, Fast Fourier Transform, Convolutions, Plancherel’s Formula, Parseval’s Formula, Inversion Formulas. Definition and Basic Properties of the Discrete Fourier Transfer (DFT), Plancherel’s Formula, Parseval’s Formula, The Fast Fourier Transform, Inversion Formulas. Fourier Transformation and frequency of a signal, Fourier basis as frequency, behavior of DFT under translation, Translation invariant linear transformation and signals, Convolution, Fast Fourier transforms (FFT).- Construction of Wavelets on Z_N (finite dimensional case ), advantage of Wavelets over Fourier Transform, Applications of Wavelets on noise removal of Signal, data compression, time and frequency analysis using Wavelets, computation of change of basis using FFT. First-stage wavelet basis, Haar basis, First-stage Shannon wavelet basis, Downsampling and Upsampling operator, Filter bank analysis, perfect reconstruction of a signal using filter bank approach, Construction of pth stage wavelet, Applications and examples, pth stage Haar System, pth stage Shannon wavelets, time/frequency analysis of signals, Daubechies’s D6 Wavelets, image compression using various wavelet basis.
**Basics of Functional Analysis:**Banach Space, Hilbert Space. Complete orthonormal set in Hilbert space, Plancherel’s Formula, Parseval’s Formula, orthogonal projection. Set of square integrable functions: L^2 space, L^p space, Fourier transformation and convolution on L^2(Z), The Wavelet Transform : Definition, Examples, Inversion Formula, Plancherel’s Formula, Discrete Wavelet Transform, Continuous Wavelet Transform, Shannon Wavelets, Daubechies’s Wavelets, Fast Wavelet Transform, Applications and Examples including denoising signal, image compression.- t-dilation, approximation identity, convolution, Fourier transformation and Fourier inversion on L^1(R), compact support, denseness, Plancherel’s Formula, Parseval’s reation, Fourier transformation and Fourier inversion on L^2(R), Complete orthonormal set in L^2(R), scaling function, scaling idenities, Multiresolution analysis (MRA), Mallat’s theorem, Wavelet system in L^2(R), Haar MRA.
**Suggested Reading:**

**Michael W. Frazier,**An Introduction to Wavelets through Linear Algebra.**D. Walnut,**An Introduction to Wavelet Analysis, Birkhauser, Boston (2001).

**Title of the Course :**Parallel Numerical Algorithms**Course Code :**5072**Prerequisites :**Basic/Numerical Linear Algebra, Programmeming Languages: C/FORTAN/C++**Proposed Curriculum:**- Fundamentals of parallel computing; Parallel techniques and algorithms; Parallel algorithms for linear algebraic equations; Design of parallel algorithms for eigen value problems; Parallel issues of factorization: singular value decomposition and related problems; Parallel implementation of classical iterative methods; Conjugate gradient method; Parallel methods for ordinary and partial differential equations.
**Suggested Readings:**

**T.L. Freeman, C. Phillips,**Parallel Numerical Algorithms, Prentice Hall International Series in Computer Science, (1992).**David E. Keyes, Ahmed Sameh, V. Venkatakrishnan,**Parallel Numerical Algorithms (ICASE Interdisciplinary Series in Science and Engineering), Kluwer Academic Publishers (1997).

**Title of the Course :**Image Processing and Computer Graphics**Course Code :**MTH5082**Prerequisite :**Consent of the instructor.**Course Description:**- Image perception; image types/applications; image restoration and enhancement; edge/boundary detection; image transformation; image segmentation; statistical and syntactical pattern recognition; image information measures and compression; Graphics hardware; raster algorithms; geometric transformations; 2D/3D interactive graphics; 3D viewing and perspective projections; color and lighting models; hidden surface removal; modeling hierarchies; fractals; curved surfaces.
**Suggested Reading:**

**Hill, F.S.,**Computer Graphics using Open GL, 3rd ed., Prentice Hall, 2007.**Milan Sonka,**Vaclav Hlavac, Roger Boyle, Image Processing, Analysis, and Machine Vision. Thomson Learning, 3 edition.**Richard Szeliski**(Microsoft Research). Computer Vision: Algorithms and Applications**Kleinberg and Tardos,**Algorithm Design, Addison Wesley, 2006.**Gonzalez and Woods,**Digital Image Processing, Prentice Hall, second edition, 2002.**Stan Z. Li,**Markov Random Field Modeling in Image Analysis, Springer, third edition, 2009.

**Title of the Course :**Mathematics in Multimedia**Course Code :**MTH5092**Pre-requisites:**Linear Algebra, Elementary numerical analysis**Proposed Curriculum:****Numbers and Arithmetic:**Integers, Modular arithmetic, Representing integers in binary computers, Integer arithmetic, Real numbers, Precision and accuracy, Representing real numbers, Propagation of error, Time and Frequency: Fourier analysis, Periodic functions, Localization, Fourier series, Discrete Fourier analysis, Discrete Fourier transform, Discrete Hartley transform, Discrete sine and cosine transforms, Sampling and Estimation: Approximation and Sampling, Polynomial interpolation, Piecewise interpolation, Sampling spaces, Measurement and Estimation, Quantization, precision, and accuracy, Estimation, Scale and Resolution: Wavelet Analysis, Haar functions, The affine group, Wavelet transforms, Discrete Wavelet Transforms, Multiresolution analysis (MRA), From MRAs to filters, From filters to discrete wavelet transforms, Lifting, Redundancy and Information: Information Source Coding, Lossless encoding, Efficient coding, Huffman’s algorithm, Error Correction and Detection, Parity bits, Hamming codes, Checksums and cyclic redundancy codes.**Suggested Readings:**

**M.V. Wickerhauser,**Mathematics for Multimedia, Birkhaeuser Verlag**M.V. Wickerhauser,**Adapted Wavelet Analysis from Theory to Software.

**Title of the Course :**Data Mining: A Mathematical Perspective**Course Code :**MTH5102**Prerequisites :**Statistical Course**Proposed Curriculum:****Introduction:**Basic Data Mining Tasks, Data Mining Issues, Data Mining Metrics, Data Mining from a Database Perspective.**Data Mining Techniques:**A Statistical Perspective on Data Mining, Similarity Measures, Decision Trees, Neural Networks, Genetic Algorithms.**Classification:**Statistical-Based Algorithms, Distance-Based Algorithms, Decision Tree-Based Algorithms, Neural Network-Based Algorithms, Rule-Based Algorithms, Combining Techniques.**Clustering:**Similarity and Distance Measures, Hierarchical Algorithms, Partitional Algorithms, Clustering Large Databases, Clustering with Categorical Attributes.**Association Rules:**Basic Algorithms, Parallel and Distributed Algorithms, Incremental Rules, Advanced Association Rule Techniques, Measuring the Quality of Rules.**Advanced Techniques:**Web Mining, Spatial Mining, Temporal Mining.**Suggested Readings:**

**J. Han and M. Kamber,**Data Mining: Concepts and Techniques, 2nd Ed. Morgan Kaufman. 2006.**M. H. Dunham,**Data Mining: Introductory and Advanced Topics, Pearson Education. 2001.**I. H. Witten and E. Frank,**Data Mining: Practical Machine Learning Tools and Techniques, Morgan Kaufmann. 2000.**D. Hand, H. Mannila and P. Smyth,**Principles of Data Mining, Prentice-Hall. 2001.

**Title of the Course :**Introduction to Stochastic Differential Equations**Course Code :**MTH5112**Pre-requisites :**Probability Theory**Proposed Curriculum:**- Random variables, mean and variance, Brownian motion, Stochastic integration and Ito

calculus, Stochastic differential equations, Random dynamics, Stochastic calculus in Hilbert spaces; Stochastic heat equation; Stochastic wave equation; Stability, distribution and moments of solutions, Analytical and approximation techniques; Stochastic numerical simulations via Matlab; Dynamical systems approach for stochastic partial differential equations, Stochastic bifurcation, Dynamical impact of noises; Stochastic flows and cocycles; Invariant measures, Lyapunov exponents and ergodicity; and applications to engineering and science and other areas. **Suggested Readings:**

**G. Da Prato and J. Zabczyk,**Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.**C. Prevot and M. Rockner,**A concise course on Stochastic Partial Differential Equations, Lecture notes in Mathematics, Springer.**Pao-Liu Chow,**Stochastic Partial Differential Equations, Chapman & Hall.

**Title of the Course :**Mechanics**Course Code :**MTH5122**Prerequisites :**Consent to teacher**Proposed Curriculum:****Generalised co-ordinates:**Degrees of freedom, Constraint, Principle of Virtual Work. Lagrangian formulation of Dynamics: Lagrange’s equations of motion for holonomic and non-holonomic systems. Ignoration of coordinates. Routh’s processs for the ignoration of co-ordinates. Rayleigh’s dissipation function. Calculus of variation and Euler-Lagrange differential equations, Brachistochrone problem. Configuration space and system point. Hamilton’s principle; Hamilton’s cannocical equations of motion. Principle of energy. Principle of least action, Canonical Transformations, Poisson Bracket. Theory of small oscillations. Normal co-ordinates. Euler’s dynamical equations of motion of a rigid body about a fixed point.**Suggested Readings:**

**H. Goldstein,**Classical Mechanics.**Synge and Griffith,**Principles of Mechanics.**K. C. Gupta,**Classical Mechanics of Particles and Rigid Bodies.**F. Gantmacher,**Lectures in Analytical Mechanics.

**Title of the Course :**Continuum Mechanics**Course Code :**MTH5132**Prerequisites :**Mechanics**Proposed Curriculum:**- Body, configuration, axiom of continuity, Motion of a body. Reference configuration, deformation. Material and spatial coordinates. Material and spatial time derivatives - relation between them. Velocity and acceleration. Physical components of acceleration in general curvilinear coordinates – in cylindrical coordinates and spherical polar coordinates, Deformation gradient tensor, Reynolds transport theorem for volume property. Principle of conservation of mass – Path lines, stream lines and streak lines. Material surface. (Bounding surface). Lagrange’s criterion for material surface, Polar decomposition theorem (Statement), polar decomposition of deformation gradient tensor – stretch and rotation tensors, principal stretches. Classical theory of infinitesimal deformations. Compatibility equations,Relative deformation gradient tensor, relative stretch tensors and relative rotation tensor. Rate-of –strain tensor-its principal values and invariants rate-of rotation tensor – vorticity vector; velocity gradient tensor, General principles of momenta balance; Euler’s laws of motion. Body forces and contact forces. Cauchy’s laws of motion: Stress equation of motion and symmetry of stress tensor for non-polar material. Energy balance–first and second laws of thermodynamics.
- Constitutive equation (stress-strain relations) for isotropic elastic solid. Elastic modulii Strain-energy function. Beltrami-Michel compatibility equations for stresses. Equations of equilibrium and motion in terms of displacement. Fundamental boundary value problems of elasticity and uniqueness of their solutions.
**Suggested Readings:**

**T. J. Chung,**Continuum Mechanics, (Prentice – Hall).**C. Truesdel,**Continuum Mechanics.**A. C. Eriengen,**Non-linear Theory of Continuous Media, (MacGraw-Hill).**Kolin, Kebel and Roze,**Theoretical Hydromechanics.

**Title of the Course :**Cryptography**Course Code :**MTH5142**Prerequisites :**Basic knowledge of Number Theory, Finite Field, Graph Theory**Proposed Curriculum:****Introduction to cryptology:**Polynomial rings over fields, Extension of fields, computation in GF(q), Root fields of polynomials, Vector space over finite fields. Some Simple Cryptosystems, Monoalphabatic and Polyalphabatic cipher, The Shift Cipher, The Substitution Cipher, The Affine Cipher, The Vigenere Cipher, The Hill Cipher, The Permutation Cipher, Stream Ciphers, Cryptanalysis, Some Cryptanalytic Attacks, Cryptanalysis of the Affine Cipher Cryptanalysis of the Substitution Cipher, Cryptanalysis of the Vigenere Cipher, Cryptanalysis of the Hill Cipher, Cryptanalysis of the LFSR Stream Cipher, Stream & Block ciphers, Mode of operations in block cipher.**Shannon's Theory of Perfect Secrecy:**Elementary Probability Theory, Perfect Secrecy, Birthday Paradox, Vernam One Time Pad, Random Numbers, Pseudorandom Numbers.**DES & AES:**The Data Encryption Standard (DES), Feistel Ciphers, Description of DES, Security analysis of DES, Differential & Linear Cryptanalysis of DES, Triple DES, The Advanced Encryption Standard(AES), Description of AES, analysis of AES.**Prime Number Generation:**Trial Division, Fermat Test, Carmichael Numbers, Miller Rabin Test, Random Primes.**Public Key Cryptography:**Principle of Public Key Cryptography, RSA Cryptosystem, Cryptanalysis of RSA: RSA & Factoring, Low exponent attack, Efficiency of RSA, RSA with Chinese Remainder Theorem (CRT).**Cryptographic Hash Functions:**Hash and Compression Functions, Compression Function from Encryption Functions, Hash Functions from Compression Functions, Security of Hash Functions, Iterated Hash Functions, SHA-1, Others Hash Functions, Message Authentication Codes.**Suggested Readings:**

**D R Stinson,**Cryptography: Theory & Practice, CRC Press.**Johannes A Buchmann,**Introduction to cryptography, Spiringer-verlag.**Bruce Schiener,**Applied Cryptography, Willey India.**William Stalling,**Cryptography and networks security: Principles and Practice, Prentice Hall.

**Title of the Course :**Graph Theory**Course Code :**MTH5152**Prerequisites :**None**Proposed Curriculum:**- Graphs, Simple Graphs, finite Graphs and Infinite Graphs, Null graphs, Degree of graphs, Isolated and pendant Vertices. Isomorphic Graphs, Subgraphs, Edge-disjoint Subgraphs, Vertex-disjoint Subgraphs, Complement of a subgraph and simple graphs. Walk, Path, Circuits and Operations on Graphs. Connected Graphs and Components. Complete Graphs, Bipartite Graphs, and regular Graphs.
- Euler graphs, Solution of Konigsberg Problem. Unicursal graphs, Randomly Eulerian graphs. Fleury’s Algorithm, Hamiltonian graphs and its applications. Chinese Postman Problem, Dijkstra’ Algorithm for shortest Path.
- Basic concepts of Trees, Minimally connected Graph. Rank and Nullity of Graphs. Spanning Trees, Minimal Spannin tree Kruskal’s Algorithm and Prim’s Algorithm. Cut-sets, Fundamental cut-sets, Edge-connectivity, Vertex Connectivity and Separability. Network-flows, Planarity, Kurotowski’s Planar graphs. Regions and Euler’s formula.
- Incidence matrix, Adjacency Matrix, Edge sequences between vertices, Path Matrix, Circuit Matrix, Fundamental Circuit Matrix, and Cut-Set Matrix. Types of Digraphs, Rooted and Binary tree, Level and Height of tree, Arborescence. Algebraic Expression and Polish Notation.
- Chromatic Number, Chromatic Polynomial, Coloring graphs, Welch Powell Algorithm. Boolean Algebra, Principal of Duality, Sub-algebra, Union and Intersection of Boolean algebra, Some questions on Boolean algebra.