Courses

Calculus: Rolle’s theorem, Lagranges theorem, Cauchy’s mean value theorem (Taylor’s and Maclaurin theorems with remainders), Indeterminate forms, Concavity and convexity of a curve, points of inflexion, maximum, minimum of a function, 2nd derivative test for max min, Asymptotes and curvature, Cartesian curve tracing, polar curve tracing.

Calculus of Several Variables: Limit, continuity and differentiability of functions of several variables, partial derivatives and their geometrical interpretation, differentials, derivatives of composite and implicit functions, derivatives of higher order and their commutativity, Euler’s theorem on homogeneous functions, harmonic functions, Taylor’s expansion of functions of several variables, maxima and minima of functions of several variables, Lagrange’s method of multipliers.

Vector Calculus: Double and triple integrals, Scalar and vector fields, level surfaces, directional derivative, Gradient, Curl, Divergence, line and surface integrals, theorems of Green, Gauss and Stokes. Beta and Gamma functions.

Ordinary Differential Equations: First order differential equations, exact, linear and Bernoulli’s form, second order differential equations with constant coefficients, Euler’s equations, particular integrals by: variation of parameters, undetermined coefficients, operator method, system of differential equations.

Linear Algebra: Vector spaces, subspaces, span, Linear dependence, independence of vectors, basis, dimension, linear transformations, range, kernel, rank, nullity of linear transformation, space of all linear transformations, Operator equations, matrix associated with a linear map, linear map associated with a matrix, elementary row operations, solution of algebraic equations, consistency conditions. Matrix inversion by row operations, Eigenvalues and eigenvectors, Hermitian and skew Hermitian matrices, orthogonal and unitary matrices, application to reduction of quadrics.

Complex Analysis: Limit, continuity, differentiability and analyticity of functions Cauchy-Riemann equations (cartesian and polar), Harmonic functions, Elementary complex functions, Line integrals, upper bounds for moduli of contour integrals, Cauchy’s integral theorem, Cauchy’s integral formula, derivatives of analytic functions, Power series, Taylor’s series, Laurent’s series, Zeros and singularities, Residue theorem, evaluation of improper integrals by residue theorem.

Unit I: Limit and continuity, Sequence and sequential criterion for limit, Differentiation, Mean value theorems, Indeterminate forms and evaluation of limits using L’Hopital rule, Riemann Integration and fundamental theorem of calculus, Improper integrals.

Unit II: Asymptotes, Increasing and decreasing functions, Concavity and Convexity, Points of inflection, Tracing of curves.

Unit III: Taylor’s expansion and error estimates, Power series and its convergence, Functions of two variables, Partial and directional derivatives, Gradient, Maxima and Minima, Lagrange’s multiplier method.

Unit IV: Vectors in three dimensional space, dot and cross product, lines and planes in space, Tangent to curves, Normal, Green’s theorem, Gauss divergence theorem, Stokes theorem.

Unit I: System of Linear Equations, Matrices and elementary row operations, Row reduced echelon form of matrices, Rank of matrices, Vector spaces, subspaces, bases and dimension, column and row spaces.

Unit II: Direct sum of vector space, quotient spaces, Linear Transformation and matrix representation of linear transformation, Rank-Nullity theorem, Dual spaces, transpose and determinants.

Unit III: Eigenvalues and eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem and Diagonalization.

Unit IV: Inner product spaces, Gram-Schmidt orthonormalization, orthogonal projections, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators.

Unit I: Basic Probability concepts, Random variables: Concept, cumulative distribution function, discrete and continuous random variables. Discrete random variables: Bernoulli’s trials, Binomial distribution, geometric distribution, Poisson distribution. Continuous random variables: Uniform distribution, exponential distribution, Gamma distribution, Normal distribution. Bivariate random variables: Joint distribution.

Unit II: Expected value of random variables: expectation, Raw and central moments of random variables, mean, variance, moment generating function, characteristics function, Median, Mode, covariance, correlation coefficient. Convergence in Probability: Tchebycheff’s inequality, properties of convergence, laws of large numbers, central limit theorem, approximation of distributions.

Unit III: Population, Finite and Infinite Population. Pseudo random variable, Theory of Sampling, parameter and statistic, estimator, Chi-square distribution, distribution of sample mean and variance, good estimator, Consistent & Unbiased estimates. Estimation of parameters: Maximum likelihood estimates, Confidence interval.

Unit IV: Testing of hypothesis: Problem, Different types of hypothesis, Formation of the problem – Probability of Type – I error, Probability of Type – II error, Best Critical region, Likelihood ratio testing, examples. Chi-square test of goodness of fit.

Unit I: Real Number system, Supremum and Infimum, Archimedean property, Sequences and Series, Cauchy sequence and completeness, Convergence tests for series.

Unit II: Riemann Integral, Riemann Integrable functions, the fundamental theorems of calculus and their consequences, Improper Integrals.

Unit III: Sequence and series of functions, Uniform convergence and its relation with continuity, differentiability and integrability.

Unit IV: Metric spaces, Interior points and limit points, open, closed and perfect sets. Connected and compact sets, Bolzano-Weierstrass theorem.

b Fundamental techniques of enumeration and construction of combinatorial structures, pigeon-hole principle, permutation, combination, summations. Introduction to recurrence relation and generating function, principle of inclusion-exclusion

Probability: Classical and axiomatic definitions of probability, Addition rule and conditional probability, Multiplication rule, Bayes' Theorem and independence, Discrete and continuous random variables: Uniform, Binomial, Geometric, Poisson, Exponential, Gamma, Normal distributions etc, Jointly distributed random variables. Mathematical expectation, Moments, Probability- and Moment-generating functions, Chebyshev's inequality, the strong and weak law of large numbers, Functions of Random Variables and their probability distributions

Statistics: The Central Limit Theorem, Distributions of the sample mean and the sample variance for a normal population, Sampling Distributions: Chi-Square, t and F distributions, Point and interval Estimation, maximum likelihood estimation, Null and alternative hypotheses, The critical and acceptance regions, Neyman-Pearson Fundamental Lemma, Tests for one sample and two samples problems for normal populations.

Unit I: Introduction to computer fundamentals, types of registers, memory unit, operating system, machine language, compiler, interpreter, Problem solving, algorithm and flowchart. Overview of C program, basic structure of C program, Constants, variables and data type.

Unit II: Operators and expression: Arithmetic, relational and logical operations. Assignment statements, conditional operators. Input and output statements. Formatted input and output. Decision making and branching – if, else if and go to statements. Looping structure: while, Do while, for statements.

Unit III: User defined function, types of functions, recursion, and concepts of parameter passing by values and by reference.

Unit IV: Arrays: single and multidimensional, searching and sorting. Characters and string and its manipulations, types of string functions. Introduction to pointers, pointer to pointer, pointer to array, array of pointers. Structure, union, file handling.

Unit I: Set theory: Sets, relations, equivalence relations, partially ordered sets, functions, countability, lattices and Boolean algebras.

Unit II: Logic: Well-formed formula, interpretations, propositional logic, predicate logic, theory of inference for propositional logic and predicate logic.

Unit III: Combinatorics: Permutations, combinations, recurrences, generating functions, partitions, principle of inclusion-exclusion, pigeonhole principle.

Unit IV: Graph Theory: Graphs and digraphs, special types of graphs, isomorphism, connectedness, Euler and Hamiltonian paths and circuits, trees, planar graphs, graph colouring.

Ordinary Differential Equations:


Unit I: First order differential equations, statement of existence and uniqueness theorem, exact, linear and Bernoulli equations, orthogonal trajectories, applications.

Unit II: Second and higher order linear differential equations with constant coefficients, method of undetermined coefficients, variation of parameters, operator method, Power series solutions, Legendre and Bessel equations.

Partial Differential Equations:


Unit III: Formulation of partial differential equations by eliminating arbitrary constants and functions, Lagrange’s linear equation, Charpit’s method, classification of second order PDEs.

Unit IV: Method of separation of variables, wave equation, heat equation, Laplace equation, Fourier series solutions.

Unit I: Errors in numerical computations, floating-point arithmetic, Solution of nonlinear equations: Bisection, Newton-Raphson, Secant, Fixed-point iteration methods, rate of convergence.

Unit II: Interpolation: Lagrange, Newton divided difference, finite difference operators, Newton forward and backward interpolation, Cubic spline interpolation. Numerical differentiation and integration: Trapezoidal, Simpson’s rules, Gaussian quadrature.

Unit III: Solution of systems of linear equations: Gauss elimination, pivoting, LU decomposition, Jacobi and Gauss-Seidel iterative methods. Eigenvalue problems: Power method.

Unit IV: Numerical solution of ordinary differential equations: Euler, Modified Euler, Runge-Kutta methods of order 2 and 4, Multistep methods. Introduction to numerical solution of PDEs.

Unit I: Definition, examples and properties of groups, Subgroups, Normal Subgroups, Quotient groups, Homomorphism, Isomorphism theorems, Order of a subgroup, Lagrange’s Theorem.

Unit II: Subgroups generated by subsets of a group, Cyclic group, Dihedral group, Group actions, Orbit - Stabilizer Theorem, Centralizer, Normalizer, Class Equation.

Unit III: Symmetric and Alternating groups, Simplicity of A_n, Cayley’s Theorem, Automorphism groups, Direct product.

Unit IV: The Sylow Theorems, Structure theorem of finite abelian groups. Introduction to rings and fields with examples.

Unit I: Overview of linear algebra: System of linear equations, consistent and inconsistent system. Basic solutions, degenerate solution, basic variables, basis of matrix.

Unit II: Convex set, extreme point. Single objective linear programming model. Graphical solution and its interpretation. Bounded and unbounded model. Feasible and infeasible solutions.

Unit III: Simplex method, revised simplex method, Duality, sensitive analysis, Complementary slackness theorem.

Unit IV: Transportation and assignment problems. Introduction to Nonlinear Optimization: Basic theories, method of Lagrange multipliers.

Project: Project in one assigned topic under the guidance of one faculty. Students undertake an independent or guided research-oriented project in a chosen area of mathematics, culminating in a written report and presentation.

Unit I: Review of Arithmetic in $\mathbb{Z}$: Divisibility, gcd and lcm, Division algorithm, Euclidean algorithm. Prime numbers, Fundamental theorem of Arithmetic, Goldberg conjecture, Euclid’s and Euler’s proof of infinitude of primes, twine prime conjecture (statement only).

Unit II: Modular Arithmetic: Congruence relations - linear and polynomial, Fermat’s little theorem, Euler’s theorem, Wilson’s theorem, Chinese Remainder Theorem, Solutions of $X^{n}=1$

Unit III: Arithmetic functions: Definition and examples of arithmetic functions like Mobius functions, Euler’s $\phi$-function, Divisor function, the prime counting function $\pi(x)$. Dirichlet multiplication and Mobius inversion formula, Multiplicative functions, Dirichlet series and Euler product.

Unit IV: Quadratic Residue and Diophantine Equations: Primitive roots, index arithmetic, structure of unit group of $\mathbb{Z}/n\mathbb{Z}$. Quadratic residue, Legendre and Jacobi symbols, Quadratic Reciprocity law (statement only); Sums of two squares and four squares. Linear Diophantine equations, Pythagorean triples, Fermat’s last theorem; Continued fractions and rational approximations.

Unit I: Isomorphism, incidence and adjacency matrices, Sperner lemma, Trees, Cayley formula, connector problem, connectivity, constructing reliable communication network.

Unit II: Euler tours, Hamilton cycle, Chinese postman and traveling salesman problems.

Unit III: Matchings and coverings, perfect matchings, edge colouring, Vizing Theorem, time table problem, Independent sets, Ramsey theorem, Turan theorem, Schur theorem, vertex colouring, Brook theorem, Hajos conjecture, chromatic polynomials, storage problem.

Unit IV: Planarity, dual graphs, Euler formula, Kuratowski theorem, five colour theorem, history of four colour theorem, nonhamiltonian planar graphs, planarity algorithm, directed graphs, job sequencing, one way road system, ranking participants in tournaments.

Unit I: Vector functions and space curves, Arc length and curvature, Motion in space, Cylindrical and spherical coordinates.

Unit II: Partial derivatives, Taylor series expansion in two variables, Tangent planes and differentials, The chain rule, Directional derivatives and the gradient vector, Total derivative, Local and absolute extrema for functions of several variables.

Unit III: Double and iterated integrals including polar coordinates, Applications of double integrals, Triple and iterated integrals, including cylindrical and spherical coordinates, Change of variable in multiple integrals.

Unit IV: Vector fields and line integrals, The fundamental theorem for line integrals, Curl and divergence, Green’s theorem, Stokes’ theorem, Divergence theorem.

Unit I: Kinematics of Fluid Motion: Solids, Liquids and Gases, Continuum hypothesis, Velocity at a point of a fluid, Lagrangian and Eulerian approaches, streamlines, pathlines and streak lines, Acceleration at a point of a fluid, Equation of continuity, Reynolds Transport theorem.

Unit II: Mechanics of Fluid Motion: Pressure (static and dynamic), Boundary conditions, Euler’s equation of motion of an ideal fluid, Bernoulli’s Equation, Kelvin's circulation theorem, Vorticity equation, Energy equation.

Unit III: Potential Flow: Analysis of Local fluid motion: Translation, Rotation, and Deformation, Vorticity, potential flow, Mean value of the velocity potential, Uniform flow, Axially symmetric flows, Sphere at rest in a uniform stream.

Unit IV: Two-Dimensional Flows: stream function, complex potential, complex velocity potential, sources, sinks, doublets, Milne-Thomson Circle Theorem, theorem of Blasius.

Unit I: Set theory and logic, Fundamental concepts, Functions and relations, Cartesian products, Countable and uncountable sets, Definitions and examples of topological spaces, Open sets, Basis for a topology.

Unit II: Order topology, Product topology on X×Y, Subspace topology, Closed sets and limit points, Continuous functions, Homeomorphisms.

Unit III: Metric topology, Examples and properties of metric spaces, Hausdorff spaces, Uniform limit theorem, Quotient map, Definition and examples of quotient topology.

Unit IV: Definition and examples of connected spaces, Intermediate value theorem, Path connectedness, Compact spaces, Tube lemma, Heine-Borel theorem, Extreme value theorem, Uniform continuity theorem.

Unit I: Complex Functions, Limit, continuity, and differentiability, Cauchy-Riemann Equations, Analytic Functions.

Unit II: Power Series, Radius of convergence, Complex Integration, Line or Path Integral, Cauchy Theorem, Cauchy Integral Formula, and their consequences.

Unit III: Elementary Conformal Mappings, Mobius Maps, Maximum Modulus Principle, Liouville Theorem, Fundamental Theorem of Algebra (proof).

Unit IV: Laurent Series, Classification of singularities: Isolated Singularities, Removable, Pole, and Essential, Residue Theorem and evaluation of certain improper integrals.

Unit I: Definition, examples and properties of rings, Subrings, Ideals, Prime and maximal ideals, Ring homomorphism, Isomorphism theorems.

Unit II: Quotient Rings, Rings of fractions, Chinese Remainder Theorem, Polynomial Rings, Integral domain, Euclidean Domain, Principal Ideal Domain, Unique Factorization Domain, Gauss’ Lemma, Irreducibility criterion, Eisenstein’s criterion.

Unit III: Definition, examples and properties of fields, Characteristic of a field, Field extensions, Algebraic and transcendental elements.

Unit IV: Splitting fields, Separable and normal extensions, Primitive roots of unity, Cyclotomic polynomials, Finite fields.

Seminar: Presentation and discussion on selected advanced topics in mathematics under the guidance of a faculty member. Students are required to prepare and deliver seminars on contemporary or classical topics, demonstrating understanding, literature survey and communication skills.

Project work under the guidance of a faculty member (Part-I).

Project work under the guidance of a faculty member (Part-II).

Vector spaces over fields, subspaces, bases and dimension; Systems of linear equations, matrices, rank, Gaussian elimination; Linear transformations, representation of linear transformations by matrices, rank-nullity theorem, duality and transpose; Determinants, Laplace expansions, cofactors, adjoint, Cramer's Rule; Eigenvalues and eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem, triangulation, diagonalization, rational canonical form, Jordan canonical form; Inner product spaces, Gram-Schmidt orthonormalization, orthogonal projections, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators, Spectral Theorem for normal operators; Rayleigh quotient, Min-Max Principle. Bilinear forms, symmetric and skew-symmetric bilinear forms, real quadratic forms, Sylvester's law of inertia, positive definiteness.

Real number system and Set theory: Completeness property, Archimedian 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, Totally boundedness, Finite intersection property.

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, Weierstrass approximation theorem.

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, pigeonhole principle, inclusion and exclusion principle, partitions, recurrence relations, generating functions; Graph Theory: paths, connectivity, subgraphs, isomorphism, trees, complete graphs, bipartite graphs, matchings, colourability, planarity, digraphs, Eulerian cycle and Hamiltonian cycle, adjacency and incidence matrices.

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.

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; Trees, Binary trees, Searching and sorting - linear search, binary search, insertion-sort, bubble-sort, selection-sort; Introduction to object oriented programming.

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.

Polar representation and roots of complex numbers; Spherical representation of extended complex plane; Elementary properties and examples of analytic functions: The exponential, Trigonometric functions, Mobius transformations, Cross ratio; Complex integration: Power series representation of analytic functions, Zeros of analytic functions, Cauchy theorem and integral formula, The index of a point with respect to a closed curve, the general form of Cauchy’s theorem; Open Mapping Theorem; Classification of singularities: Residue theorem and applications; The Argument Principle; The Maximum modulus Principle; Schwarz’s lemma; Phragmen-Lindelof theorem.

Topological spaces, Basis and subbasis, The order topology, Subspace topology, Closed sets. Countability axioms, Limit points, Convergence of nets in topological spaces, Continuous functions, homomorphisms. The product topology, box topology, Metric topology, Quotient topology. Connected spaces, Connected sets in R, Components and path components, Compact spaces, Compactness in metric spaces, Local compactness, One point compactification. Separation axioms, Uryshon’s lemma, Uryshon’s metrization theorem, Tietz extension theorem. The Tychonoff theorem, Completely regular spaces, Stone -Czech compactification.

Ordinary differential equations- first order equations, Picard’s theorem (existence and uniqueness of solution to first order ordinary differential equation). Second order differential equations- second order linear differential equations with constant coefficients. Systems of first order differential equations, equations with regular singular points, stability of linear systems. Introduction to power series and power series solutions. Special ordinary differential equations arising in physics and some special functions (e.g. Bessel’s functions, Legendre polynomials, Gamma functions) and their orthogonality. Oscillations - Sturm Liouville theory.

Definition and sources of errors, Propagation of errors, Backward error analysis, Sensitivity and conditioning, Stability and accuracy, Floating-point arithmetic and rounding errors. Nonlinear equations, Bisection method, Newton's method and its variants, Fixed point iterations, Convergence analysis. Newton's method for non-linear systems. Finite differences, Polynomial interpolation, Hermite interpolation, Spline interpolation, B-splines. Numerical integration, Trapezoidal and Simpson's rules, Newton-Cotes formula, Gaussian quadrature, Richardson Extrapolation IVP: Taylor series method, Euler and modified Euler methods, Runge-Kutta methods, Multistep methods, Predictor-Corrector method Accuracy and stability, Solution for Stiff equations BVP: Finite difference method.

Fundamentals of normed linear spaces: Normed linear spaces, Riesz lemma, characterization of finite dimensional spaces, Banach spaces. Bounded linear maps on a normed linear spaces: 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.

Bodies, Deformation, Strain, Characterization of rigid deformations, Small Deformations, Characterization of infinitesimal rigid displacements, Motions, Smoothness Lemma, Types of motions, Rate of Stretching, Characterization of rigid motions, Transport Theorems, Volume, Isochoric motions, Spin, Circulation, Vorticity; Conservation of mass, linear and angular momentum, Centre of mass; Force, Theorem of Virtual Work, Stress, Cauchy's Theorem for

Linear programming, simplex method, duality theory, transportation and assignment problems, integer programming, nonlinear programming, unconstrained and constrained optimization, Kuhn-Tucker conditions, dynamic programming, network flows.

Rings and Algebra, Monotone classes. Measures and outer measures. Measurable sets; Lebesgue Measure and its properties. Measurable functions and their properties, Convergence in measure. Integration: Sequence of integrable functions; Signed measures, Hahn and Jordan decomposition, Absolute continuity of measures, Radon-Nikodym theorem; Product measures, Fubini's theorem; Transformations and functions: The isomorphism theorem, Lp-spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces, Dual of Lp-spaces; Measure and Topology: Baire and Borel sets, Regularity of Baire and Borel measures, Construction of Borel measures, Positive and bounded linear functionals.

Mathematical models leading to partial differential equations. First order quasi-linear equations. Nonlinear equations. Cauchy-Kowalewski’s theorem (for first order). Classification of second order equations and method of characteristics. Riemann’s method and applications. One dimensional wave equation and De’Alembert’s method. Vibration of a membrane. Duhamel’s principle. Solutions of equations in bounded domains and uniqueness of solutions. BVPs for Laplace’s and Poisson’s equations. 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.

Congruences: linear and polynomial congruences; prime numbers: counting primes, numbers of special forms, pseudo-primes and primality testing; factorization: factorization algorithms; arithmetic functions: multiplicative and additive functions, Euler's phi function, sum and number of divisors functions, the Mobius function and other important arithmetic functions, Dirichlet products; primitive roots and quadratic residues: primitive roots, index arithmetic, quadratic residues, modular square roots; Diophantine equations: linear Diophantine equations, Pythagorean triples, Fermat's last theorem, Tell's, Bachet's and Catalan's equations, sums of squares; Diophantine approximations: continued fractions, convergent, approximation theorems; quadratic fields: primes and unique factorization.

Eigenvalues, eigenvectors and similarity, Unitary equivalence and normal matrices, Schur’s theorem, Spectral theorems for normal and Hermitian matrices; Jordan canonical form, Application of Jordan canonical form, Minimal polynomial, Companion matrices, Functions of matrices; Variational characterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischer theorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law of inertia; Matrix norms, Location and perturbation of eigenvalues Gerschgorin disk theorem; Positive semidefiniteness, Singular value decomposition, Polar decomposition, Schur and Kronecker products; Positive and nonnegative matrices, Irreducible nonnegative matrices.

error analysis of orthogonal matrices, Stability of QR factorization. Solution of linear least squares problems, Normal equations, Singular Value Decomposition(SVD), Polar decomposition, Moore-Penrose inverse, Rank deficient least squares problems, Sensitivity analysis of least-squares 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 non-symmetric matrices, Implementation of implicit QR algorithm. Computing the SVD, Sensitivity analysis of singular values and singular vectors. Overview of iterative methods: Jacobi, Gauss-Seidel and successive over relaxation methods, Krylov subspace method, The Arnoldi and the Lanczos iterations. Software Support: MATLAB.

Linear Lie Groups: the exponential map and the Lie algebra of linear Lie group, some calculus on a linear Lie group, invariant differential operators, finite dimensional representations of a linear Lie group and its Lie algebra. Examples of linear Lie group and their Lie algebras, e.g., Complex groups: GL(n, C), SL(n, C), SO(n, C), Groups of real39matrices in those complex groups: GL(n, R), SL(n, R), SO(n, R), Isometry groups of Hermitian forms SO(m, n),U(m, n), SU(m, n). Finite dimensional representations of su(2)and SU(2) and their connection. Exhaustion using the lie algebra su(2). Lie algebras in general, Nilpotent, solvable, semisimple Lie algebra, ideals, Killing form, Lies and Engels theorem. Universal enveloping algebra and Poincare-Birkho-Witt Theorem (without proof). Semisimple Lie algebra and structure theory: Definition of Linear reductive and linear semisimple groups. Examples of Linear connected semisimple/ reductive Lie groups along with their Lie algebras (look back at 2 above and find out which are reductive/ semisimple).Cartan involution and its differential at identity; Cartan decomposition g = k + p, examples of k and p for the groups discussed above. Definition of simple and semisimple Lie algebras and their relation, Cartans criterion for semisimplicity. Statements and examples of Global Cartan decomposition, Root space decomposition; Iwasawa decomposition; Bruhat decomposition.

Syntax of First-Order Logic: First Order Languages, Terms and Formulas of a First Order language, First Order Theories. Semantics of First-Order Languages: Structures of First-Order Languages, Truth in a Structure, Model of a Theory. Propositional Logic: Tautologies and Theorems of propositional Logic, Tautology Theorem. Proof in First Order Logic, Metatheorems of a first order theory, e.g. , theorems on constants, equivalence theorem, deduction and variant theorems etc., Consistency and Completeness, Lindenbaum Theorem. Henkin Extension, Completeness theorem, Extensions by definition of first order theories, Interpretation theorem. Model Theory: Embeddings and Isomorphisms, L¨owenheim-Skolem Theorem, Compactness theorem, Categoricity, Complete Theories. Recursive functions, Arithmatization of first order theories, Decidable Theory, Representability, Godel’s first Incompleteness theorem.

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts. Basic constructions: cones, mapping cones, mapping cylinders, suspension. Cell complexes, subcomplexes, CW pairs. Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen's Theorem, Covering spaces, lifting properties, deck transformations. universal coverings (existence theorem optional). Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular Homology. Mayer-Vietoris Sequences. Long exact sequence of pairs and triples. Homotopy invariance and excision (without proof). Degree. Cellular Homology. Applications of homology: Jordan-Brouwer separation theorem, Invariance of dimension, Hopf's Theorem for commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point Theorem. Optional Topics: Outline of the theory of: cohomology groups, cup products, Kunneth formulas, Poincare duality.

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.

Review of gradient, divergence and curl.

Elementary idea of tensors. Velocity of fluid, Streamlines and path lines, Steady and unsteady flows, Velocity potential, Vorticity vector, Conservation of mass, Equation of continuity. Equations of motion of a fluid, Pressure at a point in fluid at rest, Pressure at a point in a moving fluid, Euler's equation of motion, Bernoulli's equation. Singularities of flow, Source, Sink, Doublets, Rectilinear vortices. Complex variable method for two-dimensional problems, Complex potentials for various singularities, Circle theorem, Blasius theorem, Theory of images and its applications to various singularities. Three dimensional flow, Irrotational motion, Weiss's theorem and its applications. Viscous flow, Vorticity dynamics, Vorticity equation, Reynolds number, Stress and strain analysis, Navier-Stokes equation, Boundary layer Equations

Functions on Euclidean spaces, continuity, differentiability; partial and directional derivatives, Chain Rule, Inverse Function Theorem, Implicit Function Theorem. Riemann Integral of real-valued functions on Euclidean spaces, measure zero sets, Fubini's Theorem. Partition of unity, change of variables. Integration on chains, tensors, differential forms, Poincare Lemma, singular chains, integration on chains, Stokes' Theorem for integrals of differential forms on chains. (general version). Fundamental theorem of calculus. Differentiable manifolds (as subspaces of Euclidean spaces), differentiable functions on manifolds, tangent spaces, vector fields, differential forms on manifolds, orientations, integration on manifolds, Stokes' Theorem on manifolds.

Programming laboratory will be set in consonance with the material covered in lectures of the course "Computer Programming and Data Structures". This will include assignments in a programming language like C and C++ in GNU Linux environment.

Programming laboratory will be set in consonance with the material covered in lectures of the course "Numerical Analysis". This will include assignments in MATLAB.

Programming laboratory will be set in consonance with the material covered in lectures of the course "Optimization Techniques".

Literature survey on assigned topic and presentation.

Literature survey on assigned topic and presentation.

One has to choose one course from the list of electives or prepare a Research Review Paper on an assigned topic under faculty guidance.

Errors: Round-off error, Truncation error, Absolute error, Relative error, Percentage error;

Ordinary differential equations (ODE): Solutions of Initial Value Problems by Taylor Series, Euler, Improved Euler, Modified Euler, Runge-Kutta methods for First and second order differential equations, Multistep methods (Milne and Adams Bashforth). Consistency, stability and convergence aspects of the methods of IVP. Boundary Value Problems: Shooting and finite difference methods.

Partial Differential Equations (PDE): Classification of PDEs, Finite difference approximations to partial derivatives, Numerical solutions of Elliptic, Parabolic and Hyperbolic partial differential equations. Solutions of Laplace equation by Leibmann’s iteration procedure, Poisson equation, Explicit, Crank-Nickolson, Du Fort Frankel methods for Parabolic PDE. Explicit formula for Hyperbolic PDE and Consistency, stability and convergence aspects of these methods.

Matrices which leave a cone invariant: Introduction, Cones, Convex cones, Polyhedral cones, Solids, Spectral properties of matrices which leave a cone invariant, Cone primitivity. Nonnegative Matrices: Nonnegative matrices , Inequalities and Generalities, Positive matrices, Nonnegative Irriducible Matrices, Perron’s Theorem, Perron-Frobenius Theory, Nonsingular M-matrices. Reducible Matrices, Primitive Matrices, A Genenral Limit Theorem Stochastic and Doubly Stochastic Matrices: The Birkhoff-von Neumann Theorem, Fully indecomposable matrices, Konig's Theorem and rank. Semigroups of Nonnegative Matrices: Algebraic semigroups, Nonnegative idempotents, The semigroup Nn, The semigroup Doubly Stochastic Matrices Dn. Symmetric Nonnegative Matrices: Inverse eigenvalue problems, Nonnegative matrices with given sums, Some applications.

Series and product developments: Power series expansions, Partial fractions and factorizations; Entire functions: The Hadamard’s theorem, Jensen’s formula, The Riemann zeta function; Normal families: Equicontinuity, Normality and compactness, Arzela’s theore; Analytic continuation and Riemann surfaces: Germs and sheaves, Analytic continuations along arcs, Homotopic curves, The monodromy theorem, Branch points, Algebraic functions; Picard’s theorem: Lacunary values; The Reimann mapping theorem;The Dirichlet problem; Canonical mappings of multiply connected regions; Elliptic functions and Weierstrass Theory; Basic results on univalent functions; The range of analytic functions: Bloch’s theorem, Schottky’s theorem;

The philosophy and scope of fractal geometry, Scaling and self similarity, Hausdorff measure and dimensions, Box-counting dimensions, Techniques for calculating dimensions, Local structure and projections of fractals, The Thermodynamic Formalism: Pressure and Gibb’s measures, the dimension formula, Invariant measuresand the transfer operator, Entropy and the Variational principle; The ergodic theorem; The renewal theorem; Martingales and the convergence theorem, Bi-Lipschitz equivalence of fractals; Multifractal Analysis; Applications of fractals: Iterated function systems (IFS) and Recurrent IFS, Applications to image compression, Julia sets and the Mandelbrot set, Random fractals, Brownian motion and Random walks, Percolation, Fractal interpolation,

Topological space, subspace, base, subbase, continuous function, connectedness, paths, homotopy, homotopy of Paths and homotopy of maps, simplicial complex, polyhedral, graphs, homology theory, computation of beti numbers.

Integral Equations: Basic concepts, Volterra integral equations, relationship between linear differential equations and Volterra equations, resolvent kernel, method of successive approximations, convolution type equations, Volterra equations of first kind, Abels integral equation, Fredholm integral equations, Fredholm equations of the second kind, the method of Fredholm determinants, iterated kernels, integral equations with degenereted kernels, eigenvalues and eigenfunctions of a Fredholm alternative, construction of Green's function for BVP, singular integral equations. Calculus of variations: Euler-Lagrange equations, degenerate euler equations, Natural boundary conditions, transversality conditions, simple applications of variational formulation of BVP, minimum of quadratic functional. Approximation methods-Galerkin's method, weighted-residual methods, Colloation methods.Variational methods for time dependent problems.

Introduction to Soft Computing, Components of Soft Computing, Importance of Soft Computing, Fuzzy Set Theory - Definition, Different types of fuzzy set Membership Functions, Fuzzy Set theoretic operations, Fuzzy Rules and Fuzzy Reasoning, Fuzzy Inference Systems, GA, Simulated Annealing, Particle swarm optimization, Neural Networks- Supervised Learning, Unsupervised Learning, Hybrid Systems - Neuro Fuzzy Modeling, Fuzzy c-means, Applications in Image Processing, Neuro-fuzzy control, Data Mining. Implementation of the problems using MATLAB.

Biological and Artificial Neuron, Perceptron model, Adaline model, Different types of Activation functions, Learning Techniques: Supervised and Unsupervised, Multilayered feed forward Networks, Back propagation algorithm and its improvements, Applications of Back propagation algorithm to statistical pattern recognition, classification and regression problems, Advantages of Neural Networks over statistical classification techniques, Performance Surfaces and Optimum Points, Steepest Descent, Stable learning Rates, Minimizing Along a line ,Newton’s method and conjugate method. Recurrent networks, Radial Basis Function Networks as an interpolation model, Time delay neural networks for forecasting problems, Probabilistic Neural Networks, Kohonen’s self organizing map, Self organizing maps with quadratic functions and its applications medical imaging, Adaptive Resonance, Theory model, Applications of Art model for knowledge acquisition, Extensive sessions in MATLAB for solving statistical pattern recognition, classification, regression and prediction problems using different kinds of Neural Network models.

Probability and random variable, discrete and continuous univariate and multivariate distributions, moments, law of large numbers and central limit theorem (without proof). Poisson process, birth and death process, infinite and finite queueing models M/M/1, M/M/C, M/G/1, M/M/1/N, M/E/1, E/M/1, M/G/1/N, GI/M/1, and more complex non-Markovianqueueing models, such as GI/G/1 queues, Multiserver Queues: M/M/c, M/G/c, GI/M/c modles, Erlan’sg loss system, Queues with finite populations: M/M/1/N/K, M/G/1/N/K etc. models and Engset formula, Concept bulk queues: M[X]/M/1, M/M[Y]/1, M/M(a,b)/1, M[X]/G/1, GI[X]/M/1, M/G(a,b)/1, GI/M(a,b)/1 etc. queueing models. Priority queueing models, Vacation queueing models,Network of queues, finite processor sharing models, central server model of multiprogramming, performance evaluation of systems using queueing models. Concepts of bottleneck and system saturation point. Introduction to discrete time queues and its applications.

Markov chains, Poisson process, Birth-Death process, Simple Markovian queueing models: M/M/1, M/M/1/N, M/M/C, transient behaviour of M/M/1 and M/M/1/N queue. Models with general arrival or service: M/G/1, GI/M/1. Concept bulk queues, Some Markovian and no-Markovianbulk arrival or bulk service queueing models: M[X]/M/1, M/M[Y]/1, M/M(a,b)/1, M[X]/G/1, GI[X]/M/1, M/G(a,b)/1, GI/M(a,b)/1 etc. models. Network of Markovian queues.General concept of discrete time queues, system reliability, Reliability of Systems of Independent Components, Markovian models in reliability theory, Life testing using the exponential and Weibull models. Expected System Lifetime, Bounds on the expected system lifetime, Systems with Repair Elementary inventory models, Deterministic and Stochastic models, Inventory models with price breaks, Multi-item Economic-order-quantity (EOQ) model with storage limitation, Single- and multi-period probabilistic inventory models, Concept of just-in-time inventory.

Stochastic and deterministic mathematical models for forecasting of time series, Classical techniques in Time Series Analysis, Different Smoothing Techniques, General linear process, Autoregressive process AR(P), Moving average process Ma(q): Autocorrelation, Partial autocorrelation, and Spectrum, Linear stationary models for autoregressive moving average processes, Mixed autoregressive moving average processes, Autocovariances, Autoregressive Integrated Moving Average Processes (ARIMA). Forecasting, Calculating and updating forecast, Forecast function and forecast weight, Examples of forecast functions and their updating, Identification in time domain, Model Identification , Estimation of Parameters, Model estimation, Non-linear estimation, Model diagnostic checks, Seasonal models, Transfer function models, Discrete transfer function model, continuous dynamic models, Identification, fitting and checking of transfer function models, Intervention analysis models and outlier detection, Estimation of Autoregressive moving average (ARMA) models, Use of time series techniques in Engineering fields.

Analysis of strain, deformation, affine deformation, infinitesimal affaine deformation, geometrical interpretation of the components of strain, principal directions, invariants, general infinitesimal deformation, Examples of strain, strain compatibility equations; Analysis of stress, body and surface forces, stress tensor, equations of equilibrium, stress quadric of Cauchy, examples of stress; The strain – energy function and its connection with hooks law, Saint Venant's principle. Boundary value problems in elasticity; Extension, bending and Torsion: statement of problem, extension of beams by longitudinal forces, Beam stretched by its own weight, Bending of beams by terminal couples of elliptic cylinder, Torsion of a circular shaft, Torsion of cylindrical bars, stress function, Torsion of elliptical cylinder; Two-dimensional elastostatic problems: plane deformation, plane stress, Generalized plane stress, plane elastostatic problems, Airy’s stress function, General solution of Biharmonic equation, formulas for stresses and displacements, the structure of the functions ø(z) and φ(Z); Vibration of elastics solids, wave propagation in infinite regions, Ryleigh and Love waves .

Elements of Group theory, Operators in Hilbert Space, Representation theory – finite groups, Continuous groups, Group theory and Quantum mechanics, Crystallographic and Molecular Symmetries

Classification of Partial Differential Equations; Classification of system of first order PDE’s; Classification of system of second order PDE’s; Boundary conditions; consistency, stability and convergence of FDE; Lax’s equivalence theorem. Finite difference approximations to partial derivatives for equally spaced grid points and unequally spaced grid points. Methods to solve Parabolic partial differential equations, Elliptic and Hyperbolic partial differential equations and stability analysis; Vorticity Stream function formulation of the Navier-Stokes equations, Finite difference methods for steady, viscous incompressible N-S equations: Flow in a lid driven cavity, Viscous flow past a circular cylinder, flow past a sphere. Line Gauss-Seidal, Point Gauss Seidal , SOR schemes, Finite difference applications in Heat conduction and convection; Discussion on first, second and higher order accurate methods. Higher order compact schemes to Navier Stokes equations.

Fundamental Concepts: independent sets, matchings, spanning trees, Hamiltonian cycles, Eulerian orientations, cycle covers, etc.; Operations on Graphs and the Resulting Spectra: the polynomial of a graph, eigenvalues and eigenvectors, line graphs and total graphs. etc.; The Divisor of Graphs: The divisor and cover, symmetry properties, some generalizations; Spectral Characterizations: Eigenvalues of L-, Q-, and adjacency matrix, co-spectral graphs, graphs characterized by their spectra; Spectral Techniques in Graph Theory and Combinatorics: Computing the structures suchas, independent sets, matchings, spanning trees, Hamiltonian cycles, Eulerian orientations, etc. Additional Topics: Random Graphs, Ramsey Theory, Extremal Problems.

Normal families, Montel's theorem; Iteration of polynomials and rational maps: The structure of the Fatou set, Sullivan classification of Fatou components, External rays, Properties of the Julia set; Polynomial like mappings; Parameter spaces of rational maps: Mandelbrot set, Holomorphic motion and stability; Renormalisation; Dynamics of transcendental entire and meromorphic functions: Singular points of inverse function, Baker domains and Wandering domains, Maps of bounded type, Escaping points, Omitted values and dynamics; Conformal measure on Julia sets, Dimension of Julia sets; Kleinian groups and Sullivan’s dictionary.

Examples and Basic concepts: Dynamical system, orbits, Circle rotations, Shifts and sub-shifts, Hyperbolic toral automorphisms, The Horseshoe, The solenoid, Flows and differential equations, Chaos and Lyapunov exponents; Topological dynamics: Limit sets, Recurrence, Mixing, Transitivity, Entropy; Symbolic Dynamics: Subshifts, Sofic shifts, codes, Perron Frobenius Theorem, Data storage; Ergodic Theory: Ergodicity and mixing, Ergodic theorems (Von Neumann Ergodic Theorem, Birkhoff Ergodic Theorem), Invariant measures for continuous maps,; Unique ergodicity and Weyl’s theorem, Discrete spectrum, Weak mixing, Internet search; Hyperbolic dynamics: Stable and unstable manifolds, Anosov diffeomorphisms, Axiom A and structural stability; Ergodicity of Anosov diffeomorphisms: Holder continuity of the stable and unstable distributions, Absolute continuity of stable and unstable foliations;

Topological groups, locally compact groups, Haar measure. Modular function, convolutions, homogenous spaces , unitary representations, Gelfand-Raikov theorem Functions of positive type, GNS construction, Potrjagin dulaity, Bochner's theorem, Induced representations, Mackey's impritivity theorem.

Basic Properties of Fourier Series: Uniqueness of Fourier Series, Convolutions, Cesaro and Abel Summability, Fejer's theorem, Poisson Kernel and Dirichlet problem in the unit disc. Mean square Convergence, Example of Continuous functions with divergent Fourier series. Distributions and Fourier Transforms: Calculus of Distributions, Schwartz class of rapidly decreasing functions, Fourier transforms of rapidly decreasing functions, Riemann Lebesgue lemma, Fourier Inversion Theorem, Fourier transforms of Gaussians. Tempered Distributions: Fourier transforms of tempered distributions, Convolutions, Applications to PDEs (Laplace, Heat and Wave Equations), Schrodinger-Equation and Uncertainty principle. Paley-Wienner Theorems, Poisson Summ-ation Formula: Radial Fourier transforms and Bessel's functions. Hermite functions. Optional Topics: Applications to PDEs, Wavelets and X-ray tomography.Applications to Number Theory.

Fixed point theory, Banach contraction mapping theorem, contractive type mappings, generalization of Banach contraction mapping theorem, fixed point theorem of other types. Nonlinear operators, monotone, strictly monotone and strongly monotone operators, their properties and applications. Variational inequalities and complementarity problem. Approximation theory, theory of best approximation and farthest points. Calculus of Banach Space, Frechet and Gateaux differentiability, strict convexity and uniform convexity of norms, semi inner product space.

Introduction and motivation, Weak formulation of BVP and Galerkin approximation, Piecewise polynomial spaces and finite element method, Computer implementation of FEM, Results from Sobolev spaces, Variational formulation of elliptic BVP, Lax-Milgram theorem, Estimation for general FE approximation, Construction of FE spaces, Polynomial approximation theory in Sobolev spaces, Variational problem for second order elliptic operators and approximations, Mixed methods, Iterative techniques.

Non-Linear Programming Problems: One variable unconstrained optimization, multivariable unconstrained optimisation, Karush-Kuhn-Tucker (KKT) conditions for constrained optimization, quadratic programming, separable programming, convex and non convex programming, steepest and Quasi-Newton method. Dynamic Programming: Characteristics of dynamic problems, deterministic dynamic programming and probabilistic dynamic programming, Network analysis, Shortest path problems, minimum spanning tree problem, maximum flow problem, minimum cost flow problem, network simplex, interior point methods, stochastic programming, Nonlinear goal programming applications, Geometric Programming. Multi-objective Optimization Problems: Linear and non linear programming problems, Weighting and Epsilon method, P-norm methods, Gradient Projection Method, STEM method, Convex Optimization.

Conic programming: linear programming (LP), second-order cone programming (SOCP), semidefinite programming (SDP), linear matrix inequalities, conic duality, conic duality theorem, Applications of semidefinite programming: control and system theory, combinatorial and nonconvex optimization, machine learning, Smooth convex optimization: gradient descent, optimal first-order methods (Nesterov's method and its variants), complexity analysis, Nonsmooth convex optimization: conjugate functions, smooth approximations of nonsmooth functions by conjugation, prox-functions, Nesterov's method for composite functions, Proximal minimization and mirror-descent algorithms (MDA),Augmented Lagrangian methods and alternating direction method of multipliers (ADMM),Example problems in statistics, signal and image processing, control theory.

Definition and classification of stochastic processes, Poisson process, Flow of events, telegraph signal, Birth and death processes, application in queues, Random walk, Markov chains: classification of states, hitting times and absorption probabilities, Analysis of finite transient discrete Markov chain,finite and infinite ergodic chains, Continuous-time Markov chains, The Kolmogorov differential equations and transition probabilities, Martingales, Stopping times, Martingales to analyze random walks, application to GI/G/1 queues and ruin problems in risk theory. Renewal processes, key renewal theorem, renewal reward process, Regenerative processes, stationary point processes, Branching process,Condititional limit laws of branching process, Continuous time Markov branching process, General branching process, Elements of continuous time and continuous state space processes: Brownian motion, Diffusion processes, variations of Brownian motions and drift, backward and forward diffusion equations, applications of Kolmogorov’s equations for obtaining limiting distributions of semi-Markov processes, queues

Solutions of uncontrolled systems, spectral form, exponential matrix, repeated roots, solutions of controlled systems, time varying systems, discrete time varying systems, relation between state space and classical forms; Controllability, observability, controllability and polynomials, linear feedback, state observer realization of constant systems; Algebraic criteria for linear systems, Nyqist criteria Liapunov theory, applications to linear systems, construction of liapunov functions, input-output stability, optimal control.

Single Species Population Dynamics: Linear and Nonlinear First Order Discrete Time Models, Differential Equation Models, Evolutionary Aspects, Continuous Models and Discrete Models. Population Dynamics of Interacting Species: Host-parasitoid Interactions, The Lotka-Volterra Prey-predator Equations, Modelling the Predator Functional Response, Ecosystems Modelling, Interacting Metapopulations. Infection Diseases: The Simple Epidemic and SIS Diseases, SIR Epidemics, SIR Endemics, Eradication and Control, Age-structured Population, Basic Model for Macroparasitic Diseases. Biological motion: Macroscopic Theory of Motion; A Continuum Approach, Directed Motion, or Taxis, Steady State Equations and Transit Times. Molecular and cellular biology: Biochemical kinetics, Metabolic Pathways, Neural Modelling. Tumour modeling, Pharmaco kinetics.

Review of Arithmetical functions: Mobius functions, Euler’s φ-function, Divisor function, the von-Mangoldt function, Liouville’s function, Dirichlet multiplication and Mobius inversion formula, Multiplicative functions, Dirichlet series and Euler product. Averages of arithmetical functions: Summation formulas, Average order of divisor function, Euler φ-function, Mobius function and von-Mangoldt function, Dirichlet hyperbola method. Distribution of primes without complex analysis: Euclid’s method, Prime counting function π(x), Chebyshev’s psi and theta functions, Partial sums of reciprocal series of primes, Merten’s formula, zeta function and Euler’s proof of infinitely many primes. Distribution of primes using complex analysis: Characters of finite abelian groups, Dirichlet characters, primitive characters, Gauss sum, Riemann zeta function and Dirichlet L-functions as functions of complex variable. Functional equations and analytic continuations: Schwarz function on R, Z, R/Z and their Fourier transform, the Poisson summation formula, functional equations for zeta function and L-functions, Bernoulli numbers and its relation to Riemann zeta function. Hadamard factorization theorem (statement), product formulas for completed zeta function and L-functions, Classical zero-free region and Riemann hypothesis, Number of zeros of zeta function and L-functions, Explicit formulas, the Prime Number Theorem (PNT) and Dirichlet’s Theorem on primes in arithmetic progression (PNT in AP). Introduction to Public Key Crypography. If time permits: advance topics like Sieve Methods, the circle method, bounds of exponential sums and Modular forms.

The modular group and its subgroups, Fundamental domain for SL(2, Z). Modular functions and forms, Examples of modular forms and functions such as Eisenstein series, Poincare series, Ramanujan delta function, j-function, theta functions. Space of modular forms and cusp forms, Valence formula, Dimension formula, Fourier expansions of Eisenstein series at infinity and Fourier coefficients of cusp forms, Ramanujan tau- function and Ramanujan conjectures. Hecke theory (abstract and concrete), Petersson inner product, L-functions attached to a modular form, analytic continuation and converse theorem of Hecke, Rankin-Selberg L-function. Theory of modular forms of higher levels, Atkin-Lehner-Li theory, Applications of modular forms. If time permits Modular forms of half-integral weights, Hilbert modular forms, Siegel modular forms and automorphic forms.