The discipline conducts the following academic programmes regularly:
M.Sc. (2 years): http://www.iitgn.ac.in/jam/index.htm
Ph.D. programme: http://www.iitgn.ac.in/phd.htm
Courses Offered:
| MA 101 : Mathematics I (4-1-0-4) |
| Review of limits, continuity, differentiability; Mean value theorem, Taylor’s Theorem, Maxima and Minima; Riemann integrals, Fundamental theorem of Calculus, Improper integrals, applications to area, volume; Convergence of sequences and series, Newton’s method, Picard’s method; Multi-variable functions, Partial Derivatives, gradient and directional derivatives, chain rule, maxima and minima, Lagrange multipliers; Double and Triple integration, Jacobians and change of variables formula; Parametrization of curves and surfaces, vector fields, Line and surface integrals; Divergence and curl, Theorems of Green, Gauss, and Stokes. |
| MA 102 : Mathematics II (3-1-0-4) |
| Linear Algebra: Vectors in Rn; Vector subspaces of Rn; Basis of vector subspace; Systems of Linear equations; Matrices and Gauss elimination; Determinants and rank of a matrix; Abstract vector spaces, Linear transformations, Matrix of a linear transformation, Change of basis and similarity, Rank-nullity theorem; Inner product spaces, Gram-Schmidt process, Orthonormal bases; Projections and least-squares approximation; Eigenvalues and eigenvectors, Characteristic polynomials, Eigenvalues of special matrices; Multiplicity, Diagonalization, Spectral theorem, Quadratic forms. Differential Equations: Exact equations, Integrating factors and Bernoulli’s equation; Orthogonal trajectories; Lipschitz condition, Picard’s theorem; Wronskians; Dimensionality of space of solutions, Abel-Liouville formula; Linear ODE’s with constant coefficients; Cauchy-Euler equations; Method of undetermined coefficients; Method of variation of parameters; Laplace transforms, Shifting theorems, Convolution theorem. |
| MA 201 : Mathematics III (3-1-0-4) |
| Complex Analysis: Definition and properties of analytics functions; Cauchy-Riemann equations, Harmonic functions; Power series and their properties; Elementary functions; Cauchy’s theorem and its applications; Taylor series and Laurent expansions; Residues and the Cauchy residue formula; Evaluation of improper integrals; Conformal mappings. Differential Equations: Review of power series and series solutions of ODE’s; Legendre’s equation and Legendre polynomials; Regular and irregular singular points, method of Frobenius; Bessel’s equation and Bessel’s functions; Sturm-Liouville problems; Fourier series; D’Alembert solution to the Wave equation; Classification of linear second order PDE in two variables; Vibration of a circular membrane; Fourier Integrals, Heat equation in the half space. |
| MA 202 : Mathematics IV (3-2-0-4) |
| Probability and Statistics
Random Experiments; Events; Probability; Random variables; Probability Distributions: Discrete and Continuous Distributions, Mean and Variance of Distributions, Distributions of Several Random variables; Random sampling; Estimation of parameters; Confidence Intervals; Testing of Hypotheses; Goodness of fit – test; Quality control and Acceptance Sampling; Confidence intervals for regression parameters. Numerical Methods Interpolation: Divided differences, Lagrange and Hermite Interpolation; Weierstrass and Taylor’s theorem (statements); Numerical Quadrature (Trapezoidal Simpson’s and Gauss); Numerical Linear Algebra: Matrix norms, Condition number, Direct and Iterative Methods for Linear Systems, LU and QR Decompositions, Eigenvalue Problems, Inclusion of Matrix Eigenvalues, Eigenvalues by Iteration (Power Method); Numerical Solutions to ODE: Euler’s, Multistep, Runge-Kutta methods, BVP- Finite Difference Method, Introduction to Finite Element Method; Numerical solutions to PDE: Elliptic PDE. |
| MA 403 : Introduction to Real Analysis (3 – 0 – 0 – 6 – 4) |
| Functions, finite and Infinite sets; Real numbers as an ordered, complete, Archimedean field; Cantor set; Nested intervals theorem, Bolzano-Weierstrass theorem; Topology of Cartesian spaces, Heine-Borel theorem; Convergence of sequences and series of numbers, Bolzano-Weierstrass and Cauchy criterion; Continuous functions, global properties, uniform continuity; Sequence of functions, Weierstrass approximation theorem; Derivatives for real functions, extreme values, Rolle’s theorem, mean value theorem, Taylor’s theorem; Riemann integrals, fundamental theorem of calculus. |
| MA 404 : Introduction to Functional Analysis (3-0-0-4) |
| Metric spaces: Convergence and completeness, Uniform continuity and compactness, Baire category theorem and Ascoli-Arzela theorem, Banach’s fixed point theorem and its applications; Normed linear spaces: Finite dimensional normed spaces, Heine-Borel theorem, Riesz lemma; Continuity of linear maps, Hahn-Banach extension theorem; Banach spaces, Dual spaces and transposes; Uniform-boundedness principle and its applications; Spectrum of a bounded operator; Inner product spaces: Hilbert spaces, orthonormal basis, projection theorem and Riesz representation theorem. |
| MA 405 : Topics in Analysis (3-0-0-4) |
| Topics in Complex Analysis: Complex plane and stereographic projection, Generalized form of Cauchy’s integral theorem, Weierstrass theorems for sequences and series; Morera’s theorem and fundamental theorem of Algebra, Existence of harmonic conjugate, Zeros of analytic functions; Conformal mappings and Möbius transformation; Maximum modulus theorem, Maximum Principle, Schwarz’s lemma; Liouville’s theorem, Riemann mapping theorem, Picard’s theorem and Casorati-Weierstrass theorem. Inversion of Laplace transforms.
Integral Equations: Classification, Degenerate Kernels, Neumann and Fredholm series; Schmidt-Hilbert theory. Calculus of Variations: Euler-Lagrange equation, Generalizations of the basic problem. |
| MA-501: Basic Algebra (3-0-0-4) |
| Review of groups, subgroups, homomorphisms, finite and discrete groups of motions, group actions, class equation, Sylow theorems, groups of order 12, generators and relations, SL(R), SU(2), simplicity of alternating groups and PSL(2), Rings, ideals, quotient rings, Euclidean domains, principal ideal domains, unique factorization domains, primes in Z[i] and Fermat’s 2-square theorem, ideal classes in imaginary quadratic fields, Modules, matrices, free modules and bases, diagonalization of integer matrices, generators and relations for modules, structure theorem for abelian groups, applications to Jordan canonical forms and linear operators, Extension fields, splitting fields, fundamental theorem of Galois Theory, constructibility by ruler and compass, finite fields. |
| MA-502: Complex Analysis (3-0-0-4) |
| Spherical representation of extended complex plane, Analytic Functions, Harmonic Conjugates, Elementary Functions, Cauchy Theorem and Integral Formula, Homotopic version, Power Series, Analytic Continuation and Taylor’s theorem, Zeros of Analytic functions, Hurwitz Theorem, Maximum Modulus Theorem and Open Mapping Theorem, Laurent’s Theorem, Classification of singularities, Residue theorem and applications, Evaluation of real integrals, Argument Principle, Theorems of Rouche and Gauss-Lucas, Winding numbers, Mobius Transformations, Schwarz-Christoffel Transformation, Fractals, Algorithms to generate Sierpinski Gasket. |
| MA-503: Functional Analysis (3-0-0-4) |
| Convergence and completeness, Uniform continuity and compactness, Baire category theorem and Ascoli-Arzela theorem, Banach’s fixed point theorem and its applications, Finite dimensional normed spaces, Bounded linear maps on a normed linear spaces: Examples, linear map on finite dimensional spaces, operator norm, Heine-Borel theorem, Riesz lemma, Continuity of linear maps, Hahn-Banach theorems: Geometric and extension forms and their applications, Banach spaces, Dual spaces and transposes, Duals of classical spaces, weak and weak* convergence, BanachAlaoglu theorem, adjoint of an operator, Uniform-boundedness principle and its applications, Compact operators, eigen values, eigen vectors, Banach algebras, Spectrum of a bounded operator, Spectral theorem for compact self adjoint operators, Hilbert spaces, Orthonormal basis, Projection theorem and Riesz representation theorem. |
| MA-504: Introduction to Linear Algebra (3-0-0-4) |
| Fields and linear equations, Vector spaces, Linear transformations and projections, Determinants, Elementary canonical forms: diagonalization, triangulation, primary decomposition etc.
Secondary decomposition theorem, Rational canonical forms, Jordan canonical forms and some applications, Inner product spaces, Gram-Schmidt process, orthonormal bases, projections and least squares approximation, Eigenvalues and eigenvectors, characteristic polynomials, eigenvalues of special matrices (orthogonal, unitary, hermitian, symmetric, skew- symmetric, normal)., algebraic and geometric multiplicity, diagonalization by similarity ransformations, spectral theorem for real symmetric matrices, application to quadratic forms. |
| MA-505: Measure Theory (3-0-0-4) |
| Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theorem. Measure spaces, Outline of extension of measures from algebras to the generated sigma-algebras,Measurable sets; Lebesgue Measure and its properties. Measurable functions and their properties; Integration and Convergence theorems. Introduction to Lp-spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces. Absolute continuity of measures, Radon-Nikodym theorem. Dual of Lp-spaces, Product measure spaces, Fubini’s theorem. Fundamental Theorem of Calculus for Lebesgue Integrals. |
| MA-506: Nonlinear Functional Analysis (3-0-0-4-6*) |
| Calculus in Banach spaces, Inverse & implicit function theorem, Fixed point theorems of Brouwer, Schauder&Tychonoff; Fixed point theorems for nonexpansive& set-valued maps; Degree theory: TherBrouwer degree, The Leray-Schauder degree, Borsuk’s theorem, Genus, Index and applications to differential equations; Bifurcation theory: The Lyapunov-Schmidt method, Morse’s lemma, A perturbation method, Krasnoselskii’s theorem, Rabinowitz theorem.
Variational methods: minimization of functionals, the Palais-Smale condition, the deformation lemma, multiplicity of critical points, Mountain pass theorem, Lyusternik-Schnirelmann theorem. |
| MA-507: Ordinary Differential Equation (3-0-0-4) |
| Vector fields, Graphical representation of solutions, Lipschitz functions, Integral inequalities, Uniqueness of solutions, Boundary value problems, Green’s functions, Distribution of zeros of solutions, Functional analytical preliminaries, Existence of solutions by Picard’s method, Existence by Perron’s method, Fixed point method, Uniqueness and continuous dependence, Continuity and differentiability with respect to initial conditions and parameters, Continuation of solutions, Linear equations, general theory, Solutions of linear equations with constant coefficients, Equations with periodic coefficients, Floquet theory, Classification of stationary points and phase portraits, Oscillation and boundedness of solutions, Lyapunov theory of stability, Poincare-Bendixson theorem and applications. |
| MA-508: Partial Differential Equation (3-0-0-4) |
| First order quasi-linear equations: method of characteristics, Monge cone, Nonlinear equations, Cauchy-Kowalewski’s theorem, Higher order equations and characteristics, Classification of second order equations, Riemann’s method and applications, Wave equation and D’ Alembert’s method, Method of decent and Duhamel’s principle, Solutions of equations in bounded domains and uniqueness of solutions.
Laplace equations: mean value property, maximum principle, Poisson’s formula, BVPs for Laplace’s and Poisson’s equations, Green’s functions and properties, Existence theorem by Perron’s method, Heat equation, Uniqueness of solutions via energy method, Uniqueness of solutions of IVPs for heat conduction equation, Stability theory, Finite difference solutions of partial differential equations. |
| MA-509: Topics in Real Analysis (3-0-0-4) |
| Real number system and set theory : Completeness property, Archimedean property, Nested intervals theorem, Bolzano-Weierstrasstheorem.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, Functions of several variables: Differentiation, inverse and implicit function theorems. Riemann-Stieltjes integral: Definition and existence of the integral, Properties of the integral, Differentiation and integration. Sequence and Series of functions: Uniform convergence, Uniform convergence and uniform continuity, Uniform convergence and integration, Uniform convergence and differentiation, Equicontinuity, Ascoli’s theorem, The Stone-Weierstrass theorem. |
| MA-510: Topology (3-0-0-4) |
| Topological Spaces: open sets, closed sets, neighborhoods, bases, sub bases, limit points, closures, interiors, continuous functions, homeomorphisms, nets and filters.Examples of topological spaces: subspace topology, product topology, metric topology, order topology, quotient topology.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 UrysohnMetrization 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, Stone-CechCompactification. |
| MA 511: Graph Theory and Its Applications (3-0-0-2-3) |
| Course Contents:
Graphs, Discovery of Graphs, Finite, Infinite and Isomorphic Graphs, Connected Graphs, Walk, Path, Cycles, Shortest path, Trees, Some Properties of Trees, Spanning Trees, Cut-sets and Cut-vertices, Max-Flow Min-Cut Theorem, Special Class of Graphs: Bipartite Graphs, Eulerian Graphs, Hamilton Graphs, Planer Graphs, Euler Formula for Connected Planer Graphs, Coloring of Graphs, Five Color Theorem for Planer Graphs. |
| MA 512 : Algebraic Topology (3-0-0-4) |
| Quotient topology, Identification spaces; The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications; Simplicial Complexes, Simplicial and Singular homology – Definitions, Properties and Applications. |
| MA 513 : Numerical Analysis (3-0-0-4) |
| Module 1, Interpolation:
Interpolation Formulas of Lagrange and Newton, Error in Polynomial Interpolation, Hermite Interpolation, Interpolation by Spline Functions, Cubic Spline and B-Splines. Module 2, System of Linear Equations: Gaussian Elimination, Gauss-Jordan Algorithm, The Cholesky Decomposition, Q-R Decomposition, Least Square Approximations, Iterative Methods and Convergence Theorems. Module 3, Nonlinear Equations in R^n: Derivatives and Other Basic Concepts, Convex Functionals, Contractions, Inverse and Implicit Function Theorems, Newton’s Methods and Its Variations and Minimization Methods. Module 4, Differential Equations and Boundary Value Problems: One Step Methods with Convergence, Multi Step Methods with Convergence, Simple and Multiple Shooting Methods, Difference Method and Variational Method. Laboratory Component Each Module 1-4 consists of a mini project component involving numerical computation and analysis of a concrete problem. One of the mathematical softwares – MATHEMATICA, MATLAB, MAPLE – to be used for project investigation. |
| MA 514 : Advanced Probability Theory (3-0-0-4) |
| Probability spaces, random variables, expectations; Independence, conditional expectations and conditional probabilities; Convergence of random variables, Laws of Large Numbers, Central Limit Theorem, Large Deviations; Martingales and Markov chains in discrete time; Convergence of probability measures, Prohorov’s theorem, Wiener measure. |
| MA 515 : Stochastic Differential Equations (3-1-0-4) |
| Overview of measure-theoretic probability; Stochastic processes, filtrations, stopping times, martingales; Brownian motion, construction and properties, Kolmogorov’s extension and continuity theorems; Stochastic integrals, construction and properties, Ito versus Stratonovich, Ito’s formula, Levy’s characterisation of Brownian motion, Girsanov’s theorem; Stochastic differential equations, existence and uniqueness of solutions, strong and weak solutions, Markov property, infinitesimal generator, probabilistic representation of solutions to certain linear partial differential equations; the filtering problem, Kalman-Bucy filter. |
| MA 516 : Number Theory (3-1-0-4) |
| Divisibility, Bezout’s Identity, Linear Diophantine Equations, Prime Numbers, Congruences, Congruences with a Prime-power Modulus, Chinese Remainder Theorem, The Groups of Units Un, Quadratic Reciprocity, Finite Fields. Arithmetical functions and Dirichlet multiplication, big oh notation, Euler’s summation formula, average order of some arithmetical functions, summation by parts, Chebyshev’s functions, the Prime Number Theorem, Dirichlet characters, Gauss sums, Dirichlet’s theorem on primes in arithmetic progressions, Introduction to the theory of the Riemann zeta function, zero-free regions for zeta(s). |
| MA 601 : Mathematical Methods in Engineering (3-0-0-4) |
| Review of Linear Algebra: Vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalization; Inner product spaces, Gram-Schmidt orthonormalization, spectral theorem for real symmetric matrices. Systems of ODEs: Homogeneous and nonhomogeneous linear systems; Eigenvalue method. Nonlinear systems: qualitative approach; linearization. Series solutions of differential equations: Frobenius method, equations of Legendre and Bessel. Sturm-Liouville problems: orthogonality of eigenfunctions and eigenfunction expansions. Fourier series, Fourier integrals and Fourier transforms: basic results. Partial Differential Equations: Classification of linear second order PDEs in two variables; Modeling: vibrating string, heat conduction; solutions using Fourier series, Fourier integrals and Fourier transforms. |
| MA 602 : Advanced Numerical Methods in Engineering (3-0-2-4) |
| Interpolation and Approximation of functions. Eigensystems. Similiarity Transformations, Diagonalisation. Schur Decomposition QR algorithm. SVD. Linear Systems . Krylov Sequence Methods. Krylov Subspaces. Arnoldi Decompositions. Optimization Methods in multi-dimensions. Conjugate Gradient Method and Preconditioned variants as iterative schemes for sparse linear systems. Nonlinear Equations. in numerical handling of ordinary and partial differential equations. Numerical Software Libraries such as PETSc etc and Programming Software Environment.Computational Lab projects on all serial and parallel computer architectures. |
| MA-603: Non-linear Analysis with Applications (3-0-0-4-6*) |
| Elements of Nonlinear Analysis: Review of linear analysis, Differentforms of continuity of nonlinear operators in Banach spaces, Concepts ofGateaux and Frechet differentiability, Subdifferential of convex functionsand duality mapping, Inverse and implicit function theorems.Theory of Monotone Operators and Fixed Point Theorems:Definition with examples of monotone operators, Surjectivitytheorems,Well-known fixed point theorems, Solvability of operator equations,Hammerstein operator equations and applications to boundary valueproblems and integral equations.Variational Analysis with Optimization and Control:Bilinearandsemilinear forms, Convex functionals and minimization, Controllability infinite and infinite dimensional spaces. |
| MA 605 : Commutative Algebra (3-0-0-4) |
| Rings and Ring Homomorphism, Nil radical and Jacobson radical, Modules, Direct Sum and product, Tensor product, Exact sequence, Rings and Modules of fractions, Noetherian and Artinian Rings, Primary Decomposition, Integral extension, Krull dimension of ring, Noether Normalization lemma, Hilbert’s Nullstellensatz, Completions, Graded Rings, Artin-Rees lemma, Associated graded rings. |
| MA 606 : Introduction to de Rham Cohomology (3-0-0-4) |
| A quick review of analysis of several variables, The Alternating Algebra: Multilinear maps, Alternating multilinear maps, Exterior product. Differential forms: Exterior derivative, Pull-back of forms. de Rham cohomology: Poincaré lemma, Chain complexes and their cohomology, Long exact sequences, Homotopy, Application of de Rham cohomology, Smooth manifolds, Differential forms on smooth manifolds, Integration on manifolds. |
| MA 621 : Functional Analysis and Partial Differential Equations (3-0-0-4) |
| The Hahn-Banach theorems in extensi n and separation form , conjugate convex functions; The niform boundedness principle and the closed graph theorem; Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity; L^{p} Spaces; Hilbert Spaces, Basic variational principles; Compact operators; Distributions, Sobolev Spaces; Variational for ulation of elliptic boundary value problems; Weak solutio s of elliptic boundary value problems. |