Course Catalog

Course description not available.

Course description not available.

Course description not available.

Overview: Hermitian, Unitary and Normal operators are used widely. To understand the questions (and the answers) asked about these operators, it is essential to understand the situation in finite dimensional linear spaces. This course aims to do just that. The spectral theorems for Self-adjoint and Normal operators will be proved. If time permits, we shall look at either tensor products or quadratic forms. . Detailed Syllabus 1. Review: Vector spaces, subspaces, linear span, linear independence, quotient spaces, basis, dimension, linear transformations, linear functionals, dual spaces, annihilators, transpose, matrix of a linear transformation, determinants, characteristic polynomial, Cayley-Hamilton theorem, minimal polynomial, characteristic roots, eigenvalues, eigenvectors. 2. Linear Transformations: Range, null space, rank, nullity, row rank, column rank, projections. 3. Inner Product Spaces: Inner products, Cauchy-Schwarz Inequality, Gramian, orthogonal and orthonormal sets, Gram-Schmidt orthogonalisation, complete orthonormal sets, orthogonal complements, perpendicular projections, Complexification. 4. Adjoints: Adjoint of a linear transformation, existence in finite dimensional inner product spaces, inner products on dual space, orthogonal projections. 5. Spectral Theorem: Self-adjoint operators, matrix of self-adjoint operators, Hermitian, Unitary and Normal operators, Positive operators, Spectral theorems for self-adjoint and Normal operators, orthogonal linear transformations, polar decomposition, other consequences of the spectral theorem. 6. If time permits – Tensor Products or Quadratic Forms

Course description not available.

Course description not available.

Core course for B.Sc. (Research) programs in Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 101 (Calculus I) Overview:  Analytic Geometry:  Double and triple integrals: Double integrals over rectangles, double integrals over general regions, double integrals in polar coordinates, center of mass, triple integrals, triple integrals in cylindrical coordinates, triple integrals in spherical coordinates, change of variables. Vector Integration: Vector fields, line integrals, fundamental theorem, independence of path, Green's theorem, surface integrals, Stokes' theorem, Gauss theorem. References: Calculus, Volume II, by Tom M Apostol, Wiley. Essential Calculus – Early Transcendentals by James Stewart, Cengage, India Edition. Calculus and Analytic Geometry by G B Thomas and R L Finney, 9th edition, Pearson. Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1st edition, Springer (India), 2011. Calculus by Ken Binmore and Joan Davies, 1st edition, Cambridge, 2010.

Course description not available.

(a) Introduction to Design Theory: Basic definitions and properties, Incidence matrices, Fisher’s Inequality. (b) Symmetric BIBD’s i. Intersection Property, Residual and Derived BIBD’s, Projective planes and Geometries ii. The Bruck-Ryser-Chowla Theorem (c) Difference Sets and Automorphisms: Quadratic residue difference sets, Singer difference sets. (d) Hadamard Matrices and Designs An equivlance between Hadamard matrices and BIBD’s, Conference matrices and Hadamard matrices, Bent Functions (e) Latin Squares: Steiner Triple systems, Orthogonal Latin Squares, MOL’s, Orthogonal arrays (f) PBIBD’s: Connection of PBIBD’s to Association Schemes and Distance regular graphs. (g) Applications of Combinatorial Design Theory: Medicine, Agriculture, Visual cryptography, Information Security, Statistical designs.

Linear Algebra provides the means for studying several quantities simultaneously. A good understanding of Linear Algebra is essential in almost every area of higher mathematics, and especially in applied mathematics. Detailed Syllabus: 1. Matrices and Linear Systems 2. Vector Spaces and Linear Transformations 3. Inner Product Spaces 4. Determinant 5. Eigenvalues and Eigenvectors 6. Positive definite matrices 7. Linear Programming and Game Theory Matlab will be used in tutorial for computational purposes. Main References: • Linear Algebra and its Applications by Gilbert Strang, 4th edition. • Algebra by Michael Artin, Second edition. Other References: • Linear Algebra: A Geometric Approach by S. Kumaresan, PHI, 2011. • Advance engineering Mathematics by E. Kreyszig, 10th Edition

Major elective course for B.Sc. (Research) Mathematics. Not open to B.Tech. Computer Science majors or any other student who has taken CSD205. Credits: 3:0:1 (3 lectures and 1 tutorial weekly) Prerequisites: Class XII Mathematics Overview: This course offers an in-depth treatment of Lattice theory which will be used in areas of algebra and analysis in the subsequent semesters. Special kinds of lattices known as Boolean algebras are studied in reasonable detail and their importance is demonstrated through real life applications involving digital circuits. This course builds on the Foundations course taken by the first year students and it provides an exposure to formal proof writing. Detailed Syllabus: Theory of Relations: Types of relations, Matrix representation of relations, Equivalence classes, Operations on relations, Closure of relations, Importance of transitive closure, Warshall’s algorithm. Lattice Theory: Posets, Chains, Hesse diagram,  Extremal elements in a poset, Meet and Join operations, Lattices, General properties of lattices, isomorphism, modular lattice, distributive lattice, complements, atoms in a lattice, Boolean algebras. Finite Boolean algebras: Functions on Boolean algebras, Karnaugh maps, Logic gates, Digital circuits. References: Thomas Donnellan, Lattice Theory, Pergamon Press, Oxford. J.E. Whitesitts, Boolean Algebra and Its Applications, Addison-Wesley Publications. G. Birkhoff, Lattice Theory, American Mathematical Society, 2nd Edition. E. Mandelson, Schaum’s Outline of Boolean Algebra and Switching Circuits, McGraw Hill. Bernard Kolman, Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Pearson Education, New Delhi. Kenneth H. Rosen, Discrete Mathematics and its Applications, Tata McGraw-Hill, New Delhi. C. L. Liu, D. P. Mohapatra, Elements of Discrete Mathematics, Tata McGraw-Hill, New Delhi. J.P. Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, 1st edition, Tata McGraw-Hill, New Delhi, 2001. Past Instructors: Niteesh Sahni

A Major Elective for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 220 Real Analysis I. And one of MAT 184 Probability or MAT 205 Mathematical Methods III or CSD209. Overview: This sequel to an introductory course on probability provides a rigourous look at the subject which opens up many more applications, especially to stochastic processes. This course is compulsory for students opting for the Specialization in Mathematical Finance. Detailed Syllabus: Introduction to measures and probability triples Random variables: independence, limit events, expectation Inequalities and convergence, Laws of large numbers Distributions, change of variables Limit Theorems, Differentiation of expectation, Moment generating functions, Fubini’s theorem Weak convergence Characteristic functions, Central Limit Theorem and generalizations, Method of Moments Lebesgue and Hahn decompositions Conditional probability and expectation References: A First Look at Rigorous Probability Theory by J S Rosenthal, 2nd edition, World Scientific Publishing, 2006. Measure, Integral and Probability by M Capinski and E Kopp, 2nd edition, Springer. Probability and Random Processes by G R Grimmett and D R Stirzaker, 2nd edition, Oxford University Press. Past Instructors: Debashish Bose

Core course for BSc (Research) Economics. Students of BSc (Research) Mathematics or any B.Tech. program are not allowed to credit this course. Prerequisites: Calculus I (MAT 101) Overview: Probability is the means by which we model the inherent randomness of natural phenomena. This course introduces you to a range of techniques for understanding randomness and variability, and for understanding relationships between quantities. The concluding portions on Statistics take up the problem of testing our theoretical models against actual data, as well as applying the models to data in order to make decisions. This course is a prerequisite for later courses in Advanced Statistics, Stochastic Processes and Mathematical Finance. Detailed Syllabus: 1.    Probability: Classical probability, axiomatic approach, conditional probability, independent events, addition and multiplication theorems with applications, Bayes’ theorem. 2.    Random Variables: Probability mass function, probability density function, cumulative density function, expectation, variance, standard deviation, mode, median, moment generating function. 3.    Some Distributions and their Applications: Uniform (discrete and continuous), Bernoulli, Binomial, Poisson, Exponential, Normal. 4.    Sequences of Random Variables: Chebyshev’s Inequality, Law of Large Numbers, Central Limit Theorem, random walks. 5.    Joint Distributions: Joint and marginal distributions, covariance, correlation, independent random variables, least squares method, linear regression. 6.    Sampling: Sample mean and variance, standard error, sample correlation, chi square distribution, t distribution, F distribution, point estimation, confidence intervals. 7.    Hypothesis Testing: Null and alternate hypothesis, Type I and Type II errors, large sample tests, small sample tests, power of a test, goodness of fit, chi square test. Main References: •    A First Course in Probability by Sheldon Ross, 6th edition, Pearson. •    John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011. Other References: •    Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance by Kai Lai Chung and Farid Aitsahlia, 4th edition, Springer International Edition, 2004. •    Introduction to the Theory of Statistics by Alexander M. Mood, Franklin A. Graybill and Duane C. Boes, 3rd edition, Tata McGraw-Hill, 2001.  

Overview: Continuing the work done in MAT 220 of understanding the rigor behind one-variable Calculus, this course dwells on various aspects of functions on more general spaces, namely, metric spaces. A brief introduction to the generalities of metric spaces leads to discussions on functions on metric spaces; sequences and series of functions on metric spaces; uniform convergence and consequences; some important approximation theorems for continuous functions; rigorous discussions of some special functions; and then finally to the world of functions of several variables. Detailed Syllabus: 1. Integration: Upper and lower Riemann integrals, basic properties of Riemann integral, Riemann integrability of continuous and monotone functions, non-Riemann integrable functions, Fundamental Theorem of Calculus and consequences. 2. Topology of Rn: Open and closed sets, continuous functions, completeness, compactness, connectedness. 3. Sequences and Series of Functions: Pointwise and uniform convergence, uniform convergence and continuity, series of functions, Weierstrass M-test, uniform convergence and integration, uniform convergence and differentiation, equicontinuous families of functions, Stone-Weierstrass Theorem. 4. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions. 5. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem.

Core course for all B.Tech. Programs. Optional for B.Sc. (Research) Chemistry. Not available as UWE. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: Class XII Mathematics Overview:  We will study Ordinary Differential Equations which are a powerful tool for solving many science and engineering problems. This course also covers some basic linear algebra which is needed for systems of ODEs. Detailed Syllabus: First order ODEs: separable, exact, linear Second order ODEs: homogeneous and nonhomogeneous linear, linear with constant coefficients, Wronskian, undetermined coefficients, variation of parameters Laplace transform: definition and inverse, linearity, shift, derivatives, integrals, initial value problems, time shift, Dirac’s delta function and partial fractions, convolution, differentiation and integration of transform Matrices: operations, inverse, determinant, eigenvalues and eigenvectors, diagonalization Systems of ODEs: superposition principle, Wronskian, constant coefficient systems, phase plane, critical points, stability References: James Stewart, Essential Calculus – Early Transcendentals, Cengage. Erwin Kreyszig, Advanced Engineering Mathematics, Wiley. Past Instructors: Ajit Kumar, Neha Gupta

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 101 Calculus I or MAT 103 Mathematical Methods II Overview: Ordinary Differential Equations are fundamental to many areas of science. In this course we learn how to solve large classes of them, how to establish that solutions exist in others, and to find numerical approximations when exact solutions can’t be achieved. Further, many phenomena which undergo changes with respect to time or space can be studied using differential equations. In this course, we will also see many examples of mathematical modeling using differential equations. Detailed Syllabus: First Order ODEs: Modelling, Geometrical Meaning, Solution techniques Second and Higher Order Linear ODEs: Modelling, Geometrical Meaning, Solution techniques Numerical Techniques Existence of Solutions of Differential Equations Systems of ODEs: Phase Plane and Qualitative Methods Laplace Transforms Series Solutions References: Erwin Kreyszig, Advanced Engineering Mathematics, 9th edition, Wiley India, 2012. G.F. Simmons and S. Krantz, Differential Equations: Theory, Technique, and Practice, McGraw Hill Publishing Company, 2006. J. Polking, D. Arnold, A. Boggess, Differential Equations, Pearson, 2005. C. Henry Edwards and David E. Penney, Differential Equations and Boundary Value Problems: Computing And Modeling, 3rd edition, Pearson, 2010. Hirsch, Morris W., Stephen Smale, and Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2012. Past Instructors: Ajit Kumar

Course description not available.

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: Calculus I (MAT 101) Overview: This course provides a rigorous base for the geometric facts and relations that we take for granted in one-variable Calculus. The main ingredients include sequences; series; continuous and differentiable functions on R; their various properties and some highly applicable theorems. This is the foundational course for further study of topics in pure or applied Analysis, such as Metric Spaces, Complex Analysis, Numerical Analysis, and Differential Equations. Detailed Syllabus: Fundamentals: Review of N, Z and Q, order, sup and inf, R as a complete ordered field, Archimedean property and consequences, intervals and decimals. Functions: Images and pre-images, Cartesian product, Cardinality. Sequences: Convergence, bounded, monotone and Cauchy sequences, subsequences, lim sup and lim inf. Series: Infinite Series: Cauchy convergence criterion, Infinite Series of non-negative terms, comparison and limit comparison, integral test, p-series, root and ratio test, power series, alternating series, absolute and conditional convergence,  rearrangement. Continuity: Limits of functions, continuous functions, Extreme Value Theorem, Intermediate Value Theorem, monotonic functions, uniform continuity. Differentiation: Differentiable functions on R, local maxima, local minima, Mean Value Theorems, L'Hopital's Rule, Taylor's Theorem. References: A Basic Course in Real Analysis by Ajit Kumar and S Kumaresan. CRC Press, 2014. Introduction to Real Analysis by R G Bartle & D R Sherbert, John Wiley & Sons, Singapore, 2/e (or later editions), 1994. Elementary Analysis: The Theory of Calculus by Kenneth A Ross. Springer India, 2004. Analysis I by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002. Calculus, Volume 1, by Tom Apostol, Wiley India. 2nd Edition, 2011. Past Instructors: Pradip Kumar

A Major Elective for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. And MAT 160 Linear Algebra I. Overview: This course combines the traditional approach to learn the basic concepts of curves and surfaces with the symbolic manipulative abilities of Mathematica. Students will learn and study the classical curves/surfaces as well as more interesting curves/surfaces using computer methods. For example, to see the effect of change of parameter, the student will explore and observe with the help of Mathematica and then the mathematical proof of the observation will be developed in the class. Detailed Syllabus: 1- Curves in the plane: Length of a curve, Vector fields along curves. Famous plane curves: cycloids, lemniscates of Bernoulli, cardioids, catenary, cissoid of Diocles, tractrix, clothoids, pursuit curves. 2- Regular curve, curvature of a curve in a plane, curvature and torsion of a curve in R3. Determining a plane curve from given curvature. 3- Global properties of plane curves: Four vertex theorem, Isoperimetric inequality. 4- Curves on Sphere. Loxodromes on spheres, animation of curves on a sphere. 5- Review of calculus in Euclidean space. 6- Surfaces in Euclidean spaces: Patches in R3, local Gauss map, Regular surface, Tangent vectors. 7- Example of surfaces: Graphs of a function of two variables, ellipsoid, stereographic ellipsoid, tori, paraboloid, seashells. 8- Orientable and Non-orientable surfaces. Mobius strip, Klein Bottle. 9- The shape operator, normal curvature, Gaussian and mean curvature, fundamental forms. 10- Surfaces of revolution   References: Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition by Elsa Abbena, Simon Salamon, Alfred Gray. Elementary Differential Geometry by A.N. Pressley, Springer Undergraduate Mathematics Series. Differential Geometry of Curves and Surfaces by Manfredo DoCarmo. Past Instructors: Pradip Kumar

Major Elective for BSc (Research) Mathematics. Available as UWE. Prerequisites: MAT240 Algebra I

Course description not available.

Major Elective for BSc (Research) Mathematics. Available as UWE. Prerequisites: MAT330 PDE.

Overview Ordinary Differential Equations are fundamental to many areas of science. In this course we learn how to solve large classes of them, how to establish that solutions exist in others, and to find numerical approximations when exact solutions can’t be achieved. Further, many phenomena which undergo changes with respect to time or space can be studied using differential equations. In this course we will also see many examples of mathematical modeling using differential equations.

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab) = 3:0:1 (3 lectures and 1 two-hour lab weekly) Prerequisites: Class XII Mathematics Overview: Numerical Analysis takes up the problems of practical computation that arise in various areas of mathematics, physics and engineering. The focus is on analyzing the numerical methods and algorithms for obtaining approximate solutions, error estimates and rate of convergence, and implementation of computer programs. Detailed Syllabus: Solving Equations: Iterative methods, Bisection method, Secant method, Newton-Raphson method, Rates of convergence, Roots of polynomials. Interpolation: Lagrange and Hermite interpolation, Interpolating polynomials using difference operators. Numerical Differentiation: Methods based on interpolation, methods based on finite difference operators. Numerical Integration: Newton-Cotes formula, Gauss quadrature, Chebyshev’s formula. Systems of Linear Equations: Direct methods (Gauss elimination, Gauss-Jordan method, LU decomposition, Cholesky decomposition), Iterative methods (Jacobi, Seidel, and Relaxation methods) Labs: Computational work using C, Python or Matlab. References: E. Suli and D. Mayers, Introduction to Numerical Analysis, Cambridge University Press, 2003. R.L. Burden and J.D. Faires, Numerical Analysis, Cengage Learning, 9th Edition, 2010. M.K. Jain, S.R.K. Iyengar, and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, New Age International Ltd., 1999. J.H. Mathews and K. Fink, Numerical Methods using Matlab, PHI Learning, 4th Edition, 2003.

A Major Elective for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 160 Linear Algebra I Overview: Graphs are fundamental objects in combinatorics. The results in graph theory, in addition to their theoretical value, are increasingly being applied to understand and analyze systems across a broad domain of enquiry, including natural sciences, social sciences and engineering. The course does not require any background of the learner in graph theory. The emphasis will be on the axiomatic foundations and formal definitions, together with the proofs of some of the central theorems. Few applications of these results to other disciplines would be discussed. Detailed Syllabus: Unit 1 Definitions of Graph, Digraph, Finite and Infinite Graph, Degree of a Vertex, Degree Sequence, Walk, Path, Cycle, Clique. Operations on graphs, Complement of a graph, Subgraph, Connectedness, Components, Isomorphism. Regular graph, Complete graph, Bipartite graph, Cyclic graph, Euler graph, Hamiltonian path and circuit, Tree, Cut set, Spanning tree. Unit 2 Planar graph, Colouring, Covering, Matching, Factorization, Independent sets. Unit 3 Graphs and relations, Adjacency matrix, Incidence matrix, Laplacian matrix, Spectral properties of graphs, Matrix tree theorem, Automorphism group of a graph. Unit 4 DFS, BFS for minimal spanning tree, Kruskal, Prim and Dijkstra algorithms. References: D. West, Introduction to Graph Theory, 2nd ed., PHI Learning, New Delhi, 2009. N. Deo, Graph Theory: With Application to Engineering and Computer Science, PHI Learning, New Delhi, 2012. C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, New Delhi, 2013. B. Kolman, R.C. Busby, S.C. Ross, Discrete Mathematical Structures, 6th ed., PHI Learning, New Delhi, 2012. F. Harary, Graph Theory, Narosa, New Delhi, 2012. J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, New Delhi, 2013. R.J. Wilson, Introduction to Graph Theory, 4th ed., Pearson Education, New Delhi, 2003. Past Instructors: Sudeepto Bhattacharya

Geometry of Curves and Surfaces

Core course for B.Sc. (Research) Biotechnology. Only available as UWE with prior permission of Department of Mathematics. Does not count towards a Minor in Mathematics. Credits (Lec:Tut:Lab): (3:1:0) Overview: This course is targeted at undergraduates who did not take Mathematics at +2 level in school, and now need to quickly acquire basic calculus skills in order to satisfy their major requirements. For example, the purpose may be to enable them to take a Probability course which requires basic concepts from calculus. The course will emphasize geometric meaning rather than formal proof. Students who need greater rigour, as well as more computational skills, should take MAT101 Calculus I. Detailed Syllabus: Functions: Real line and its subsets, real functions, graphs, polynomials, rational functions, real powers, trigonometric functions, roots, boundedness, monotonicity, composition of functions, inverse functions. Limits and Continuity: Algebra of limits, left and right limits, limits involving infinity, continuity, left and right continuity, types of discontinuity. Differentiation: Rates of change, tangents to graphs, first and higher derivatives, algebra of differentiation, chain rule, exponentials and logarithms. Applications of differentiation: Exponential growth and decay, intervals of increase and decrease, first and second derivative tests, curve sketching. Integration: Definite and indefinite integrals, Fundamental Theorem of Calculus, substitution, integration by parts, trigonometric integrals, improper integrals. First-Order Differential Equations: Separable differential equations, logistic growth. References: Short Calculus, by Serge Lang, Springer. Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition. Past Instructors: L M Saha

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: Real Analysis I (MAT 220) Overview: Continuing the work done in MAT 220 of  understanding  the rigor behind one-variable Differential Calculus, this course dwells on various aspects of Integration as well as functions on higher dimensional spaces. We discuss sequences and series of functions; uniform convergence and consequences; some important approximation theorems for continuous functions; rigorous discussions of some special functions; and finally the world of functions of several variables. Detailed Syllabus: Integration: Upper and lower Riemann integrals, basic properties of Riemann integral, Riemann integrability of continuous and monotone functions, non-Riemann integrable functions, Fundamental Theorem of Calculus and consequences. Sequences and Series of Functions: Pointwise and uniform convergence, uniform convergence and continuity, series of functions, Weierstrass M-test,  uniform convergence and integration, uniform convergence and differentiation,  equicontinuous families of functions, Stone-Weierstrass Theorem. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions. Topology of Rn: Open and closed sets, continuous functions, completeness, compactness, connectedness. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem.  References: Analysis II by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006. Real Mathematical Analysis by Charles C Pugh. Springer India. 2004. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002. Calculus, Volume 2, by Tom Apostol, Wiley India. 2nd Edition, 2011.  

Core course for B.Sc. (Research) Biotechnology. Only available as UWE with prior permission of Department of Mathematics. Does not count towards Minor in Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures +1 tutorial weekly) Prerequisites: Class XII Mathematics or MAT 020 (Elementary Calculus) or MAT 101 (Calculus I) Overview: Probability is the means by which we model the inherent randomness of natural phenomena. This course provides an introduction to a range of techniques for understanding randomness and variability, and for understanding relationships between quantities. The concluding portions on Statistics take up the problem of testing our theoretical models against actual data, as well as applying the models to data in order to make decisions. This course will act as an introduction to probability and statistics for students from natural sciences, social sciences and humanities. Detailed Syllabus: Describing data: scales of measurement, frequency tables and graphs, grouped data, stem and leaf plots, histograms, frequency polygons and ogives, percentiles and box plots, graphs for two characteristics Summarizing data: Measures of the middle: mean, median, mode; Measures of spread: variance, standard deviation, coefficient of variation, percentiles, interquartile range; Chebyshev’s inequality, normal data sets, Measures for relationship between two characteristics; Relative risk and Odds ratio Elements of Probability: Sample space and events, basic definitions and rules of probability, conditional probability, Bayes’ theorem, independent events Sampling: Population and samples, reasons for sampling, methods of sampling, standard error, Population parameter and sample statistic Special random variables and their distributions: Bernoulli, Binomial, Poisson, Uniform, Normal, Exponential, Gamma, distributions arising from the Normal: Chi-­‐square, t, F Distributions of Sampling statistics: Sampling distribution of the mean, The central limit theorem, Determination of sample size, standard deviation versus standard error, the sample variance, sampling distributions from a normal population, sampling from a finite population Estimation: Maximum likelihood estimator; Interval estimates; Estimating the confidence interval for population mean, variance and proportions; Confidence intervals for the difference between independent means Hypothesis testing: Null and alternate hypothesis; Significance levels; Type  I  and  Type  II  errors;  Tests  based  on  Normal,  t,  F  and  Chi-­‐Square distributions for testing of mean, variance and proportions, Tests for independence  of  attributes,  Goodness  of  fit;  Non-­‐parametric  tests:  the sign test, the Signed Rank test, Wilcoxon Rank-­‐Sum Test. Analysis of variance: Comparing three or more means: One-­‐way analysis of variance, Two-­‐factor analysis of variance, Two-­‐way analysis of variance with interaction Correlation and Regression: Correlation, calculating correlation coefficient, coefficient of determination, Spearman’s rank correlation; Linear regression, Least square estimation of regression parameters, distribution of the estimators, assumptions and inferences in regression; analysis of residuals: assessing the model; transforming to linearity; weighted least squares; polynomial regression Main References: Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press. Other References: Basic and Clinical Biostatistics by Beth Dawson-­‐Saunders and Robert G. Trapp, 2nd edition, Appleton and Lange. John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011. Past Instructors: Sneh Lata, Suma Ghosh

Core course for B.Sc. (Research) Mathematics. Not available as UWE. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: None Overview: Introduction to modern mathematical language and reasoning: Sets and Logic, Proof strategies, Functions, Induction. Detailed Syllabus: Sentential Logic: Deductive reasoning, negation of a sentence, conjunction and disjunction of sentences, equivalence of sentences, truth tables, logical connectives. Sets: Operations on sets, Venn diagrams, cartesian product, quantifiers. Proof Strategies: Direct proofs, proofs involving negations, conditionals, conjunctions, and disjunctions, existence and uniqueness proofs, proofs involving equivalence. Relations and Functions: Ordered pairs, equivalence relations, equivalence classes, partitioning of a set, functions as many-one relations, graphs of functions, one-one functions, onto functions, inverse of a function, images and inverse images of sets. Mathematical Induction: Division algorithm, principle of mathematical induction, well ordering principle, strong induction, principle of recursive definition. More on Sets: Finite and infinite sets, countable and uncountable sets. References: Book of Proof by Richard Hammack, 2nd edition, Richard Hammack. Mathematical Thinking by Keith Devlin, Lightning Source. How to Prove It by Daniel J. Velleman, Cambridge University Press. Mathematical Writing by Franco Vivaldi, Springer. Proofs and Fundamentals by Ethan D. Bloch, Springer. Introduction to Logic and to the Methodology of Deductive Sciences, Alfred Tarski, Oxford University Press. Past Instructors: Amber Habib, Priyanka Grover

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: Linear Algebra I (MAT 160) Overview: In MAT 260 we studied real and complex linear transformations up to the diagonalizability of symmetric operators. In this course we take up vector spaces over arbitrary fields and more advanced results on expressing linear transformations by simple matrices. Detailed Syllabus: Linear Equations – Systems of linear equations, matrices, elementary row operations and row reduction. Vector spaces – Abstract vector spaces, subspaces, dimension, coordinates. Linear transformations – Matrix representations, change of basis, linear functionals and the double dual, transpose. Determinants – Commutative rings, determinant function, permutations, properties. Canonical Forms – Characteristic values, invariant subspaces, simultaneous diagonalization and triangulation, invariant direct sums, Primary Decomposition Theorem, cyclic subspaces, Rational Form, Jordan Form. Inner Product Spaces – Linear functionals and adjoints, unitary and normal operators, spectral theory. References: Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd edition, PHI Learning. Friedberg, Insel and Spence, Linear Algebra, 4th edition, PHI Learning Sheldon Axler, Linear Algebra Done Right, 2nd edition, Springer International Edition Paul Halmos, Finite Dimensional Vector Spaces, 2nd edition, Springer International Edition Paul Halmos, Linear Algebra Problem Book, Mathematical Association of America, 1995. Past Instructors: Neha Gupta

Core course for B.Sc. (Research) programs in Mathematics, Physics and Economics. Optional course for B.Sc. (Research) Chemistry. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: Class XII mathematics or MAT 020 (Elementary Calculus) Overview:  This course covers one variable calculus and applications. It provides a base for subsequent courses in advanced vector calculus and real analysis as well as for applications in probability, differential equations, optimization, etc. One of the themes of the course is to bring more rigour to the formulas and techniques students may have learned in school. Detailed Syllabus: Real Number System: The axioms for N and R, mathematical induction. Integration: Area as a set function, integration of step functions, upper and lower integrals, integrability of bounded monotone functions, basic properties of integration, polynomials, trigonometric functions. Continuous Functions: Functions, limits, continuity, Intermediate Value Theorem, Extreme Value Theorem, integrability of continuous functions, Mean Value Theorem for integrals. Differentiation: Tangent line, rates of change, derivative as function, algebra of derivatives, implicit differentiation, related rates, linear approximation, differentiation of inverse functions, derivatives of standard functions (polynomials, rational functions, trigonometric and inverse trigonometric functions), absolute and local extrema, First Derivative Test, Rolle's Theorem, Mean Value Theorem, concavity, Second Derivative Test, curve sketching. Fundamental Theorem of Calculus: Antiderivatives, Indefinite Integrals, Fundamental Theorem of Calculus, Logarithm and Exponential functions, techniques of integration. Polynomial Approximations: Taylor polynomials, remainder formula, indeterminate forms and L'Hopital's rule, limits involving infinity, improper integrals. Ordinary Differential Equations: 1st order and separable, logistic growth, 1st order and linear. References: Calculus, Volume I, by Tom M Apostol, Wiley. Introduction to Calculus and Analysis I by Richard Courant and Fritz John, Springer Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition. Calculus with Analytic Geometry by G F Simmons, McGraw-Hill Past Instructors: Amber Habib, Debashish Bose

Cryptography

Course description not available.

Error Correcting Codes

Core course for B.Tech. except Computer Science. Not available as UWE. Credits (Lec:Tut:Lab)= 3:0:0 (3 lectures weekly) Prerequisites: MAT 103 (Mathematical Methods I) Overview:  Probability is the means by which we model the inherent randomness of natural phenomena. This course introduces you to a range of techniques for understanding randomness and variability, and for understanding relationships between quantities. The concluding portions on Statistics take up the problem of testing our theoretical models against actual data, as well as applying the models to data in order to make decisions. Detailed Syllabus: Probability: sample space and events, classical and axiomatic probability, permutations and combinations, conditional probability, independence, Bayes’ formula  Random Variables: discrete and continuous probability distributions, mean and variance, binomial and Poisson, normal, joint distributions, covariance, correlation and regression (linear)  Mathematical Statistics: exploring data, random samples, point estimation, Central limit theorem, Maximum likelihood, chi-square, t and F-distributions, confidence intervals, hypothesis testing References: Advanced Engineering Mathematics by Erwin Kreyszig, Wiley. Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press. Theory and Problems of Beginning Statistics by L. J. Stephens, Schaum’s Outline Series, McGraw-Hill John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011. Past Instructors: Charu Sharma, Niteesh Sahni, Suma Ghosh

Project I

Programming in Excel VBA

Core course for B.Sc. (Research) Mathematics. Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. MAT 230 Ordinary Differential Equations or MAT 104 Mathematical Methods II. Overview: Many physics principles like conservation of mass, momentum, energy, when applied to real life scenarios, take the form of PDEs. In this course we will learn how basic physics concepts together with simple calculus translate into mathematical models of many engineering problems in the form of PDEs. We will learn some well-known techniques to solve these problems in simple settings. We will also learn approximation techniques which will be needed in cases where it is impossible to get analytical solutions. Detailed Syllabus: Essentially Chapters 1, 2, 4, 5, 6, and 8 of the book by Strauss. This material will be supplemented with exercises from other prescribed texts and Matlab exercises. The list of topics covered is: Definition of PDEs, well-posedness, initial value and boundary value problems Examples of PDEs, classification of PDEs Wave equation, diffusion equation Source terms Boundary conditions and their impact on solution Fourier Series and their use in solving PDEs Harmonic equations and their solution Numerical methods References: Partial Differential Equations, an Introduction, Second Edition, by Walter A. Strauss Applied Partial Differential Equations by Paul DuChateau, David Zachmann Partial Differential Equations for Scientists and Engineers, by Stanley J. Farlow Past Instructors: Ajit Kumar, Samit Bhattacharyya, Srinivas VVK

Course description not available.

Course description not available.

Evolutionary Game Theory and Applications

Optimal control of systems governed by PDEs

Advanced Linear Algebra

Reading Course: Probabilistic models and Statistical inference

Numerical PDE

Numerical Linear Algebra

Project

Graph Theory and Complex Networks

Signal and Image Processing

Application of Complex Networks to Landscape Ecology

Reading course

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: Overview: Detailed Syllabus: Formal Logic: Statements and notation, Connectives, Normal Forms, Predicate Logic and Inference Theory, Propositional Logic, Proof in Propositional Logic Syntax of First-Order Logic: First-Order Languages, Formulas of a Language, First-Order Theories Semantics of First-Order Languages: Structures of First-Order Languages, Truth in a Structure, Model of a Theory, Embeddings and Isomorphisms Regular Languages and Regular Grammars: Regular Expressions, Regular Languages, Properties of Regular Languages, Regular Grammars Finite Automata: Finite State machines, Finite Automata, Deterministic Finite Automata, Nondeterministic Finite Automata Context-Free Grammars and Languages: Context-Free Grammars, Parsing, Ambiguity in Grammars and Languages, Pushdown Automata, Chomsky Normal Forms Computability: Turing Machines, Decidable Languages, the Halting Problem, Undecidability, Reducibility Time Complexity: Measuring Complexity, the Complexity Class P, the Complexity Class NP, NP-Completeness References: S.M. Srivastava A Course on Mathematical Logic, Springer. J. E. Hopcroft, R. Motwani, J. D. Ullman Introduction to Automata Theory, Languages and Computations, Pearson. J. P. Tremblay, R. Manohar Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw-Hill.

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 623 Analysis II Overview: Detailed Syllabus: Measure and Integration Measure Spaces Measurable functions Integration – Fatou’s Lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem. General Convergence Theorems Signed Measures – Hahn Decomposition Theorem, Jordan decomposition of a measure, Radon-Nikodym Theorem, Lebesgue Decomposition Theorem. The Lp spaces – Riesz Representation Theorem. Measure and Outer Measure Outer measure and measurability The extension theorem – Caratheodory Theorem The Lebesgue-Stieltjes Integral Product measures – Fubini’s Theorem, Tonelli Theorem, Lebesgue Integral on Rn, change of variable. Inner Measure Measure and Topology Baire sets and Borel sets Regularity of Baire and Borel measures Construction of Borel Measure Positive linear functionals and Borel Measures – Riesz Markov Theorem (Dual of Cc(X)). Bounded linear functionals on C(X) – Riesz Representation Theorem   References: Real Analysis by H. L. Royden and P. Fitzpatrick. 4th edition, Prentice-Hall India, 2010. Measure Theory and Integration by G. de Barra, New Age International, reprint 2006. Real Analysis: Modern Techniques and their Applications by G. B. Folland, Wiley, 2nd edition, 1999. Measure Theory by Paul Halmos, Springer, 1974.

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: Linear Algebra, Group Theory Overview:  An overview of graduate algebra with an emphasis on commutative algebra. The instructor may choose 3 or 4 topics from the following, depending on the background and interest of the students. Detailed Syllabus: Group Actions: Orbit-stabilizer theorem, Centralizers, Normalizers, Class Equation, Sylow Theorems. Rings: Homomorphisms and ideals, quotient, adjunction, integral domains and fraction fields, maximal ideals, factorization, unique factorization domains, principal ideal domains, Euclidean domains, factoring polynomials. Modules: Submodules, quotient modules, homomorphisms, isomorphism theorems, direct sums, simple and semisimple modules, free modules, finitely generated modules, Schur’s Lemma, Jordan-Holder Theorem, Modules over a matrix algebra. Fields: Algebraic and transcendental elements, degree of field extension, adjunction of roots, finite fields, function fields, transcendental extensions, algebraic closure. Galois Theory: Galois group,  Galois extension, cubic equations, symmetric functions, primitive elements, quartic equations, Kummer extensions, quintic equations. Homological Algebra: Categories, monomorphisms and epimorphisms, projective and injective modules, left/right exact functors, additive functors, the Hom functor, diagram chasing, push-outs and pull-backs, tensor product, natural transformations, adjoint functors, flat modules. References: Algebra by M Artin, 2nd edition, Prentice-Hall India, 2011. Abstract Algebra by D S Dummit and R M Foote, 2nd edition, Wiley, 1999. Algebra by S. Lang, 3rd edition, Springer, 2005. Algebra by T. W. Hungerford, Springer India, 2005.

Complex Networks

Hardy-Hilbert Spaces and Applications 

Discrete Mathematics

Algebraic Number Theory

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: None. Not open to undergraduates. Overview: The theory of vector spaces is an indispensable tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications  of the language of vector spaces and morphisms among such spaces. Detailed Syllabus: Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem;  statements of various versions of Axiom of Choice. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces. Systems of linear equations: Elementary matrix operations and systems of linear equations. Determinants: Definition, existence, properties, characterization. Diagonalization: Eigenvalues and eigenvectors; diagonalizability;  invariant subspaces; Cayley-Hamilton Theorem. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form. References: Friedberg, Insel and Spence: Linear Algebra, 4th edition, Prentice Hall India Hoffman and Kunze: Linear Algebra, 2nd edition, Prentice Hall India Paul Halmos: Finite Dimensional Vector Spaces, Springer India Sheldon Axler: Linear Algebra Done Right, 2nd edition, Springer International Edition S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India

Research Methodology

Analytic Number Theory

Algebraic Graph Theory

Computational Statistics using R

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 634 (Differential Geometry) and MAT 640 Graduate Algebra I Overview: Representations of groups realize the group elements as linear transformations on vector spaces, or even more concretely, as matrices. This enables the use of linear algebra to study algebra, and connects group theory with other areas such as geometry, harmonic analysis and number theory. In this course, we will first study the representation theory of finite groups, and then that of compact groups. Detailed Syllabus: Part 1: Representations of Finite Groups Review of group actions, representations,  unitarizability and unitary equivalence of finite dimensional representations of finite groups, complete reducibility, group algebra as a *-algebra, regular representations, matrix coefficients, Schur’s lemmas, tensor products of representations, orthogonality of matrix coefficients, orthogonality of characters, direct sum decompositions, projection formulas, dimension theorem, character tables, Frobenius-Schur theorem on real and quaternionic representations, Fourier analysis on finite groups, subgroups of index 2, induced representations, Frobenius character formula, Frobenius reciprocity, Mackey irreducibility criterion. Part 2: Representations of Compact Groups Review of manifolds and Lie groups, the classical compact groups, topological properties of G and G/H, invariant forms and integration, Haar measure, examples of Haar measure for matrix groups, matrix coefficients, characters, Schur orthogonality, review of spectral theory, Schur’s lemma, regular representations, Frobenius reciprocity, Peter-Weyl theorem, representations and harmonic analysis of SU(2), Fourier theory. References: Representation Theory of Finite Groups by Benjamin Steinberg, Springer. Representations of Finite and Compact Groups by Barry Simon, Graduate Studies in Mathematics, American Mathematical Society. Representation Theory – A First Course by William Fulton and Joe Harris, Springer. A First Course on Representation Theory and Linear Lie Groups by S C Bagchi, S Madan, A Sitaram, and U B Tewari, Universities Press. Compact Lie Groups by Mark R Sepanski, Springer. Short Courses in Mathematics by S Kumaresan, Universities Press.

Biomathematics

Formal Languages and Automata Theory II

Algebras of Operators

Operations Research

Business Statistics

Core course for M.Sc. Mathematics Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 621 (Analysis I). Undergraduates not allowed. Overview: A graduate course of one variable complex analysis. “The shortest path between two truths in the real domain passes through the complex domain” – Jacques Hadamard. Detailed Syllabus: The complex number system: The field of Complex numbers, the complex plane, Polar representation and roots of complex numbers, Line and Half planes in the Complex plane, the extended plane and its Stereographic representation. Metric spaces and Topology of complex plane. Open sets in Complex plane, Few properties of metric topology, Continuity, Uniform convergence Elementary properties of Analytic functions. Analytic functions as mapping. Exponential and Logarithm Complex Integration: Basic review of Riemann-Stieltjes integral (without proof), Path integral, Power series representation of an analytic function, Liouville’s theorem and Identity theorem, Index of a closed curve, Cauchy theorem and Integral Formula, Open mapping theorem. Singularities: Removable singularity and Pole, Laurent series expansion, Essential singularity and Casorati-Weierstrass theorem Residues, Solving integral, Argument Principle, Rouche’s Theorem, Maximum modulus theorem. Harmonic Functions: Basic properties, Dirichlet problem, Green function. References: Functions of One Complex Variable by John B Conway, 2nd edition, Narosa. Complex Analysis by Lars Ahlfors, 3rd edition, McGraw Hill Education India. Introduction to Complex Analysis by H A Priestley, Oxford University Press. Complex Function Theory by D Sarason, 2nd edition, TRIM Series, Hindustan Book Agency. Complex Analysis by T W Gamelin, Springer. Complex Variables by M J Ablowitz and A S Fokas, 2nd edition, Cambridge University Press.

Core course for M.Sc. Mathematics Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 230/430 ODE (for undergraduates) Overview: Detailed Syllabus (a) Review of Solution Methods for first order and second order linear equations. (b) Existence and Uniqueness of Initial Value Problems: Lipschitz and Gronwall's inequality, Picard’s Theorem, dependence on initial conditions, continuation of solutions and maximal interval of existence. (c) Higher Order Linear Equations and Linear Systems: fundamental solutions, Wronskian, variation of constants, matrix exponential solution, behaviour of solutions. (d)* Two Dimensional Autonomous Systems and Phase Space Analysis: critical points, proper and improper nodes, spiral points, saddle points, Limit cycles, and periodic solutions. (e)* Asymptotic Behavior: Stability (linearized stability and Lyapunov methods). (f) Sturm-Liouville Boundary Value Problems: Sturm-Liouville problem for 2nd order equations, Green's function, Sturm comparison theorems and oscillations, eigenvalue problems. Sections (d) and (e) will also be explored by computer implementation using MATLAB or other software. References: M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and Introduction to Chaos, Academic Press, 2004. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol. 7, 2nd edition, Springer Verlag, New York, 1998. G. F. Simmons and S. G. Krantz, Differential Equations, Theory, Technique, and Practice, 4th edition, McGraw Hill Education, New Delhi, 2013. William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Wiley, New York, 1992. Past Instructors: Ajit Kumar, Samit Bhattacharyya

Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 621 (Analysis I) or MAT 332/432 (Geometry of Curves and Surfaces) Overview: Differential Geometry generalizes the calculus of several variables on Euclidean spaces to `differential manifolds’. This enables the use of analysis and linear algebra to study geometry. The two highlights of this course are the study of Lie groups and of differential forms, with the latter leading to the general Stokes’ theorem in integration. Detailed Syllabus: Part 1: Calculus in Rn Rn as a normed linear space, derivative, chain rule, mean value theorem, directional derivatives, inverse mapping theorem, implicit function theorem, immersions and submersions, integration, higher derivatives, maxima and minima, existence of solutions of ODE. Part 2: Differential Manifolds and Lie Groups Differential manifolds, smooth maps and diffeomorphisms, Lie groups, tangent spaces, derivatives, immersions and submersions, submanifolds, vector fields, Lie algebras, flows, exponential map, Frobenius theorem, Lie subgroups and subalgebras. Part 3: Differential Forms and Integration Multilinear algebra, exterior algebra, tensor fields, exterior derivative, Poincare lemma, Lie derivative, orientable manifolds, integration on manifolds, Stokes' theorem. References: An Introduction to Differentiable Manifolds and Riemannian Geometry, by William M Boothby, 2nd edition, Academic Press. A Course in Differential Geometry and Lie Groups, by S Kumaresan, TRIM Series, Hindustan Book Agency. Analysis on Manifolds, by James Munkres, Addison-Wesley. Calculus on Manifolds, by Michael Spivak, Addison-Wesley.

Course description not available.

Topics in Mathematics

Core course for M.Sc. Mathematics Credits: 4 (3 lectures and 1 tutorial weekly) Prerequisites: MAT 231/320 Real Analysis II (for undergraduates) Overview: The aim of this course is to build a rigorous base for advanced topics such as Complex Analysis, Measure & Integration, Numerical Analysis, Functional Analysis, and Differential Equations. The course design also attempts to take into account the diverse backgrounds of our students. Detailed Syllabus: Topics under each section are divided in two parts. Part (a) contains topics that will be covered only briefly whereas topics in part (b) will be done in detail. Real number system Archimedean property, density of rationals, extended real numbers, countable sets, uncountable sets. Cauchy completeness of reals, Axiom of Choice, Zorn’s Lemma, equivalence of AC & ZL. Metric spaces Definitions and examples, open sets, closed sets, limit points, closure,equivalent metrics, relative metric, product metric, convergence, continuity, connectedness, compactness. Uniform continuity, completion of a metric space, Cantor’s intersection property, finite intersection property, totally bounded spaces, characterization of compact metric spaces. Sequences and Series of Functions Pointwise and uniform convergence, uniform convergence and continuity, uniform convergence and integration, differentiation, Weierstrass M-test. Power series, exponential and logarithmic functions, Fourier series, equicontinuous family of functions, Stone-Weierstrass approximation theorem, Arzela-Ascoli theorem. References: Principles of Mathematical Analysis by Walter Rudin, Tata McGraw-Hill Mathematical Analysis by Tom M. Apostol, Narosa Topology of Metric Spaces by S. Kumaresan, Narosa Introduction to Topology & Modern Analysis by G. F. Simmons, Tata McGraw-Hill Real Analysis by N. L. Carothers, Cambridge University Press

Mathematical Computing