Course Details:

Unit I: Simple linear regression, multiple linear regression, model adequacy checking, transformations and weighting to correct model inadequacies.
Unit II: Polynomial regression models, orthogonal polynomials. dummy variables, variable selection and model building, multicollinearity.
Unit III:Nonlinear regression. Generalized linear models, autocorrelation, measurement errors, calibration problem, bootstrapping.

Course References:

Text Books:
1. 1. Montgomery, D. C., Peck, E. A., and Vining, G. (2012), Introduction to Linear Regression Analysis (5th ed.), Hoboken, NJ: Wiley.

References:
1. Draper, N. R., and Smith, H. (2003), Applied Regression Analysis, New York: Wiley.
2. Sen, A. A. and Srivastava, M. (1990). Regression Analysis Theory, Methods & Applications, Springer-Verlag, Berlin.
3. Bowerman, B. L. and O'Connell, R. T. (1990). Linear Statistical Models: An Applied Approach, PWS-KENT Pub., Boston.

Course Details:

Discrete-Time Markov Models:
Discrete-Time Markov Chains, Transient Distributions, Occupancy Times, Limiting Behavior, First-Passage Times.
Poisson Processes:
Poisson Processes, Superposition of Poisson Processes, Thinning of a Poisson Process, Compound Poisson Processes.
Continuous-Time Markov Models:
Continuous-Time Markov Chains, Transient Analysis: Uniformization, Occupancy Times, Limiting Behavior, First-Passage Times.
Generalized Markov Models:
Renewal Processes, Cumulative Processes, Semi-Markov Processes.
Queueing Models:
Queueing Systems, Single-Station Queues, Birth and Death Queues.
Brownian Motion:
Standard Brownian Motion, Brownian Motion, First-Passage Times, Martingales and Semimartingales, Black Scholes Formula.

Course References:

Text Books:
1. V. G. Kulkarni, Introduction to modeling and analysis of stochastic systems, second edition, Springer, 2011.

References:
1. S. M. Ross, Stochastic processes, second edition, Wiley, 1996.
2. S. Karlin and H. M. Taylor, A first course in stochastic processes, second edition, Academic Press, 1975.
3. S. M. Ross, Introduction to Probability Models, tenth edition, Academic Press, 2009.

Course Details:

Basic Concepts " Ergodicity, Freeness, Poincare Recurrence, Almost Periodicity, Weak mixing, Mixing; Structure Theorems: Furstenberg-Zimmer structure theorem, Hilbert modules, Extensions; Amenablity: Ameanble groups, Ergodic Theorems, Quasi Tilings, Rokhlin Lemma. Borel Equivalence Relations: Connes-Feldman-Wiess Theorem.

Course References:

1. Peter Walters -- An Introduction to Ergodic Theory, Springer, ISBN 978-0-387-95152-2
2. Eli Glasner " Ergodic Theory via Joinings, AMS, ISBN-10: 0-8218-3372-3
3. Ergodic Theory ?" Hanfeng Li and David Kerr, online book.
Reference Books: Online Lecture Notes --
1. Jesse Peterson Vanderbilt University " Ergodic Theory
2. Blog/Course Notes of Terrence Tao " Ergodic Theory

Course Details:

Representation of finite groups: Definitions, Basic examples, sub-representations, irreducible representations, the tensor product of two representations, character of representation, Schur's lemma, orthogonality of characters, regular representation, induced representation, representation of compact groups, various examples.

Course References:

Text Books:
Introduction to representation theory- J.P. Serre, Springer GTM

Reference Books:
Representation Theory- Fulton and Harris, Springer GTM
Prerequisite: ma5310-linear algebra,ma5320-algebra-1

Course Details:

CourseContent:
General solution of wave equation in R, energy methods, waves with a source, Duhamel’s Principle, reflections of waves, wave equation on an interval: separation of variables, wave equation in higher dimensions, Shock wave: weak solution, entropy condition and Riemann’s problem, shock structure.Sound waves: acoustic energy, simple source, acoustic dipole, and source regions in different fields, ripple, scattering, and radiation from various bodies. One-dimensional waves in fluids: theory of longitudinal waves, transmission of waves, and propagation through branching systems. Water waves: surface gravity waves, internal waves, sinusoidal waves on deep water, ripples, group velocity, wave patterns, Fourier analysis of dispersive systems.

Course References:

TextBooks:
1. Evans, Partial differential equations, AMS, 1997,
2. Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973,
3. Waves in fluids, James Lighthill, Cambridge, 2001,
4. Linear and nonlinear waves, Whitham, Wiley-Interscience, 1999,

ReferenceBooks:
1. Geophysical Fluid Dynamic, Joseph Pedlosky, Springer, 1987,
2. Internal Gravity waves, B. R. Sutherland, Cambridge University Press, 2010,
Prerequisite:Nil

Course Details:

Some Basic Topological Notions: Homotopy, Convexity, Contractibility, Paths and Path Connectedness ; Fundamental group and Covering Spaces: The Fundamental Group of the circle, Induced Homomorphisms� Free Products of Groups, Seifert Van Kampen's Theorem, Covering Spaces, Lifting Properties, The Classification of Covering Spaces, Deck Transformations and Group Actions.

Course References:

Books:
1. Allen Hatcher, Algebraic Topology, Cambridge University Press, New York 2002.
2. W. S. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics 127, Springer Verlag, New York 1991.

Course Details:

Algebra of Cartesian Tensors, Index notation, Isotropic tensors, Invariants of a tensor, Polar decomposition theorem.
Course contents: Continuum Hypothesis, Kinematics of deformation, Compatibility conditions, Balance principles, Euler's laws of motion, Stress-Laws of Thermodynamics, Energy Balance, Constitutive equations of fluids and elastic materials, principles of material objectivity, Applications (simple problems of fluid dynamics and elasticity)

Course References:

Books:
D.S. Chandrasekhraiah and Loknath Debnath, Contiuum Mechanics, Academic Press (1995)

Course Details:

Introduction: Physical systems, models and representations.
Linear system representation: Definitions, state transition map, response map, impulse response matrix, adjoint equations, linear quadratic optimization, applications.
Linear time invariant systems: General properties, minimal polynimial, decomposition theorem, the linear map X—› AX+XB.
Stability: Input-output stability, state related stability concepts and applications.
Controllability and Observability: Controllability and observability of dynamical systems, Controllability of the pair (A(.),B(.)), observability of the pair (C(.),A(.)), duality, Kalman decomposition theorem, minimal realization, controllable canonical form.
Linear state feedback and estimation: Linear state feedback, linear output injection and state estimation, feedback of the estimated state, infinite time horizon linear quadratic optimization.

Course References:

Text Books:
1. L.A. Zadeh and C.A. Desoer, Linear System Theory: The state space approach, Dover, 2008.
Reference:
1. P.J. Antsaklis and A. N. Michel, Linear Systems, Birkhauser, 2005.
2. Chi-Tsong CHen, Linear System Theory and Design, Oxford Univ Press, 3rd edition, 1998.
3. W. J. Rugh, Linear System Theory, Prentice Hall, 2nd edition, 1995.
4. T. Kailath, Linear Systems, Prentice Hall, 1980.

Course Details:

Introduction: Stochastic analogs of classical differential equations.
Mathematica; preliminaries: Probability space, random variable, stochastic process, Brownian motion.
Ito Integral: Definition, Properties, extensions.
Ito formula and Martingale representation Theorem: One-dimensional Ito formula, Multi-dimensional Ito formula, Martingale representation Theorem.
Stochastic differential equations: Examples and some solution methods, Existence an Uniqueness result, weak and strong solutions.
Applications: Boundary value problems, filtering, optimal stopping, stochastic control, mathematical finance.

Course References:

Text Books:
1. B. K. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Apringer, 2010.
Reference:
1. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calxulus, SPringer, 1991.
2. P. Protter, Stochastic Integration and Diferential Equations, Springer, 2nd edition. 2010.
3. I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, Springer, 2010.
4. S. Watanabe and N. Ikeda, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981.

Course Details:

Unit I: Distributions of Random Variables: The probability set function; random variables; probability density function; distribution function; certain probability models; mathematical expectation; Chebyshevs inequality; conditional probability; marginal and conditional distributions; correlation coefficient; stochastic independence.
Unit II: Special Discrete and Continuous Distributions and the Distributions of Functions of Random Variables. Sampling Theory; Transformation of variables: Discrete and Continuous; Distribution of order statistics; moment generating function technique; Distribution of X̄ and S2 and expectations.
Unit III: Limiting Distributions and Estimation: Limiting distributions; stochastic convergence; limiting moment-generating functions; central limit theorem; point estimation, measures of quality of estimators; Confidence intervals: mean, differences of means; variances; Bayesian estimators.
Unit IV: Statistical Hypotheses and other Statistical Tests: Testing of hypotheses, Neyman-Pearson lemma, tests for one sample and two sample problems; chi-squared tests; analysis of variance; regression problem; test for stochastic independence.

Course References:

Text Books:
1. Hogg R. V. and Craig, A. T., Introduction to Mathematical Statistics, Macmillan Publishing Co., Fourth Edition, (1989).

References:
1. G. Casella and R. L. Berger. Statistical Inference. Duxbury Press, (2001).
2. V. K. Rohtagi and A. K. Md. E. Saleh. An Introduction to Probability and Statistics. Wiley Eastern, Second Edition (2002).

Course Details:

Revision of multivariate calculus , inverse and implicit function theorems Smooth manifolds, manifolds with boundary : definitions ,examples, smooth maps, diffeomorphisms, tangent vector, tangent plane, derivative (differential) of a smooth map immersions, submersions, embeddings and submanifolds orienation on manifolds vector fields and Lie bracket, Lie derivative Riemmanian metric, Riemmanian manifold, definition, example, pullback of a metric, vector bundles, tangent and cotangent bundle, Lie groups definition and examples, left , right and bi-invariant Riemmanian metric, connections on manifolds, Riemmanian (Levi-Civita) connections, covariant derivative, connections on vector bundles geodesics, geodesic flow, exponential maps, map, minimizing properties of geodesics curvature, sectional curvature, scalar curvature, tensors on Riemmanian manifolds

Course References:

Text Books:
Manifolds and Differential geometry by J.M. Lee, publisher AMS,Roode Island,2009

Reference Books:
1) Riemmanin geometry by Manfredo Peridiago Do Carmo, Publisher Birkhauser, 1992
2) Foundations of Differentiable manifolds and Lie groups F.K.Warner, publisher Springer,1983.

Prerequisite:
MA5310,MA5370,MA5380

Course Details:

Description:
To introduce manifolds, differential (smooth) structures on manifolds and classifying smooth manifolds up to the notion of diffeomorphisms

Course Content:
Revision of multivariate calculus, inverse and implicit function theorems, Smooth manifolds, manifolds with boundary, definitions ,examples, smooth maps, diffeomorphisms, tangent plane, derivative of smooth maps, immersions, submersions, local immersion and submersion theorems, embeddings and submanifolds, regular and critical points of a smooth map, regular and critical values of smooth maps, Transversality, homotopy and stability of transversality, Sards theorem, Whitney embedding theorem ( We need to define a partition of unity and its existence) The degree modulo 2 of a mapping, Oriented manifolds, The Brouwer degree Vector fields and Euler number, Poincare-Hopf index theorem Differential forms, integration on manifolds, Stoke's theorem on manifolds with boundary and introduction to De-Rham cohomology

Course References:

Text Books:
Differential topology: Victor Guillemin and Alan Pollack, Publisher: Prentice-Hall 1974

Reference Books:
1) Topology from Differentiable viewpoint : John W Milnor , Princeton University Press, 1997
2) Differential Manifolds : Antoni A Kosinski Dover Publications, Inc, New York 1993

Prerequisite:
MA5310, MA5370,MA5380

Course Details:

Description:
The goal of this course is to show how Combinatorial and topological ideas met and grew together into an important branch of mathematics. The course also conveys the fun and adventure that can be part of a mathematical investigation. Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations, computer science, graph theory. Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics — analysis, geometry, and algebra — working together amicably on important problems."

Course Content:
Combinatorial study of convex polytopes: (14/15 lectures)
Basics of Convex Polytopes, Combinatorial type of a polytope, Classification of convex polytopes up to homeomorphism, Polyhedral complex, The classical concept of Euler characteristic and a geometric proof of Euler’s formula for convex polytopes, Classification of Regular 3-Polytopes upto combinatorial equivalence.
Triangulations of manifolds:(14/15 lectures)
Principal Fibrations, a brief recollection on Rn bundles, spherical fibrations and their classifying spaces, PL triangulation on Manifolds and Fundamental Theorem of PL triangulation, Statement of the Product Structure Theorem, Existence and uniqueness of triangulations and PL triangulations of topological manifolds.
Smoothings of piecewise linear manifolds: (14/15 lectures)
Piecewise differentiable, Whitehead triangulation of smooth manifolds, Statement of the Fundamental Theorem of Smoothing and its applications, existence, and uniqueness of smooth structures of the underlying piecewise linear structure on topological manifolds.

Course References:

Text Books:
1. Yuli Rudyak, Piecewise Linear Structures on Topological Manifolds, World Scientific, 2016.
2. Morris W. Hirsch, Barry Mazur, Smoothings of Piecewise Linear Manifolds, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 80


Reference Books:
1. R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton Univ. Press, Princeton, N.J., 1977.
2. Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag,New York, 1977, Graduate Texts in Mathematics, Vol. 47. MR 0488059 (58 #7631)
3. C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. MR 0350744 (50 #3236)

Prerequisite:
Topology (MA 5380)

Course Details:

Linear Programming: Basic feasible solutions and extreme points of a linear polytope, Change of basic feasible solutions and simplex algorithm, the time complexity of the simplex algorithm.

Duality theory, Complementary slackness conditions, Farka's Lemma,

LP-Based ALgorithms: Min-Max relations and LP-duality, Maximum flow Minimum Cut Theorem

Primal-dual algorithms for Minimum Ste cover, Minimum Vertex cover (half-integrality), Minimum Feedback Vertex Set, Algorithms for Maximum Satisfiability

Course References:

Text Books:
1. Combinatorial Optimization: Algorithms and Complexity, C. H. Papadimitrou, K. Steiglitz, Dover Publications, Inc, New York, 1998.
2. Approximation Algorithms, V. V, Vazirani, Springer 2003

Reference Books:
1. Linear Programming, K. G. Murthy, Springer, 1983.
2. The Design of Approximation Algorithms, D.P. Williamson, D. B. Shmoys, Cambridge University Press, 2010.

Course Details:

Module Theory: Right and Left modules ; Examples over Z , polynomial rings and other simple rings ; Quotient modules ; module homomorphisms ; homomorphism theorems.

Ring Theory: Structure Theorem of modules over a PID ; a polynomial ring over a UFD is again a UFD ; basic properties of Noetherian rings and modules ; Universal Property of a Polynomial Ring; Quotient field and localization for integral domains.

Field Theory: Criteria for Irreducibility ; Classical Straightedge and Compass construction and examples ; normal and separable extensions.

Basic Non-Commutative Algebra: Linear Maps and Modules over non-commutative rings ; Simple and Semi-simple modules and rings

Course References:

Text Books:
1. D. S. Dummit and R. M. Foote: Abstract Algebra, 2nd Edition, John-Wiley, 1999.
2. S. Lang: Algebra 3rd Edition, Addison-Wisely, 1999.

Reference Books:
1. J.A. Gallian: Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.
2. M. Artin: Algebra, Prentice Hall inc 1994.
3. I.N. Herstein: Topics in Algebra, John-Wiley, 1995.
4. T. A. Hungerford: Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, 1980.

Course Details:

Combinatorics: Numbers and counting, partitions and permutations, principle of inclusion and exclusion, pigeonhole principle, recurrence relations, generating Functions.
Number Theory: Primes, divisibility and the fundamental theorem of arithmetic, prime number theorem, Euclidean algorithm, congruences, ring of integers mod n, Chinese remainder theorem, arithmetic functions, Fermat's last theorem, Mobius inversion formula, quadratic residues, quadratic reciprocity law, binary quadratic forms, continued fractions, Pell's equation, Diophantine equations

Course References:

Texts books:
1. J. L. Mott, A. Kandel, and T. P. Baker, Discrete Mathematics for Computer Scientists and Mathematicians, PHI Learning, 2003.
2. I. Niven, H.S. Zuckerman, and H.L. Montgomery. An Introduction to the Theory of Numbers, Wiley, 1991.

References:
1. T. Koshy, Discrete Mathematics with Applications, Elsevier, 2004.
2. K.F. Ireland and M.I. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1990.

Course Details:

Riemann - Lebesgue Lemma, localization lemma, Fourier integral theorem, Fourier Transform, Inversion, Convolution and Parseval's Theorem, Applications to Partial Differential Equations.
Laplace transform :Definition, properties, Complex inversion, Applications to initial and boundary value problems.
Z - Transform and Difference equations

Course References:

Text Books:
1. I.N. Sneddon, The Use of Integral Transforms, Tata Mc-Graw Hill (1974)
2. J.L. Schiff, The Laplace Transform, Springer (1999)
3. Michael Frazier, An Introduction to wavelets through Linear Algebra, Springer, 1999 (Chapter 2)
4. A.V. Oppenheim & R.W. Schafer, Digital Signal Processing, Prentice-Hall, 1975.

References:
1. M.R. Spiegel, Laplace Transforms (Schaum's Series), McGraw-Hill, 1965.
2. M.R. Spiegel, Advanced Mathematics for Engineers and Scientists, (Schaum's Series), McGraw Hill, 1983.
3. Jeffery M. Cooper, Introduction to PDE with MATLAB (Chapter 6) Birkhauser, 1997.

Course Details:

Kinematics of Fluid flow, Laws of fluid motion, Inviscid incompressible flows, two and three-dimensional motions, Airfoil theory, inviscid compressible flows;
Viscous incompressible flows, Navier-Stokes equations of motion and some exact solutions; Flows at small Reynolds numbers; Boundary layer theory.

Course References:

Text Books:
1. M.E.O Neill and F. Chorlton, Ideal and Incompressible Fluid Dynamics, John Wiley and Sons, New York, 1986.
2. M.E.O Neioll and FD. Chorlton, Viscous and compressible Fluid Dynamics, John Wiley and Sons, New York, 1989.

References:
1. F. Chorlton, Textbook of Fluid Dynamics, CBS Publishers, 1998.
2. A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid mechanics, Springer-Verlag, 1999.
3. G-Emanuel, Analytical Fluid Dynamics, CRC Press, 2001.
4. P.K. Kundu and I.M. Cohen, Fluid Mechanics (3rd edition) Elsevier Science & Technology, 2002.

Course Details:

Prereq: MA5350
The Methods of Linear Algebra, Polynomials, Nullstellensatz and the Incidence Geometry.
Poset counting using group theory.
Formal Powerseries ring and species of structures.
Probabilistic Method.

Course References:

Text Books:
1. Extremal combinatorics -Stasys Jukna - second edition, Springer. 2001
2. Generatingfunctionology - Hilbert Wilf - CRC Press (available also from his home page) https://www.math.upenn.edu/~wilf/gfologyLinked2.pdf

Reference Books:
1. Combinatorial Species and Treelike Structures - Bergerone and Labelle - Cambridge (1998)
2. Algebraic Combiantorics, Walks, Trees, Tableaux and More - R Stanley - Springer (2013).
3. The Probabilistic Method - Alon and Spencer - Third Edition – Wiley. 2008
4. Linear Algebra Methods in Combinatorics - Babai and Frankl - draft version

Course Details:

Review of probability distributions and statistical inference, Design of experiments, Single-factor, Randomized Blocks, Lotin Square. Regression, linear, multiple, curvilinear, Nonparametric texts, sign, signed-rank, randomness and other nonparametric tests. Statistical Quality control, Control charts for measurements and for attributes, Tolerance limits, Acceptance Sampling. Reliability and life testing.

Course References:

Course Details:

Linear Optimization: Formulation and Geometrical Ideas of Linear Programming Problems, Simplex Method, Revised Simplex Method, Duality, Sensitivity Analysis, Transportation and Assignment Problems, Introduction to Interior-Point Methods. (Ellipsoid Method, Karmarkar's Method).
Unconstrained optimization of functions of several variables: Basic theory, Classical techniques and numerical methods for unconstrained optimization (Gradient methods, Newton's method, Conjugate Direction methods, and Quasi-Newton methods).
Constrained nonlinear optimization of functions of several variables: Method of Lagrange multipliers, Kuhn-Tucker theory, Convex optimization, Quadratic optimization, Numerical methods for constrained optimization, Dynamic programming.

Course References:

Texts books:
1. E.K.P. Chong, and S.H. Zak: An Introduction to Optimization, 3rd Edn., Wiley Interscience, 2008.
References:
1. D. G. Luenberger, Linear and Nonlinear Programming, 2nd Edn., Kluwer, 2003.
2. N. S. Kambo, Mathematical Programming Techniques, East West Press, 1997.
3. M. S. Bazarra, H.D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd Edn., John Wiley, 1993. (also available as WSE (2004) edition).
4. D.P. Bertsekas, Nonlinear Programming, 2nd Edn., Athena Scientific, 1999.

Course Details:

Description:
The course will focus on designing algorithms for matrix computations, analysing these algorithms (in terms of complexity, communication costs, stability, performance in finite precision and exact arithmetic), implementation on different computer architectures.
Course Content:
Floating point arithmetic (1 lecture)
Stability of algorithms (2 lectures)
conditioning of a problem (2 lectures)
perturbation analysis (2 lectures)
algorithmic complexity (1 lecture)
Matrix decomposition including LU, Cholesky, QR, SVD, etc. (12 lectures)
Iterative techniques mainly focussing on Krylov subspace methods including Lanczos, Arnoldi, Conjugate Gradient, GMRES, etc. (12 lectures)
Preconditioning (2 lectures)
structured matrix computations (4 lectures)
designing matrix algorithms on modern computer architectures (3 lectures)

Course References:

Text Books:
1. James W. Demmel, Applied Numerical Linear Algebra, Publisher : Society for Industrial and Applied Mathematics, Year : 1997
2. N. Trefethen & David Bau III, Numerical Linear Algebra, Publisher : Society for Industrial and Applied Mathematics, Year : 1997
ReferenceBooks:
1. Biswa Nath Datta, Numerical Linear Algebra and applications, 2nd Edition, Publisher : Society for Industrial and Applied Mathematics, Year : 2010
Prerequisite:
Linear Algebra at undergraduate level

Course Details:

Introduction to Stochastic Processes, Poisson Process, Brownian Motion, Martingales Present Value Analysis, Interest rate analysis, Market Model Specification problems. Arbitrage Theorem, Multi-period binomial Model, Block - Scholes formula. Valuing investments by expected utility, Portfolio selection problem, Capital Assets Pricing model, Rates of return, Single period and geometric Brownian motion, Mean-variance analysis of risk - neutra-lpriced call options, Autoregressive models and mean regression, Other pricing options and applications.

Course References:

1. S.M.Ross, "An Introduction to Mathematiacl Finance", Cambridge University Press, 1999
2. A.J.Prakash, R.M.Bear, K.Dandapani, G.L.G.T.E.Pactwa and A.M.Parchigari, "The return Generating Models in Global Finance", Pergamon Press, 1998
3. S.M.Ross, "Applied Probability models with Optimization Applications", Holdenday, 1980.

Course Details:

Introduction to linear codes and error correcting codes. Encoding and decoding of a linear code, dual codes. Hamming codes and perfect codes. Cyclic codes. Codes with Latin Squares, Introduction to BCH codes. Weight ennumerators and MDS codes. The main linear coding theory problems and conclusions.

Course References:

Books:
1. Raymond Hill, A first course in Coding Theory, Clarandon Press, Oxford (1986).
2. J. H. Van Lint, Introduction to Coding Theory, Springer (1998).

Course Details:

Approximation of Periodic Functions, Approximation by Algebraic Polynomials, Convergence of Bernstein Polynomials, Korovkin's Theorem, The Stone- Weierstrass Theorem
Approximation in Normed Linear Spaces, Linear Chebyshev Approximation of Vector - valued Functions, Uniqueness Results and Haar Subspaces.
Strong Uniqueness and Continuity of Metric Projection, Upper Semicontinuity of Metric Projections, Lower Continuity, Continuous Selections and Lipschitz Continuity of Metric Projections.

Course References:

1 N. Mhaskar and Devidas V.Pai , Fundamentals of Approximation Theory, Revised Edition Narosa , 2007.

Course Details:

Unit 1: A quick review of Fourier series in L2(T) and Fourier transform on L2(R), Genesis of Wavelets, Short time Fourier transform, Properties and Orthogonality relation, Inversion formula Continuous wavelet transform, Orthogonality relation and Inversion formula (13 lectures)
Unit 2: Discretization, Frames and Riesz bases, Frame operators, Gabor frames and Wavelet frames, Balian-Low theorem (12 lectures)
Unit 3: Multiresolution analysis, System of translates, Construction of wavelets from Multiresolution analysis, Daubechies compactly supported wavelets, Some well known examples of wavelets (15 lectures)

Course References:

Text Books:
1. O. Christensen, Frames and Bases : An Introductory Course, Springer, 2008.
2. A. Henandez and G. Weiss, A First Course on Wavelets, CRC Press, 1996.
Reference Books:
1. I. Daubechies , Ten Lectures on wavelets, SIAM, 1992.
2. M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, GTM vol 102, AMS, 2008.
3. M. W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer, 1999.
4. K. Groechenig, A Foundation of Time-Frequency Analysis, Springer, 2001.

Course Details:

Classical Fractals, Self-similarity, Metric Spaces, Equivalent Spaces, The Space of Fractals,
Transformation on Metric Spaces, Contraction Mapping and Construction of fractals from IFS. Fractal Dimension, Hausdorff measure and dimension, Fractal Interpolation Functions, Hidden Variable FIF, Fractal Splines, Fractal Surfaces, Measures on Fractals, Complex Dynamics, Mandelbrot Set, Julia Set and Fatou Set.

Course References:

Books:
1. M.F. Barnsley, Fractals Everywhere, Academic Press.
2. K. Falconer, Fractal Geometry (Mathematical Foundations and Applications), John Wiley \& Sons.
3. Robert Devaney, A First Course in Dynamical Systems, Addison-Wesley

Course Details:

Review of Schur’s unitary triangularization theorem, its consequences: Denseness of diagonalizable matrices, simultaneous triangularization, continuity of eigenvalues, eigenvalues of rank one perturbation, Various characterizations of normal matrices.
Singular value decomposition and its applications including Moore-Penrose inverse, Polar decomposition.
Hermitian matrices and congruences: Characterizations, Variational principles of eigenvalues, Courant Fischer minimax theorem, Weyl’s inequality and Cauchy’s interlacing theorem.
Algebraic, Analytic and geometric properties of vector norms, various matrix norms, Gershgorin’s theorem, spectral radius and Gelfand’s formula.
Positive definite matrices: Various characterizations, Schur product, Hadamard and Fischer inequalities.
Non-negative matrices: Perron-Frobenius theory, Birkhoff theorem for doubly stochastic matrices.

Course References:

Text Book:
1. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge university press, 2013.
References:
1. F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd Ed., Springer, 2011.
2. M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Dover Publications, 2008.

Course Details:

Classification of Integral Equations, various examples, Abel's problem, 2nd order ordinary differential equations and Integral Equations, Initial and boundary value problems, singular boundary value problems.
Integral Equations of second kind: degenerate kernels, Neumann series.
Compact self-adjoint operators: Structure theorem, spectrum, Applications to integral equations, positive operators and Integral equations arising from Sturm-Liouville theory.
Approximate methods for eigenvalues and eigenvectors of self-adjoint operators, Approximation of integral equations based on variational principles.
Singular integral equations: introduction, solution methods, applications.

Course References:

Books:
David Porter & David S.G. Stirling, Integral Equations: A Practical Treatment, from Spectral Theory to Applications, Cambridge texts in Applied Mathematics, 1990.

Course Details:

1.Review of first order equations and characteristics
2.Weak solutions to hyperbolic equations- discontinuous solutions, shock formation, a formal approach to weak solutions, asymptotic behaviour of shocks
3.Diffusion Processes-Similarity methods, Fisher's equation, Burgers' equation, asymptotic solutions to Burgers' equations
4.Reaction diffusion equations-traveling wave solutions, existence of solutions, maximum principles and comparison theorem, asymptotic behaviour
5.Elliptic equations-Basic results for elliptic operators, eigenvalue problems, stability and bifurcation
Hyperbolic system

Course References:

Books:
J David Logan, An introduction to nonlinear partial differential equations, John Wiley and Sons, Inc., 1994

Course Details:

One-dimensional flows: fixed points and stability, linear stability analysis, Saddle- node bifurcation, Transcritical bifurcation, Pitchfork bifurcation, Flows on the circle
Two-dimensional flows: Linear systems, nonlinear autonomous systems, phase portraits, Fixed points and linearization, conservative systems, index theory, limit cycles, Poincare Bendixson theorem, Lienard systems, Saddle-node, Transcritical and Pitchfork bifurcations, Hopf bifurcation
Chaos
Discrete dynamical systems

Course References:

Books:
(1) Steven H Strogatz, Nonlinear dynamics and Chaos, Perseus books publishing, 1994
(2) James T Sandefur, Discrete dynamical systems Theory and applications, Clarendon press, 1990

Course Details:

Integral and differential forms of conservation laws, Burgers equation, Shock formation, Weak solutions, Entropy conditions, Riemann problem, scalar and non-linear vector examples, Euler equations of an ideal gas, Courant Fredrick Levy conditions, Upwind methods, Conservative schemes, Godunov methods, Riemann solvers.

Course References:

Books:
R.J. Leveque, Numerical methods for conservation Laws, Bikhauser Verlag, 1992.

Course Details:

Unit 1: The genesis of Fourier Analysis, Fourier Series, Uniqueness of Fourier coefficients, Pointwise and Uniform convergence of Fourier series, Convolution in L1(T), Cesaro and Abel summability of Fourier series, Fourier series in L2(T), A continuous function with a divergent Fourier series, Applications to Weierstrass Approximation theorem and The Isoperimetric Problem. (15 lectures)
Unit 2: Schwartz space on R, Fourier transform on the Schwartz space, Fourier transform on L2(R); Fourier transform on L1(R), Density of D(R) in Lp(R); 1 ≤ p < ∞. (12 lectures)
Unit 3: Poisson summation formula, Heisenberg uncertainty principle, Hardy's theorem, Paley-Wiener theorem, An introduction to tempered distributions, Differentiation, convolution and Fourier transform on the class of tempered distributions. (13 lectures)

Course References:

Text Books:
1.E. M. Stein and R. Shakarchi, Fourier Analysis - An Introduction, Princeton Lectures in Analysis, Volume 1, 2003.
2.H. Dym and H. P. McKean, Fourier series and Integrals, Academic Press, 1972.
Reference Books:
1.R. Radha and S. Thangavelu, Fourier Analysis, Lecture Notes, 2012, https://nptel.ac.in/courses/111106046/
2.M.A. Pinsky, Introduction to Fourier Analysis and Wavelets, GTM vol 102, AMS, 2008.
3.E. M. Stein and G. Weiss, An introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, 1971.
4.Y. Katznelson, An introduction to Harmonic Analysis, Cambridge University Press, 3rd Edition, 2002.

Course Details:

Distribution theory: The space of test functions and the convergence, the space of distributions, support and singular support of distributions, the convolution of distributions, fundamental solutions, tempered Distributions. [12]
Sobolev spaces: Definition and basic properties Sobolev spaces, extension theorems, Sobolev embedding theorems, Rellich-Kondrasov compactness theorems, dual spaces, fractional order Sobolev spaces, trace theorems. [16]
Weak Solutions: Abstract variational problems, Lax-Milgram theorem, weak solutions of second order elliptic equations, regularity of weak solutions, maximum principle, the eigenvalues of Laplacian. [12]

Course References:

Text Books:
1. S. Kesavan, Topics in Functional Analysis and Applications, New Age International Publishers, 2015.
2. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
Reference Books:
1. R. A. Adams and J. F. Fournier, Sobolev Spaces, Academic Press, 2003.
2. W. Rudin, Functional Analysis, Tata McGraw-Hill, 2006.
3. R. S. Strichartz, A guide to Distribution Theory and Fourier Transforms, World Scientific, 2008.
4. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004.

Course Details:

Unit-1: Review of general positive measure and integral; Signed measures; Hahn decomposition and Jordan decomposition; Lebesgue Radon-Nykodym theorem; Complex measures and Radon-Nykodym theorem for complex measures; Total variation norm. (15 Lectures)}
Unit-2: Stone Weiestrass theorem; Lp spaces on a general measure space, and its completeness; Vitali's convergence theorem; Dual of Lp for 1 ≤ p < ∞; Chebychev's inequality. (10 Lectures)
Unit-3: Positive linear functionals on Cc(X), where X is a locally compact Hausdorff space; Riesz representation theorem; Luzin's theorem; Vitali Caratheodory theorem; Denseness of Cc(X) in Lp(X) for 1 ≤ p < ∞; Dual of C0(X), Riesz-Thorin interpolation theorem. (15 Lectures)

Course References:

Text Books:
1. W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, International Editions, 1987.
2. G. Folland}, Real Analysis: Modern techniques and Their Applications, Second Edition, John Wiley & Sons, INC, 1999.
Reference Books:
1. H.L. Royden, Real Analysis, Third edition, Prentice-Hall of India, 1995.
2. S. Kesavan, Measure and Integration, TRIM, 2019.

Course Details:

Unit I: Analytic Continuation, Monodromy theorem, Hurwitz' theorem, Inverse function theorem, Winding number, Simply connected domains. (15 Lectures)
Unit II: Automorphisms of the upper half plane, the unit disc, Schwarz-Pick Lemma, Montel’s theorem, Riemann mapping theorem. (10 Lectures)
Unit III: The Poisson Integral formula, Characterization of harmonic functions, Schwarz Reflection principle, Runge’s theorem, Mittag-Lefler theorem, Infinite products, Weierstrass’ product theorem, Gamma and Zeta functions – a brief introduction. (15 Lectures)

Course References:

Text Books:
1. T.W. Gamelin, Complex Analysis, Springer-Verlag, 2001.
2. S. Ponnusamy and H. Silverman, Complex Variables with Applications, 2006, 524 pp, Birkhaeuser, Boston.
Reference Books:
1. L. Ahlfors: Complex Analysis, 2nd ed., McGraw-Hill,New York, 1966.
2. J.B. Conway, Functions of One Complex Variable, 2nd ed., Springer, 2002.

Course Details:

Contraction Principle, and its variants and applications; Fixed points of nonexpansive maps and set valued maps, Brouwer -Schauder fixed point theorems, Ky Fan Best Approximation Theorem, Principle and Applications of KKM -maps, their variants and applications.
Fixed Point Theorems in partially ordered spaces and other abstract spaces.
Application of fixed point theory to Game theory and Mathematical Economics.

Course References:

Books:
1. M.A.Khamsi and W.A.Kirk, An introduction to Metric Spaces and Fixed Point Theory, Wiley - Inter Sci. (2001)
2. Sankatha Singh, Bruce Watson and Pramila Srivastava, Fixed Point Theory and Best Approximation: The KKM - map Principle, Kluwer Academic Publishers, 1997.
3. Kim C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, 1985.

Course Details:

Unit I: Operators on Hilbert spaces: self-adjoint, normal, unitary, isometry, partial isometry, projections, positive operators. (7 Lectures)
Unit II: Spectral Results: Eigen spectrum and spectrum; spectral radius formula; spectral mapping theorem; spectrum of various operators on Hilbert spaces. (12 Lectures)
Unit III: Finite rank operators, compact operators; Riesz-Schauder theory for compact operators; spectral theorem for compact self-adjoint and compact normal operators; singular value decomposition of compact operators; Trace class & Hilbert Schmidt operators. (11 Lectures)
Unit IV: Continuous functional calculus; Polar decomposition; Borel functional calculus; Spectral theorem for self-adjoint & normal operators - sketch of the proofs. (10 Lectures)

Course References:

Text Books:
1. M. T. Nair, Functional analysis: A First Course, PHI-Learning, New Delhi, 2002 (Fourth Print: 2014).
2. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1962.
Reference Books:
1. J. B. Conway, A Course in Functional Analysis, Springer, 1997.
2. C.D. Kubrusly, Elements of Operator Theory, Birkhauser, 2001.
3. B.D. MacCluer, Elementary Functional Analysis, Springer, 2009.
4. V. S. Sunder, Functional Analysis: Spectral Theory, Hindustan Book Agency (TRIM Series), 1997.

Course Details:

AFFINE AND PROJECTIVE VARIETIES: Noetherian rings and modules; Emmy Noether's theorem and Hilbert's Basissatz; Hilbert's Nullstellensatz; Affine and Projective algebraic sets; Krull's Hauptidealsatz; topological irreducibility, Noetherian decomposition; local ring, function field, transcendence degree and dimension theory; Quasi-Compactness and Hausdorffness; Prime and maximal spectra; Example: linear varieties, hypersurfaces, curves.
MORPHISMS: Morphisms in the category of commutative algebras over a commutative ring; behaviour under localization; morphisms of local rings; tensor products; Product varieties; standard embeddings like the segre- and the d-uple embedding.
RATIONAL MAPS: Relevance to function fields and birational classification; Example: classification of curves; blowing-up.
NONSINGULAR VARIETIES: Nonsingularity; Jacobian Criterion; singular locus; Regular local rings; Normal rings; normal varieties; Normalization; concept of desingularisation and its relevance to Classification Problems; Jacobian Conjecture; relationships between a ring and its completion; nonsingular curves.
INTERSECTIONS IN PROJECTIVE SPACE: Notions of multiplicity and intersection with examples.

Course References:

Books:
1. Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics (GTM) Vol.52, Corr. 8th Printing, 1997, Springer-Verlag .
2. C. Musili, Algebraic Geometry for Beginners, Texts and Readings in Mathematics 20, Hindustan Book Agency, India, 2001

Course Details:

Formal proofs, resolution, Axiom systems, strong completeness and compactness of propositional logic.

First order with equality: First order structures in mathematics, Propositional reduction, completeness and compactness.Variants of Lowenheim-Skolem theorem. Some complete axiom systems. Isomorphisms and equivalence of structures. Expressive and distinguishing power of First order logics. EF games and 0-1 law. Proof sketch of Incompleteness theorems. Undecidability.

Course References:

Text Books:
1. A. Singh, Logics for Computer Science, PHI Learning. 2003
2. Lecture notes by Madhavan Mukund and S P Suresh (https://www.cmi.ac.in/~madhavan/papers/pdf/logic-aug2011.pdf)
Reference Books:
1. A Friendly Introduction to Mathematical Logic - Christopher C. Leary, Lars Kristiansen - Milne Library 2nd edition 2015.
2. A course in mathematical logic - Yu I Manin – Springer. 1977
3. S. Bilaniuk, A Problem Course in Mathematical Logic, GNU Free Documentation, http://euclid.trentu.ca/math/sb/pcml/, 2003.
4. Leonid Libkin, Elements of Finite Model Theory, Springer, 2004. (https://homepages.inf.ed.ac.uk/libkin/fmt/fmt.pdf)

Course Details:

Grammars and Languages: Language basics, Regular expressions, Regular grammars, Contextfree grammars, context-sensitive grammars, unrestricted grammars, Chomsky hierarchy.
Automata: Finite automata, pushdown automata, Pumping Lemmas and Closure properties, Turing machines and recursively enumerable languages.
Computability: Computable functions, non-recursively enumerable languages, Undecidability, Rice's theorem, Post's correspondence problem, Undecidability of validity problem of First Order Logic.
Complexity: Asymptotic order symbol, Space and Time complexity, Classes P and NP, NP-completeness, Cook-Levin tehorem, Other NP-complete problems.

Course References:

Text Books:
1. K.Krithivasan and R.Rama, Introduction to Formal Languages, Automata and Computation, Pearson Education, 2009.
2. A.Singh, Elements of Computation Theory, Springer (In: Texts in Computer Science Series),2009.

Course Details:

Complexity of optimization problems: Complexity classes P and NP; Karp reduction; decision, solution and evaluation vertions of an optimization problem.
Design techniques for approximation algorithms: Greedy methods for knapsac, independent set and TSP; sequential algorithms for scheduling, bin packing and graph coloring; local search algorithms for max-cut and TSP.
Approximation classes: Approximate solutions with guaranteed absolute error and relative error; approximability and non-approximability of TSP; limits to approximability (gap technique); complexity classes PTAS and FPTAS; strong NP-completeness and pseudo-polynomiality.
Approximation algorithms for various problems: Set-cover, graph coloring, minimum multi-cut, edge coloring, bin packing.

Course References:

Text Books:
1. G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. M. Protasi, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer, 1999.
Reference Books:
1. V. Vazirani, Approximation Algorithms, Springer 2005.
2. C. H. Papadimitriou, K. Steiglitz, Combinatorial optimization: Algorithms and Complexity, PHI 2001.
3. J. Kleinberg, E. Tardos, Algorithm Design, Pearson India, 2007.

Course Details:

Basics: subgraphs, isomorphism, automorphism group, matrices associated with graphs, degrees, walks, connected graphs, shortest path algorithms.
Connectivity: Connectivity and Mengers theorem; Structure of 2-connected and 3-connected graphs, Maders theorem.
Matchings: Berge's theorem, Hall's theorem, Tutte's perfect matching theorem, k-matchings (reduction to perfect matching problem), job-assignment-problem.
Ramsey Theory: Pigeonhole Principle, Ramsey Theorem, Ramsey Numbers.
Eulerian and Hamiltonian graphs: characterization of Euler graphs, necessary/ sufficient conditions for the existence of Hamilton cycles, Fleury's algorithm for eulerian trails, Chinese-postman-problem (complete algorithmic solution), approximate solutions of traveling salesman problem.
Planar Graphs: Euler's formula, Dual graphs, Characerization of planar graphs, planarity testing. Coloring: Brooks' Theorem, Graphs with large chromatic number, Turan's theorem.

Course References:

Text Books:
1. D.B. West, Introduction to graph theory, PHI Learning, New Delhi, 2004.
Reference Books:
1. J.A. Bondy and U.S.R. Murty, Graph Theory and Applications, GTM Vol. No. 244, Springer 2008.
2. J.A. Bondy and U.S.R. Murty, Graph Theory and Applications, GTM Vol. No. 244, Springer 2008.

Course Details:

Parabolic Equations: explicit and implicit finite difference approximations to one - dimensional heat equation, Alternating Direction Implicit (ADI) method, Hyperbolic Equation: Characteristic method, finite difference solution of second order wave equation, Elliptic equations: finite difference method in polar coordinates, techniques near curved boundaries, improvement of accuracy, methods to accelerate the convergence, Convergence, consistency and stability analysis, Finite element method: types of integral formulations, one and two dimensional elements, Galerkin formulation, application to Dirichlet and Neumann problems.

Course References:

Books:
1. G D Smith, Numerical solution of partial differential equations: Finite difference methods, Oxford University press, 1977.
2. G. Evans, J. Blackledge, P. Yardley, Numerical Methods for Partial Differential Equations, 2nd edition, Springer, 2001.

Course Details:

PART I: STATIC GAMES
1. Two-person zero-sum finite games: Matrix games, Pure and mixed strategies, saddle point equilibrium strategies, extensive form games.
2. N-person finite games: Nash equilibrium, Refinements of Nash equilibirum, N-person games in extensive form.
3. Infinite games: Equilibrium strategies, continuous-kernel games, stackelberg equilibirum.
PART II: DYNAMIC GAMES
3. Formulation of dynamic games: Discrete-time dynamic games, continuous-time dynamic games, mixed and behaviour strategies in dynamic games, representations of strategies along trajectories, time-consistency of optimal policies.
5. Equilibria of dynamic games: Open-loop and feedback equilibria for dynamic games in discrete-time, informational properties of Nash equilibrium in discrete-time dynamic games, Open-loop and Feedback equilibria for differential games.
6. Pursuit-evasion games: Necessary and sufficient conditions for equilibrium, captuability, singular surfaces.

Course References:

TEXT BOOKS:
T.Basar and G.J.Olsder, Dynamic Noncooperative Game Theory, SIAM, 2nd edn, 1999
REFERENCES:
1. R. B. Myerson, Game Theory: Analysis of Conflict, Harvard University Press, 1999
2. M. J. Osborne and A.Rubinstein, A course in Game Theory, MIT Press, 1994
3. D. Fudenberg and J. Tirole, Game Theory, MIT Press, 1991

Course Details:

Hilbert spaces: The projection theorem - Approximation - Minimum norm problems. Calculus of variations: Extrema functionals - Gateax variatin - Euler-Lagrange equations - Problems with variable end points - Isoperimetric problem - Direct methods for solving variational problems.
Optimal Control Theory: Pontryagin's maximum principle - Time optimal problems.

Course References:

Course Details:

Automata and words: Descriptional complexity of finite state automaton, local languages, syntactic semigroups, tree automata, words, Fine-Wilf theorem.
Codes: Basics of coding, types of codes, test for codes, flower automaton, finite transducers for codes.
Shift spaces: Finite and infinite, sofic shifts, entropy, Perron-Frobenius theory, finite state codes, cellular automata, dynamical systems.

Course References:

Text Books:
1. J. Berstel, D. Perrin, C. Reutenauer, Codes and Automata, Cambridge University Press, 2010.
2. D. Lind, B. Marcus, An introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.
Reference Books:
1. M. Ito, Algebraic theory of Automata and Formal Languages, Cambridge University Press, 2004.
2. M. Lothaire, Combinatorics on Words, Cambridge University Press, 1997.
3. A. Salomaa, G.Rozenberg, Handbook of Formal Languages (Vol I), Springer, 1997.
4. J-P. Allouche, J.Shallit, Automatic Sequences, Cambridge University Press, 2003.
5. S. Ginsberg, Algebraic and Automata-Theoretic Properties of Formal Languages, North Holland Publishing Co., 1975.
6. R. Tao, Finite Automata and Applications to Cryptography, Springer, 2009.

Course Details:

Review of the governing equations of Incompressible viscous flows, Stream function -vorticity approach, upwind schemes, Primitive variables, Staggered grid, Artificial compressibility, pressure correction and vortex methods; Compressible inviscid flows, central schemes with combined and independent space time discretisation, Compressible viscous flows, Explicit, implicit and PISO methods; Grid generation: Structured and unstructured grid generation methods; Finite volume method: Finite volume method to convection-diffusion equations.

Course References:

Books:
1. P Wessling, Principles of Computational Fluid Dynamics, Springer, 1991.
2. John D Anderson, Jr., Computational Fluid Dynamics, The Basics with Applications, McGraw-Hill, 1995.

Course Details:

Rings and Modules, Localisation of Rings and Modules, Noetherian Rings and Modules, Primary Decomposition, Artinian rings, Integral Extensions, Going-up, Lying-over and Going-down Theorems, Hilbert's Nullstellensatz, Noether's Normalisation, Dimension Theorem, Krull's Principal Ideal Theorem, Dedekind Domains.

Course References:

Books:
1.M. F. Atiyah and I. G. Macdonald: Introduction to Commutative Algebra, Addision-Wesley , 1969.
2.N. S. Gopalakrishnan: Commutative Algebra, Oxonian Press, 1984.

Course Details:

Algebraic Extensions of Fields, Splitting Fields and Algebraic Closures, Separable and Inseparable Extensions, Cyclotomic Polynomails and Extensions, Basics of Galois theory, Fundamental Theorem and Galois Theory, Finite Fields, Composite Extensions and simple Extensions, Cyclotomic Extensions and Abelian Extensions over q, Galois Group of Polynomials, Solvable and Radical extensions, Insolvability of the quintic equations.

Course References:

Books:
1. D.S. Dummit and R.M Foote: Abstract Algebra,2nd Edition, John-Wiley, 1999.
2. S. Lang: Algebra. Revised third edition. Graudate Texts in Mathematics, 211. Springer- Verlag, New York, 2002.

Course Details:

Commutative Algebra (General Theory of Dedekind Domains): Localization, Integral Dependence, DVR and Dedekind Domains, Modules over Dedekind Domains, Decomposition of primes in extensions.
Class Groups and Dirichlet's Unit Theorem: Class groups, Norms and Traces, Discriminant, Norms of Ideals, Computing ring of integer in Quadratic Fields and Cyclotomic Fields, Quadratic Reciprocity, Minkowski's Bound, Dirichlet's Unit Theorem.
Zeta and L-functions and applications: L-functions, Riemann Zeta functions, Dedekind Zeta functions, Class number formula, Frobenius Density Theorem, Dirichlet L-functions, Dirichlet's theorem on arithmetic progression.
Theory of p-adic numbers: Valuations, Non-Archimedean absolute values, Completion and local fields, p-adic integers, Hensel's Lemma and applications, Ostrowski's theorem.

Course References:

Text Books :
Gerald J. Janusz: Algebraic Number Fields, Second Edition. Graduate Studies in Mathematics, Volume 7. (American Mathematical Society, 1996)
Reference Books:
Daniel A. Marcus, Number Fields,Springer, New York (1977). Universitext
Serge Lang, Algebraic Number Theory, Second Edition. Graduate Texts in Mathematics, Volume 110 (Springer, New York, 1994) J. Neukirch,
Algebraic Number Theory. Grundlehren Math. Wiss. (Springer, Berlin, 1999)
J.W.S. Cassels and A. Frohlich, Algebraic Number Theory, Second Edition. London Mathematical Society, 2010.

Course Details:

Introduction: Brief introduction to number theory, Euclidean algorithm, Euler’s totient function, Fermat’s theorem and Euler’s generalization, Chinese Remainder Theorem, primitive roots and discrete logarithms, Quadratic residues, Legendre and Jacobi symbols.
Private key cryptography: Stream ciphers, Block ciphers, DES and differential and linear cryptanalysis, Advanced encryption standards, Collision resistant hashing, Authenticated encryption: security against active attacks.
RSA public key cryptosystems: RSA system, primality testing, survey of factoring algorithms. Other public key cryptosystems: El Gamal public key cryptosystem, algorithms for discrete log problem.

Course References:

Text Books:
1. Introduction to Modern Cryptography by J. Katz and Y. Lindell.
Reference Books:
1. Cryptography: Theory and Practice by D. Stinson.
2. Handbook of Applied Cryptography by A. Menezes, P. C. Van Oorschot and S. A. Vanstone.
3. A Course in Number Theory and Cryptography by N. Koblitz.

Course Details:

Eigenvalues and Walks on graphs.
Radon transform and hypercubes.
Sperner property, lattices and boolean algebra.
Enumeration under group actions.
Ferrer's diagram, Young tableaux and Matrix Tree theorem.
Applications to Electrical networks, planar graphs.
Introduction to combinatorial commutative algebra.

Course References:

Text Books:
1. Richard P Stanley, Algebraic Combinatorics : Walks - Trees - Tableaux and More, Springer, 2013.
2. A preliminary copy of the book is available from Stanley's webpage.
Reference Books:
1. Richard P Stanley, Enumerative Combinatorics - Volume 1, Springer. 2001
2. Richard P Stanley, Enumerative Combinatorics - Volume 2, Springer, 2001.
3. Combinatorial Commutative Algebra. Erza Miller and Bernd Sturmfels. Springer, 2005.
4. Rafael H Villarreal, Monomial Algebras, CRC Press, 2015.
5. R B Bapat, Graphs and Matrices, Springer, 2014.