Annexure-V. Revised Syllabus for course work of Integrated M. Phil/Ph. D. Programme in Mathematics (Applicable w.e.f July 2016)

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1 1 Annexure-V Revised Syllabus for course work of Integrated M. Phil/Ph. D. Programme in Mathematics (Applicable w.e.f July 2016)

2 2 Paper-I Research Methodology-I UNIT-I Meaning of Research Methodology ; Difference(s) between Methods of Research & Research Methodology ; Motivation for Research, Research approaches and Related Tools, Conditions and criteria for good research. UNIT -II Choosing the research area, Factors leading to the choice, defining the concrete research problem and focusing on it. UNIT III Importance of communication skill in Research- Development of power of expression in both speaking and writing, Mastery of presentation techniques. Progress-report writing on the Research topic(s). UNIT IV Review of research work: the relevance of the research work from the perspective of the Subject possible ways to apply the research work in future. Objectivity, Reality and Ethics in Research, Scientists in the Social World, The Dilemmas and the Decision Makings vis-à-vis the Reality. 1. Research Methodology Methods and Techniques, C R Kothari, New age International publications. 2. Research Methodology a step by step guide for beginners, Ranjit Kumar, SAGE. 3. Research Methodology, R. Panneerselvam, PHI.

3 3 Paper-II Research Methodology-II UNIT I Programming in C: Constants, variables, data-types; operators: arithmetic, relational, logical, problems involving: assignments, increment / decrement, formatted input and formatted output, decision making and branching, creation of loops : For loop, While loop, and do-while loop. ID and 2D arrays, reading strings from terminal, writing strings to the screen. Unit-II Numerical Analysis: Significant digits, round-off errors, Finite computational processes and computational errors, Floating- point arithmetic and propagation of errors. Loss of significant digits. Interpolation with one variable finite differences, divided differences. Lagrangian and Newtonian methods. Iterative methods. Aitken Neville s iterative scheme. Spline interpolation. Errors and remainder terms. Inverse interpolation. Interpolation with two variables. Unit-III Mathematical Soft wares: Latex and Beamer (10 Hrs), Solving linear and non linear equations using Mathematica. Differential Geometry of curves and surfaces using Mathematica. Working knowledge of MathSciNet, JSTORE and other online Journals. UNIT- IV PRESENTATION: A Student will give a 30 minute presentation on any Mathematical topic of his/her choice. In advance of the presentation, he/she will submit outline and abstract of the presentation. REVIEWS OF MATHEMATICAL ARTICLES: A student will choose two articles from a list of given articles and write a critical review of each one. 1. Schaums outline of Theory and Problems of programming with C : Gottfried 2. Mastering C by Venugopal, Prasad TMH 3. Programming in ANSI C, Balaguruswamy Scientific Computing and Differential Equations, Gene H. Golub and James M. Ortega, Academic Press. 4. Computational Methods for Applied Science and Engineering, M. G. Ancon. Rinton Press. 5. Numerical methods for scientific and engineering computation, M. K. Jain, S. R. K. Iyengar and R.K. Jain, Wiley Eastern Ltd. 6. Applied Numerical Methods, C.F. Gerald and P.O. Wheatley, Pearson Education Asia. 10. CalcLabs with Mathematica for Single Variable Calculus, Selwyn Hollis. 11. Principles of Linear Algebra with Mathematica, Kenneth Shiskowski, Karl Frinkle.

4 4 11. Sage Math for Undergraduates, Gregory V Bard, The American Mathematical Society Paper-III (Common to all) UNIT-I: (Representation Theory of finite groups): Group actions and permutation representations, Groups acting on themselves: by left multiplication and by conjugacy. Linear actions and modules over group rings. Maschke s Theorem. Wedderburn s Theorem and some Consequences. Character Theory and orthogonality relations. Characters of groups of small order. Theorems of Burnside and Hall. UNIT-II: Real and Complex Analysis Weistrass approximation Theorem and Muntz Theorems, The Gamma and Beta functions, Riemann zeta function, Complex Analysis and Prime number theorem, Integral theorem. Riemann Mapping Theorem. UNIT-III: Topology Tyconoff Theorem, Stone-Cech compactification, Metrization Theorems and Paracompactness. Homotopy of Paths, The Fundamental Group, Covering Spaces, The fundamental Group of Circles. UNIT-IV: Banach Spaces. Weak and weak star topologies, Banach-Alaoglu theorem, Goldstine theorem, Krien- Milman theorem, Reflexive Banach spaces and its characterization. Lp(μ) space and its dual, Radon Nikodym Theorem. 1. Algebra, M. Artin, Prentice Hall of India. 2. Topology, J. M. Munkers, (2nd Edition). Pearson Education India. 3. Functional Analysis, J. B. Conway, Springer Verlag. 4. Abstract Algebra, Dumit and Foote, Wily and Sons. 5. Mathematical Analysis, T M Apostal, Narosa Publications. 6. Complex Analysis, L. Ahlfors, Springer. 7. Theory of Functions, E.C. Titchmarsh Oxford University Press

5 5 PAPER-IV (Research Specific) (Theory of Semi-groups) UNIT-I Basic definitions and examples of semi-groups and sub semi groups, Direct products, Monogenic semi groups, homomorphism and Transformations. Partial orders, semi lattices and lattices, Equivalences and congruences. Homomorphism theorems, Ideals and Riez Congruences. Free semi groups and Monoids: Presentations. UNIT-II Green's Equivalences, The structure of D-classes, Green's lemma and its corollaries. Regular D-classes, Regular semi groups, The Sandwich set. Simple, 0-simple and completely 0-simple semi groups. The Rees Theorem. UNIT-III Completely simple semi groups. Isomorphism and normalization. Congruences on completely simple semi groups. Lattice of congruences on completely simple semi groups. UNIT-IV Completely regular semi groups, Clifford semi groups and bands. Inverse semi groups: Representation by injective partial mappings, Congruences on inverse semi groups and free inverse semi groups. 1. Fundamentals of semigroup theory, John M Howie Clarendon press. Oxford. 2. The Algebraic theory semigroups, Vol. I, Vol. II, A H Clifford and G B Preston, Mathematical surveys of the American Mathematical Society. 3. Techniques of Semi Group Theory, P M Higgins, Oxford University Press.

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Department of Mathematics. Revised Syllabus of M Phil/P hd Course work In operation w.e.f Academic Session 2017

Department of Mathematics. Revised Syllabus of M Phil/P hd Course work In operation w.e.f Academic Session 2017 Department of Mathematics Revised Syllabus of M Phil/P hd Course work In operation w.e.f Academic Session 2017 Paper-I Research Methodology-I Unit-I: Introduction and Motivation Meaning of research, research