Manonmaniam Sundaranar University Syllabus For Distance Education

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As you want I am here giving you syllabus for MSC Mathematics Distance Education course of Manonmaniam Sundaranar University.

MSC Mathematics Syllabus :

1st semester
Algebra-I
Analysis-I
Differential Equations
Elective-I
Elective-II
(Computer based)
Practical-I

2nd semester
Algebra-II
Analysis-II
Graph Theory
Supportive Course-I
Elective-III

3rd semester
Topology
Complex Analysis
Supportive Course-II
Elective –IV
Elective –V
(Computer Based)
Practical – II

4th semester
Functional Analysis
Elective-VI
Elective-VII
Elective-VIII
Project and viva-voce
MSC course syllabus
SYLLABUS
First Semester

Algebra – I

Unit I: Introduction to groups – Dihedral groups - Homomorphisms and Isomorphisms – Group axioms- Subgroups: Definition and Examples – Centralizers and Normalizer, Stabilizers and Kernels - Cyclic groups and Cyclic subgroups of a group – Subgroups generated by subsets of a group.

Unit II: Quotient Groups and Homomorphisms: Definitions and Examples – more on cosets and Lagrange’s Theorem – The isomorphism theorems - Composition series and the Holder program–Transpositions and the Alternating group.

Unit III: Group Actions: Group actions and permutation representations – Groups acting on themselves by left multiplication- Cayley’s theorem – Groups acting on themselves by conjugation – The class equation – Automorphisms –The Sylow theorems – The simplicity of An – Direct and semi-direct products and abelian groups: Direct Products – The fundamental theorem of finitely generated abelian groups.

Unit IV: Introduction to Rings: Basic definitions and examples - Examples: Polynomial rings, Matrix rings and group rings - Ring Homomorphisms and quotient rings – Properties of Ideals - Rings of fractions – The Chinese remainder theorem.–

Unit V : Euclidean domains, principal ideal domains and unique factorization domains: Principal ideal domains – Unique factorization domains – Polynomial rings: Definitions and basic properties – Polynomial rings over fields- Polynomial rings that are unique factorization domains – Irreducibility criteria – Polynomial ring over fields.

Text Book: Abstract Algebra (Second Edition) by David S. Dummit and Richard M. Foote, Chapter 1 (Sections 1.2 1.6 and 1.7 only), Chapter 2 (Sections 2.1 to 2.4), Chapter 3, Chapter 4, Chapter 5 (Sections 5.1 and 5.2 only), Chapters 7, 8 and 9.


Analysis – I

Unit I: Basic Topology: Finite, Countable and uncountable sets – Metric Spaces – Compact Sets – Perfect sets – Connected Sets.

Unit II: Numerical sequences and series: Convergent sequences – Subsequences – Cauchy sequences – Upper and lower limits – Some special sequences - Series – Series of nonnegative terms – The number e – The root and ratio tests – Power series – Summation by parts – Absolute convergence – Addition and multiplication of series – Rearrangements.

Unit III: Continuity: Limits of functions – Continuous functions – Continuity and compactness –Continuity and connectedness - Discontinuities – Monotonic functions – Infinite limits and limits at infinity.

Unit IV: Differentiation: The Derivative of a real function – Mean value theorems - The continuity of derivatives – L’ Hospital’s rule – Derivatives of Higher order – Taylor’s theorem –Differentiation of vector valued functions.

Unit V: The Riemann-Stielitjes integral: Definition and existence of the integral- Properties of the integral - Integration and Differentiation - Integration of vector - Valued functions-Rectifiable Curves.

Text Book: Principles of Mathematical Analysis (Third edition) by Walter Rudin, Chapters 2, 3, 4, 5 and 6.


Differential Equations

Unit I: Power Series Solutions and Special Functions.

Unit II: Some Special functions of Mathematical Physics.

Unit III: Nonlinear equations.

Unit IV: Laplace Transforms.

Unit V: The Existence and Uniqueness of solutions.

Text Book: Differential Equations with Applications and Historical Notes (Second Edition) by George F. Simmons, Sections 26 to 31, 44 to 52, 58 to 61 and 68 to 70.


Second Semester

Algebra - II

Unit I: Introduction to Module Theory: Basic definitions and examples – Quotient modules and module homomorphisms – Generation of modules, direct sums and free modules.

Unit II: Vector Spaces: Definitions and basic theory – The Matrix of a linear transformation – Dual vector spaces – Determinants.

Unit III: Field theory: Basic Theory of field extensions – Algebraic Extensions.

Unit IV: Splitting fields and Algebraic closures – Separable and inseparable extensions - Cyclotomic polynomials and extensions.

Unit V: Galois Theory: Basic definitions – The fundamental theorem of Galois theory – Finite Fields.

Text Book: Abstract Algebra (Second Edition) by David S. Dummit and Richard M. Foote, Chapter 10 (sections 10.1 to 10.3 only), Chapter 11(sections 11.1 to 11.4), Chapter 13 (section 13.1, 13.2 and 13.4 to 13.6), Chapter 14 (sections 14.1 to 14.3)


Analysis - II

Unit I: Sequences and Series of functions: Discussion of Main problem - Uniform convergence - Uniform convergence and continuity-Uniform convergence and Integration.

Unit II: Uniform convergence and differentiation- Equicontinuous families of functions-The Stone-Weierstrass theorem.

Unit III: Functions of Several Variables: Linear transformations - Differentiation -The Contraction Principle –The Inverse function theorem-The Implicit function theorem.

Unit IV: Determinants - Derivatives of higher order – Differentiation of Integrals -Integration of Differential forms: Integration - Primitive Mappings-Partitions of unity - Change of Variables.

Unit V: Differential forms - Simplexes and Chains – Stokes’ theorem –Closed forms and exact forms – Vector Analysis.

Text Book: Principles of Mathematical Analysis (Third Edition) by Walter Rudin, Chapters 7, 9 (except 9.30, 9.31 and 9.32) and 10.


Graph Theory

Unit I: Graphs and subgraphs: Graphs and simple graphs - Graph isomorphism-Incidence and adjacency matrices – Subgraphs - Vertex degrees - Path and Connection cycles – Applications: The shortest path problem – Trees: Trees - Cut edges and bonds - Cut vertices-Cayley’s formula.

Unit II: Connectivity: Connectivity – Blocks - Euler tours and Hamilton cycles: Euler tours – Hamilton cycles – Applications: The Chinese postman problem.

Unit III: Matchings: Matchings - Matching and coverings in bipartite graphs-Perfect matchings –. Edge colorings: Edge chromatic number - Vizing’s theorem- Applications: The timetabling problem.

Unit IV: Independent sets and cliques: Independent sets-Ramsey’s theorem-Turan’s theorem-Vertex colorings: Chromatic number-Brook’s theorem-Hajos’ conjecture-Chromatic polynomials-Girth and chromatic number.

Unit V: Planar graphs: Plane and planar graphs -Dual graphs-Euler’s formula- Bridges - Kuratowski’s Theorem (statement only) – The Five color theorem and The Four color conjecture -Non Hamiltonian planar graphs.
Text Book: Graph theory with Applications by J.A.Bondy and U.S.R.Murty, Chapters 1 (except 1.9), 2 (except 2.5), 3 (except 3.3.), 4 (except 4.4), 5 (except 5.5), 6, 7 (except 7.4 and 7.5) 8 (except 8.6) and 9 (except 9.8)

Third Semester

Topology

Unit I: Topological spaces and continuous functions: Topological spaces - Basis for a topology – The order topology – The product topology on X x Y – The subspace topology - Closed sets and limit points.

Unit II: Continuous functions – The product topology – The quotient topology.

Unit III: Connectedness and Compactness: Connected spaces, components and local connectedness - Compact spaces.

Unit IV: Local compactness – Countability and Separation axioms: The Countability axioms – The Separation axioms- Normal spaces.

Unit V: The Uryshon lemma- The Tietze Extension theorem - The Tychonoff theorem: Tychonoff theorem - The Stone Cech compactification.

Text Book: Topology (second edition) by J. R. Munkres, Sections 12 to 19, 22, 23, 25, 26, 29 to 33, 35, 37 and 38.

Complex Analysis

Unit I: Analytic Functions-Power Series.

Unit II: Conformality -Linear Transformation - Elementary Conformal mapping.

Unit III: Fundamental Theorems-Cauchy’s Integral formula-Local properties of Analytic Functions.

Unit IV: General form of Cauchy’s theorem (except proof of Cauchy’s theorem) -Calculus of Residues – Harmonic functions.

Unit V: Power Series Expansions – Partial fractions and factorizations – Entire functions.

Text Book: Complex Analysis (Third edition) by Lars V. Ahlfors, Chapters 2 to 4 (except section 4.5) and Chapter 5 (sections 5.1, 5.2 and 5.3 (except 5.2.4 and 5.2.5)).

Fourth Semester

Functional Analysis

Unit I: Banach spaces: The Definition and Some examples - Continuous linear transformations – The Hahn-Banach theorem -The natural embedding of N in N**.

Unit II: The Open mapping theorem – The Conjugate of an operator - Hilbert spaces: The Definition and some simple properties - Orthogonal complements – Orthonormal sets – The Conjugate space H*.

Unit III: The adjoint of an operator - Self-adjoint operators - Normal and unitary operators - Projections.

Unit IV: Finite dimensional spectral theory: Determinants and the spectrum of an operator-The spectral theorem.

Unit V: The Classical Banach Spaces: The Lp spaces – The Minkowski and Holder inequalities – Convergence and completeness – Approximation in Lp – Bounded linear functionals on the Lp spaces.

Text Book (i): Introduction to Topology and Modern Analysis by G.F.Simmons, Sections 46 to 59, 61,62. (For Units I to IV).

(ii) Real Analysis (Third edition) by H.L. Royden, Chapter 6

Address:
Manonmaniam Sundaranar University
Tirunelveli, Tamil Nadu 627012


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