MSc mathematics second semester syllabus: Model science college Jabalpur

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. II Semester (Mathematics)

Unit Paper I : Advanced Abstract Algebra MM : 35

I. The elements of Galois theory: Automorphism of a field, Group of automorphisms of a field, Fixed field, Normal extension, Galois group of a polynomial, Fundamental theorem of Galois theory, Solution of polynomial equations by radicals, Insolvability of the general equation of degree 5 by radicals.

II. Introduction to Modules, Examples, Submodules and Direct sum of submodules, R-homomorphisms and Quotient modules, Finitely generated modules, Cyclic module.

III. Simple modules, Semi-simple modules, Schur's lemma, Free modules, Rank of a module.

IV. Noetherian and Artinian modules, Ascending and Descending chain condition (ace & dcc), Noetherian and Artinian rings, Examples, Hilbert basis theorem.

V. Fundamental Structure theorem of finitely generated modules over a Principal Ideal Domain and its applications to finitely generated abelian groups

Unit Paper II : LEBESGUE MEASURE AND INTEGRATION MM : 35

I. Lebesgue outer measure. Measurable sets. Regularity. Measurable functions. Borel and Lebesgue measurability. Non-measurable sets.

II. Integration of ,Non-negative functions. The General integral. Integration of Series, Reimann and Lebesgue Integrals.

III. The Four derivatives. Functions of Bounded variation. Lebesgue Differentiation Theorem, Differentiation and Integration.

IV. The LP-spaces, Convex functions, Jensen's inequality. Holder and Minkowski inequalities. Completeness of LP.

V. Dual of space when 1 ≤ P ∞, Convergence in Measure, Uniform Convergence and almost Uniform Convergence.

Unit Paper III(A) : Topology MM : 35

I. The Tychonoff Theorem, Completely regular spaces, The Stone-Cech compactification.

II. Metrization Theorems and Paracompactness, Local finiteness, The Nagata Smirnov Metrization Theorem (Nec & suff), Paracompactness

III. Complete Metric spaces & function spaces, Complete metric spaces, A spacefilling curve, Compactness in metric spaces Ascoli's theorem, Pointwise and Compact convergence.

IV. The Fundamental Group and covering spaces, Homotopy of paths, the Fundamental Group, Homomorphism, covering spaces, The Fundamental Group of the circle

V. Nets & filters : Topology and Convergence of Nets, Hausdorffness & nets, Compactness & nets, Filters and their convergence, Ultra Filters and compactness

Unit Paper IV : Complex Analysis -II MM : 35

I. Weierstrass factorization theorem, Gamma function and its properties, Riemann zeta function, Riemann’s functional equation

II. Rung’s theorem, Mittage-Leffler’s theorem, Schwartz Reflection principle, Analytic Continuation, Analytic Continuation along a path..

III. Monodromy theorem and its consequences, Harmonic function, Harmonic functions on a disk, Harnack’s inequality and theorem.

IV. Dirichlet’s problem, Green’s function, Jenson’s formula, Poisson-Jenson Formula.

V. Order of an entire function, Hadamard’s three circle theorems, Hadamard’s factorization theorem, Bloch’s theorem, The Little Picard theorem.

Unit Paper V: ADVANCED DISCRETE MATHEMATICS MM : 35

I. Algebraic Structures : Introduction, Algebraic Systems : Examples and General Properties : Definition and examples, Some Simple Algebraic Systems and General properties, Homomorphism and isomorphism, congruence relation, Semigroups and Monoids : Definitions and Examples, Homomorphism of Semigroups and Monoids.

II. Lattices: Lattices as Partially Ordered Sets : Definition and Examples, Principle of duality, Some Properties of Lattices, Lattices as Algebraic Systems, Sublattices, Direct product, and Homomorphism.

III. Some special Lattices, e.g. Complete, Complemented and Distributive Lattices, Boolean Algebra: Definition and Examples, Subalgebra, Direct product and Homomorphism, join irreducible, atoms and antiatoms

IV. Graph Theory: Definition of a graph, applications, Incidence and degree, Isolated and pendant vertices, Null graph, Path and Circuits: Isomorphism, Subgraphs, Walks, Paths and circuits, Connected graphs, disconnected graphs, and components, Euler graph.

V. Trees: Trees and its properties, minimally connected graph, Pendant vertices in a tree, distance and centers in a tree, rooted and binary tree. Levels in binary tree, height of a tree, Spanning trees, rank and nullity.