MSc mathematics first semester syllabus: Model science college Jabalpur

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. I Semester (Mathematics)

Unit Paper I : Advanced Abstract Algebra MM : 35

I. Another counting Principle, Conjugency relation Normalizer, Class Equation Cauchy's theorems, Sylow's theorems, Double cosets, Second and third part of Sylow's theorems, Application of Sylow's theorems in Finite Groups. 

II. Series of Group : Normal and subnormal series, Composition series, Zassenhaus, Schreir refinement, Jordan-Holder theorem. 

III. Solvable groups and its properties Commutator subgroup, Nilpotent groups and its properties. 

IV. Fields: Extension field, Finite extension, Algebraic element, Algebraic and transcendental extension, Roots of polynomials, Splitting field. 

V. More about roots: Derivative of a polynomial, Simple extension, Primitive element, Separable and inseparable extension, Perfect field, Finite field. 

 

Unit Paper II : Real Analysis MM : 35

I. Definition and existence of Riemann-Stieltjes integral and its Properties, Integration and differentiation, The fundamental theorem of Calculus. 

II. Integration of vector-valued functions, Rectifiable curves. Rearrangements of terms of a series. Riemann's theorem. 

III. Sequences and series of functions, pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass M-test, Abel's and Dirichlet's tests for uniform convergence, uniform convergence and continuity, uniform convergence and Riemann-Stieltjes integration, uniform convergence and differentiation, Weierstrass approximation theorem, Power series, uniqueness theorem for power series, Abel's and Tauber's theorems. 

IV. Functions of several variables, linear transformations, Derivatives in an open subset of Rn, Chain rule, Partial derivatives, interchange of the order of differentiation, Derivatives of higher orders, Taylor's theorem, Inverse function theorem. 

V. Implicit function theorem, Jacobians, extremum problems with constraints, Lagrange's multiplier method, Differentiation of integrals, Partitions of unity, Differential forms, Stokes theorem. 

Unit Paper III : Topology MM : 35

I. Definition and examples of topological spaces, Basis for a topology, Standard topology, lower limit topology, The order topology, The product topology on X ×Y. Projections, The Subspace topology, Closed sets and limit points, Closure and interior of a set. 

II. Continuous functions, Equivalence with ε-δ condition, Examples of continuous functions, Homeomorphisms, topological property, topological Imbedding, Examples of Homeomorphisms, Rules for Constructing continuous functions, The Pasting lemma, Maps into products, The product topology, Box topology, Projection mapping, comparison of the product topology and the box topology. 

III. The Metric topology, Metrizable space, Standard bounded metric, The spaces $R^n$ and $R^w,$ Euclidean metric, square metric, Metrizability of $R^n$ and $R^w,$ Uniform metric, The sequence lemma, Uniform limit theorem. 

IV. Connected space, Separation, Definition and examples, Cartesian product of connected spaces, Connected sets in the real line, Linear continuum, Intermediate value theorem, Path connectedness. 

V. Compact spaces, Finite product of compact spaces, The Tube Lemma, Finite intersection condition, compact sets in the real line, Maximum and minimum value theorem, Limit point compactness, The Lebesgue number Lemma, Second countable and first countable spaces, Separation Axioms ($T_1,$ $T_2,$ $T_3$ spaces). 

Unit Paper IV : Complex Analysis -I MM : 35

I. Complex integration, Cauchy Goursat theorem, Cauchy integral formula, Higher order derivatives 

II. Morera's theorem, Cauchy’s inequality, Liouville's theorem, The fundamental theorem of algebra, Taylor's theorem.  

III. The maximum modulus principle, Schwartz lemma, Laurent series, Isolated singularities, Meromorphic functions, The argument principle, Rouche's throrem, Inverse function theorem. 

IV. Residues, Cauchy's residue theorem, Evaluation of integrals, Branches of many valued functions with special reference to $arg z,$ $logz$, $z^a$. 

V. Bilinear transformations, their properties and classification, Definitions and examples of conformal mappings. 

Unit Paper V : FUNCTIONAL ANALYSIS MM : 35

I. Convergence, Completeness and Baire's Theorem, Complete metric spaces. Limit & Limit point. Cantors intersection Theorem, Continuous mappings, Uniformly continuous mapping, examples. 

II. Spaces of continuous functions, Euclidean and Unitary spaces, Cauchy, Minkowski and Holders inequalities, Normed linear spaces, Examples and elementary properties, quotient spaces. Equivalence of normsm, 2-norm, supnorm, p-norm, 1≤ p < ∞, Minkowski and Holders inequalities for p-norm, examples. 

III. Banach space and examples, Continuous linear transformations, norm of an operator, Banach spaces B(N, N’). Functionals and their extensions, related Lemma, Hahn- Banach Theorem for normed linear spaces. Conjugates of normed linear spaces. 

IV. The natural embedding of normed linear space in its second conjugate space, Reflexive Banach spaces, open mapping theorem, Closed graph theorem. Conjugate of an operator, Uniform boundedness principle, examples. 

V. Inner product spaces, examples, elementary properties of Inner Product, Parallelogram law, Schwartz inequality and polarization identity, Hilbert Space and examples. orthogonal complements in Hilbert spaces. Orthonormal sets, Bessel's inequality, Gram Schmidt orthonormalization process. Conjugate space $H^*$, Riesz Representation Theorem.