University of Mysore - MSc Maths syllabus
M101-Algebra I: Max. Marks:80
Congruence, residue classes, Theorems of Fermat, Euler and Wilson, Linear Congruence. Elementary Arithmetical Functions. Primitive roots. Quadratic residues and the low of Quadratic reciprocity. Groups, Lagrange's Theorem, Homomorphism and isomorphism Normal subgroups and factor groups. Fundamental theorem of homomorphism. Permutation groups and Cayley's Theorem. Sylow's theorems. Two laws of isomorphism. Normal series and composition Series.
I. N. Herstein-Topics in Algebra (Vikas Publishing House Pvt., Ltd., New Delhi).
I. Niven and H. S. Zuckerman-An introduction to the Theory of Numbers (3/e. New York, John Wiley and Sons, Inc.)
J. B. Fraleigh- A First course in Abstract Algebra (Addison-Wesley, Publishing Company).
Emil Grosswald- Topics from the theory of numbers (The Macmillan Co., New York).
M102-Linear Algebra I: Max. Marks.80
Vector Space and Modules: Elementary basis concepts. Finite dimensional Vector space Linear dependence and independence of vectors, bases, Dual spaces. Inner product spaces. Modules. Properties of Linear transformation. The Algebra of linear transformation Rank and nullity. Algebra characterization of Algebra.
I. N. Herstein-Topics in Algebra
Hoffman and Kunze-Linear Algebra
P. R. Halmos Finite Dimensional Vector space, D. Vannostrand, 1958.
Kumeresan -Linear Algebra.
M103-Real Analysis I Max.Marks:80
Dedekind's construction of the real number system. The extended real number system. Complex numbers. Euclidean space R(n). The binomial inequality. The inequality of the Arithmetic and Geometric means. The inequality of the power means. Cauchy's and Holder's inequality and Minkowski's inequalities.
Numerical sequences and series of real and complex terms. Convergent sequences, Cauchy's sequences upper and lower limits. Series of nonnegative terms. The number `e', Tests of convergence.
W. Rudin-Principles of Mathematical Analysis (3/e, International Student edition, Mc-Graw Hill).
T. M. Apostal - Mathematical Analysis (2/d, Addison Wesley, Narosa, New Delhi).
R. R. Goldberg-Methods of real Analysis (Oxford and IBH. Publishing company, New Delhi).
M104-Real Analysis II: Max. Marks.80
The topology of the real line, continuity, Uniform continuity. Properties of continuous functions, Discontinuities. Monotonic functions. Differentiability, Mean value theorems. L' hospital rule. Taylor's Theorem. Maxima and Minima. The Riemann-Stieltije's integral. Criterion for integrabiltiy. Properties of the integral. Classes of integral functions. The integral as a limit of a sum. First and second mean value theorems. Integration and differentiation. Functions of bounded variation.
W. Rudin-Principles of Mathematical Analysis (3/e, International Student edition, Mc-Graw Hill).
D. V. Widder-Advanced Calculus (2/e Prenctice hall of India, New Delhi).
M105-Complex Analysis I Max. Marks.80
Algebra of complex numbers. The geometric representation of complex numbers. Elementary theory of power series-sequences, series, uniform convergence of power series. Abel's limit theorem. The elementary functions.
Topology of the complex plane. Linear transformations, Elementary conformal mappings. Complex integration-Cauchy's theorem. Cauchy's integral formaula. Local properties of analytic functions.
L. V. Ahlfors-Complex Analysis (Mc-Graw Hill, Kogakusha, 1979).
J. B. Conway-Functions of one complex variable, (Narosa, New Delhi).
M201- Algebra II: Max.Marks:80
Rings. Integral domains and Fields. Homomorphisms. Ideals and Quotient Rings. Prime and Maximal ideals. Euclidean and principal ideal rings. Polynomials, Zeros of a polynomial. Factorization. Irreducibility criterion. Symmetric functions. Resultant of two polynomials as a symmetric function of the roots Prime fields, adjunction of roots. Algebraic and transcendental extensions, Finite fields. Separable and inseparable extensions. Perfect and imperfect fields. Theorem on the primitive element.
I.N. Herstein - Tpoics in Algebra (Vikas publishing House Pvt.Ltd.,New Delhi.
J.B. Fraleigh - A first course in Abstract Algebra (Addison - Wesley, Publishing Company).
M202-Linear Algebra II: Max.Marks:80
Linear tranformations: Characteristic roots. Matrices, Canonical forms: Triangular forms. Nilpotent transformations. Jordan forms, rational canonical form. Treace and transpose, Determinants. Hermitian, unitary and normal transformation. Real quadratic forms. Sylvester's law of Inertia, Criterion positive definiteness.
I.N. Herstein Topics in Algebra
Hoffman and Kunze, Linear Algebra
P.R. Halmos, Finite Dimensional Vector space, D.Vannostrand, 1958.
Kumeresan Linear Algebra.
M203- Real Analysis III: Max.Marks:80
Multiplications of series, rearrangements. Double series, Infinite products, Sequences and series of functions, Uniform convergence. Uniform convergence and continuity Univorm convergence and integration, Unform convergence and differentiation. Power series. The exponential and logarithmic functions. The trigonometric functions.
W.Rudin - Principles of Mathematical Analysis (3/e International Student edition, McGraw Hill).
T.M. Apostal - Mathematical Analysis (2/d, Addison Wesley, Narosa, New Delhi).
R.R. Goldberg - Methods of Real Analysis, (Oxford and IBH, Publishing Company, New Delhi).
M 204- Real Analysis IV: Max.Marks:80
Improper integrals, Tests of convergence. The beta and Gamma functions. Improper integrals with integral containing a parameter - their uniform convergence, continuity, integrability and differentiability with respect to a parameter. Functions of several variables, partial derivatives, continuity and differentiability. Chain rule, Euler's theorem on homogeneous functions. Jacobians. Implicit function theorem. Taylor's theorem. Maxima and Minima. Lagrange's multipliers. Multiple integrals (forms) treatment. Applications.
W.Rudin - Principles of Mathematical Analysis (3'e International Student edition McGraw Hill)
2. D.V. Widder - Advanced Calculus (2/e Prenctice Hall of India, New Delhi).
M 205 - Complex Analysis II: Max.Marks: 80
The Taylor's and Laurent expansions. The calculus of residues. Harmonic functions - Definition and basic properties, the mean value property . Poisson's formula. Schwarz's theorem. The reflection principles. Mittag-Leffer's theorem. Canonical products. Weirstoss's. The Gamma functions. Jenson's formula.
L.V. Ahlfors - Complex Analysis (McGraw Hill, Kopgakusha, 1979)
J.B. Conway - Functions of one complex variuable, (Narosa, New Delhi).
M 301 - MEASURE AND INTEGRATION I Max.Marks:80
Lebesgue measure - outer measure, measurable sets and Lebessgue measure, a non-measurable set, measurable functions. The Lebesgue integrals are Lebesgue Integral of a bounded function over as set of finite measure the integral of non-negative function, the general Lebesgue integral. Differentiation and integration - Differentiation of monotonic functions, functions of bounded variation, differentiation of an integral, absolute continuity. The classical Banach space the Lp-spaces. The Holder and Minkowski inequalities, convergence and completeness, bounded linear functionals on the Lp-spaces.
Books Recommended for study/Reference.
l. H.L. Rody n-Real Analysis.
M 302 - TOPOLOGY I: Max.Marks: 80
Set theoretic preliminaries, Topological spaced and continuous - topological spaces, basic for a topology, the order topology, the product topology on X x -, the subspace topology, closed sets and limit points, continuous functions, the product topology, the metric topology, the quotient topology, connectedness and compactness connected spaces, connected sets on the real line path connectedness,compact spaces, compact sets on the line, limit point compactness, local compactness.
James Munkers - Topology
M 303- ELEMENTS OF FUNCTIONAL ANALYSIS I: Max.Marks:80
Equivalence of Compactness, Sequential compactness and completeness plus total boundedness for a metric space. Metric completion. Banach's contraction mapping theorem and applications. Baire category theorem. Ascoli Arzela theorem. Picard's theorem on the existence and Uniqueness of solutions of an ordinary differential equation of this the first order. Linear spaces and linear operators. Norm of a bounded operator. The Hahn Banach extension theorems. Stone weirstrass theorem.
Books recommended for study/references.
G.F. Simmons - Introduction to Topology and Modern Analysis
A.E. Taylor- Introduction to fundamental Analysis
A Page and A.L. Brown - Elements of Fundamental Analysis.
M 304- Differential Equations. Max.Marks:80
Linear Second Order Equations.
Initial value problem, wronskin, separation and comparison theorems, Poincare phase plane, Adjoint equation, Lagrange identity, Green's function, variation of parameters.
Boundary value problems.
Strum Liouvile system, eigen values and eigen functions, simple properties expansion in eigen functions, parsebal's identity.
Power series solutions:
Solution near ordinary and regular singular point. Convergence of the formal power series, applications to legendre, Bessel, Hermite, Laguerre and hypergeometric differential equations with their properties.
Second order partial differential equations.
Characteristic curves, reduction to canonical forms, derivation of the equations of mathematical physics and their solutions.
E.A. Coddington and N.Levinson, Theory of ordinary Differential equations.
R. Courant and D.Hilbert, Methods of Mathermatical Physics,
Vol. I. II
G>F> Simmons, Differential Equations with applications and Historical Notes.
I.N. Sneddon, Theory of partial differential equations.
M 305-GRAPH THEORY Max.Marks:80
1.1 The Konigsberg bridge problem
Around the world
The four color conjecture
Graph Theory in the 20th century
2.l Verieties of graphs
Walks and connectedness
Operations on graphs
3.l Culpoints, bridges and blocks
Block graphs and cutpoint graphs
4.l Elementary properties of trees
5.l Connectivity and line connectivity
7.l Coverings and independence number
F.Harary, Graph Theory, Addition Wesley Reading Mass (1969)
N.Deo, Graph Theory and its applications
K.R. Parthasarathy, Graph Theory
G.Chartand and L. Lesniak Graphs and Diagraphs (23nd Edition) Qwadsworth and Brooks) Cole. Mathematics series.
Clark and D.A. Holton, A Firrst Look at Graph Theory, Allied publishers.
M 305-b GALOIS THEORY Max.Marks:80
Algebraically closed fields and algebraic closures. The existence of an algebraic closure.
The basic isomorphisms of algebraic field theory. Automorphisms and fixed fields. The Frobenius automorphism. The isomorphism extension theorem. The index of a field extension. Splitting fields, separable extensions, perfect fields normal externsions. Galois theory - the main theorem of Galois theory Galois groups over finite fields. Symmetric functions, Cyclotomic extensions. Constrictible numbers, the impossibility of certain geometrical constructions, constructible polygons. Subnormal and normal series, the Jordan - Holder theorem, Radical extensions and solution of equation by radicalss. The isolvability of the quintic.
A first Course in abstract Algebra - J.B. Fraleigh Pub. Narosa publishing House.
Galois Theory - Ian Steward Pub. Chapman and Hall.
M 305-c THEORY OF NUMBERS. Max.Marks:80
Fermat and Mercenn's numbers.
Farey series, Farey dissection of the continuum Irrational numbers-Irrationality of m N,e and
Approximation Irrational numbers. Hur Witz's Theorem. Representation of a number by two or four squares. A number n is the sum of two squares if and only if all prime factors of the form 4m+3 have even exponents. tn the standard term of n. every positive integral is the sum of four squares. Definition g(K) and G(K). proof of g(4)<50, perfect numbers-An even perfect number is of the form 2n(2n+1)-1) where 2n+1-1 is prime.
Continued fractions-Finite continued fractions, convergent to a continued fraction, continued fractions with positive quotients. Simple continued fractions, The representation of an irreducible rational fraction by a simple continued fraction. The continued fraction algorithm and Euclid's algorithm. The difference between the fraction and its convergence, Infinite simple continued fractions, the representation of an irrational number by an infinite continued fraction, Equivalent numbers periodic continued fractions, some special quadratic surds, and the series of Fibonacci and Lucas.
Books for references
G.H.Hardy and E.M.Wright-An introduction to Theory of Numbers.
Niven and Zuckerman-Elementary Number Theory.
Bruce Burndt-Ramanujan's Note Books Volume-1 to 5.
G.E.Andrews-Encyclopedia of Mathematic and its applications.
M 401-MEASURE AND INTEGRATION II: Max.Marks:80
Measure and integration Measure spaces, Measureable functions, integration, Signed measures, the Radon Nikodyn theorem Measure and outer measure outer measure and measurability, the extension theorem, the Lebesgue Stieltje's integral, product measures.
Books recommended for study/reference:
H.L.Royden Real Analysis.
M 402-TOPOLOGY II: Max.Marks:80
Countability and separation axioms-the countability axioms, the separation axioms, normally of a compact Hausdorff space. Urysohn's lemma, Tietze's extension theorem, Urysohn's matrization theorem, Partitions of unity. Tychonoff's theorem on the topological product of compact spaces. Local finiteness. Paracompactness-Normality of a paracompact space. The Fundamental Group and Fundamental Group of Circle, The Fundamental Group of the Fundamental plane, Essential and Inessential Maps, The Fundamental Theorem of Algebra.
Text book: James Munkers-Topology.
M 403- Elements of Functional Analysis: Max.Marks:80
Open mapping and closed graph theorems. The Banach Steinhaus Principle of Uniform boundedness. Hilbert spaces- The orthogonal projection. Nearly orthogonal elements (Riesz's lemma). Bessel's representation theorem.
Banach algebra's-Algebras and Banach algebras. Homomorphisms and isomorphosms-The spectrum and Gelfand-Mazur theorem, the Gelfand Representation theorem.
Books recommended for study/ Reference:
G.F.simmons-Intraduction to topology and Modern Analysis.
A.E. Yaylor-Introduction to functional Analysis.
A Page and A.L. Brown-Elements of Functional Analysis.
M 405-a Theory of Partitions. Max.Marks:80
Partitions-Partitions of numbers, the generating function of p(n), other generating functions, Two Theorem of Euler, Jacobi's triple product identity, its proof and its applications. 1-1 summation of Ramanujan and its applications combinatorial proofs of Euler's identity, Euler's pentagonal number theorem. Franklin's combinatorial proof, proof of p(5n+4)=(mod5) and p(7n+5)=0(mod7). The Rogers-Ramanujan Identities. Elementary series-product Identities-p 17-22 of Encyclopaedia of Mathematics and its applications- the theory of partitions Vol. 2 by G.E. Andrews and its applications. Deduction of Euler's Gauss Heine's, Jacob's identities.
Books for reference:
1. G.H.Hardy and E.M.Wright-An introduction of Topology of Numbers.
Niven and Zuckarman-Elementart Number Theory.
Bruce C Bandit-Ramanujan's notebooks-volmes 1 to 5
G.E. Andrews-Encyclopaedia of Mathematics and its applications.
M 405-b Lie Algebra: Max.Marks:80
Definition and example, construction of lie and associative algebras, Linear Lie algebras. Derivations, inner derivations of Lie algebras. Determination of the Lie algebras of low dimensionalities. Representations and modules, some basix module operations, ideals and homomorphisms, Fundamental thorem of homomorphisms, Solvable Lie algebras, properties of solvable Lie algebras, Nilpotent Lie algebras, properties if nilpotent Lie algebras, Engle's theorem Lie's theorem, Cartants criterion for solvability.
James E. Humphreys, Introuction to Lie algebras and representation theory. Springer Verlag, New Delhi.
N. Jacobson, Lie algebras, John Wiley and Sons, New York, London, Sydney.
V. S. Varadarajan, Lie groups, Lie algebras and their representations, Prentice Hall, New Jercy.
M405-c Algorithms and Computations: Max.Marks:80
Introduction to Computers, Flowcharts, Algorithms and their features, Languages, Types of language and translators.
Numerical Algorithms-Solving a simultaneous system of linear equations using interactive and direct methods, Interpolation algorithms: equal, unequal intervals, central difference and inverse interpolation. Numerical differential and integration and their errors calculations. Graph theoretical algorithms-Connectivity, finding shortest path between two vertices, enumeration of all paths, construction of minimum spanning tree, cutest, cut vertex, coding and decoding.
Computation-Algorithms complexities, strategies, -Divide and conquer, greedy technique, Introduction to NP hard problems.
Numerical Algorithms-Conte and D'bear
Graph theory and applications-N, Deo.
Introductory theory of Computer Science-E. V. Krishnamurthy.
Fundamentals of Computer Algorithms-Horowitz and Sahni.
Numerical Algorithms-Chapre and Canale.
Computer Oriented Numerical methods-Rajarama, V.
Numerical Analysis-G. Shenkar Rao.
M405-d Advanced Functional Analysis Max.Marks:80
Bounded linear operators on Hilbert spaces, the adjoint of an operator, self adjoint operators, positive operators, properties of normal and unitary operators. One to one correspondences between projections on a Banach space and pairs of closed linear subspaces of the space, properties of orthogonal projections on Hilbert spaces. Spectral resolution of an operator on a finite dimensional Hilbert space H and the spectral theorem of a normal operator on H.
The structure of commutative Banach algebras-properties of the Gelfand mapping, the maximal ideal space, Multiplicative functional and maximal ideal. Applications of spectral radius formula. Involutions in Banach algebras, the Gelfan-Meumark theorem.
G. F. Simmons-Introduction to Topology and Modern Analysis
A. E. Taylor. Introduction to Fundament Analysis.
M 405-e GRAPH THEORY II: Max. Marks: 80
Some properties of line graphs
Characterisation of line graphs
Special line graphs
Line graphs and traversability
Plane and planar graphs
Characterisations of planar graphs
The chromatic number
Five color theorems
The adjacency matrix
The incidence matrix
The cycle matrix
Some elementary properties.
F. Harary, Graph Theory. Addition Wesley Reading Max (1969)
N. Deo, Graph Theory and its applications
K. R. Parthasarathy, Graph Theory
G. Chartrand and L. Lesniak, Graphs and Digraphs (2nd Edition) Wardsworth and Brooks/ Cole, Mathematics series.
J. Clark and D. A. Holton, A first look at Graph Theory, Allied publishers.
M 405-f COMMUTATIVE ALGEBRA Max.Marks:80
Rings and ideals. Rings and ring homomorphisms Ideals, Quotient rings, zero divisors, nilpotent elements, units, prime ideals and maximal ideals. The prime spectrum of a ring. Nil radical and Jacobson radicals. Operation on ideals, extension and contraction.
Modules: Modules and modules homomorphisms, submodules and quotient modules. Direct sums. Free modules Finitely generated modules. Nakayama Lemma, Simple modules. Exact sequences of modules.
Modules with chain conditions: Artinia and Noetherian modules, modules of finite length, Artnion rings, Noetherian rings, Hilbert basis theorem.
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra-Addision-Wesley Pub. Company.
C. Musili, Introduction to Rings and Modules- Narosa publixhing House.