# M.Sc. Syllabi

The students enrolled in the Integrated Ph. D. program, who are candidates for receiving an M.Sc. degree from the School of Mathematics are expected to have mastered the following topics while satisfying the more advanced course requirements of the school.

## Algebra

• Group Theory: Basic notions and examples, subgroups,normal subgroups, quotients, products, semi-direct product, group acting on sets, Lagrange theorem, Cauchy's theorem, Sylow theorems, p-groups; Examples to include symmetric and alternating groups, , .

References: [Herstein, Ch. 2], [Dummit & Foote, Ch. 4-5].

• Ring Theory: Elementary notions of rings and modules; basic examples and constructions. Notions of ideals, prime ideals, maximal ideals, quotients, integral domains, ring of fractions, PID, UFD.

References: [Herstein, Ch. 3], [Dummit & Foote, Ch. 7-8].

• Linear Algebra: Vector Spaces: Basis, independence of the number of elements in a basis, direct sums, duals, double dual.

Matrices and Linear transformation: Linear map as a matrix, rank of a linear map, nullity, Eigenvalues, eigenvectors, minimal and characteristic polynomials, Cayley-Hamilton theorem, Triangulation and diagnolisation, Jordan canonical form.

Modules over Principle Ideal Domains.

Elementary notions of quadratic and hermitian forms.

References: [Herstein, Ch. 4 & 6], [Dummit & Foote, Ch. 10-12]

• Field Theory: Elementary notions of algebraic and transcendental extensions, splitting fields, structure theory of finite fields.

References: [Bhattacharya etc., Ch. 15-16], [Artin, Ch. 13]

• Basic Texts
• I.N. Herstein: Topics in Algebra.
• D.S. Dummit and R.M. Foote: Abstract Algebra.
• M. Artin: Algebra.
• P.B. Bhattacharya, S.K. Jain, S. R. Nagpaul: Basic Abstract Algebra.
• S. Lang: Algebra.

## Real Analysis

• Countable and uncountable sets; basic notions of metric space and its topology including compactness and connectness.
• Sequences in a metric space, series of complex numbers, completeness, limsup and liminf. Basic convergence tests: comparison, root and ratio; absolute and conditional convergence. Riemann's theorem about re-arrangement of conditionally convergent series; product of series.
• Continuity, uniform continuity, compactness and connectness under continous map, application to existence of maxima, minima, intermediate value; discontinuties of a monotone function.
• Differentiation of a function on , chain rule, Mean value Theorem, L'Hospital's rule, Taylor's theorem.
• Riemann-Stieltjes integral, upper and lower sums as area under a curve. Interability of continuous functions, Fundamental theorem of Calculus, integration by parts, rectifiable curve.
• Sequence and series of functions, examples and counter-examples. Uniform convergence, limit of continuous and differentiable functions under uniform convergence, integration under uniform convergence. Stone-Weierstrass theorem.
• Power series: Basic theorems about convergence and continuity of a power series, radius of convergence, behaviour at the end points. Exponential and trignometric functions. Fourier series, basic convergence theorem, Parseval's theorem.
• Functions of Several variables: Derivative of a function from to as a linear map; partial derivative, relation between the two. Chain rule, inverse and implicit function theorems.
• Lebesgue Integration: Construction of Lebesgue measure on , integration; Lebesgue monotone and dominated convergence theorems; comparison of Lebesgue and Riemann integration, -space.
• Basic Texts:
• W. Rudin: Principles of Mathematical Analysis.
• T. Apostol: Mathematical Analysis.
• H.L. Royden: Real Analysis.

## Topology and Functional Analysis

• Basic Set Topology:
Notion of a topological space, continuity, compactness, connectness. Heine-Borel Theorem, Tychonoff theorem, sequential compactness, Lebesgue covering lemma, Equi-continuity, Ascoli-Arzela theorem. and spaces, normal space, Urysohn's lemma, Tietze Extension theorem.
• Fundamental Groups: Homotopy of paths, Fundamental group, Convering spaces, Fundamental groups of circle and torus, homotopy lifting, fundamental group of .
• Functional Analysis: Normed linear spaces and continuous linear maps between them. Banach spaces and basic theorems about them: Hahn-Banach, Open mapping and Uniform boundedness theorem. Weak* topology on the dual, and the compactness of the sphere under the weak* topology.
• Functional Analysis: Basic notions about Hilbert spaces. Complete Orthonormal bases, Dual space of a Hilbert space, Notion of the adjoint of an operator, Unitary and Normal Operator, Compact operator; Spectral theory of compact self-adjoint operator.
• Funtional Analysis: Basic notions of Banach Algebras, spectrum. Structure of commutative Banach algebras.
• Basic Texts:
• J.R. Munkres: Topology.
• G.F. Simmons: Introduction to topology and modern analysis.
• B.V. Limaye: Functional Analysis.

## Complex Analysis

• Differentiation, Cauchy-Riemann equations, power series and its derivative, Harmonic functions.
• Cauchy's theorem for a convex domain, Cauchy's integral formula, Cauchy estimate, power series expansion, Morera's theorem, Liouville theorem, Fundamental theorem of Algebra.
• Laurent expansion, Singularities, Meromorphic functions.
• Residue calculus, application to some explicit integrals.
• Maximum modulus principle, Phragman-Lindelof theorem.
• Harmonic functions, Poisson integral formula, Harnack's theorem, mean value property, the Schwartz reflection principle.
• Basic Texts:
• L. Ahlfors: Complex Analysis.
• J.B. Conway: Functions of one complex variable.