- 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.
- 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.
- 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.