Integrated Ph.D Mathematics entrance exam Syllabus

Give me general syllabus for Integrated Ph.D Mathematics entrance examination ?

[Kiran Chandar]

Here I am giving you general syllabus for Integrated Ph.D Mathematics entrance examination below :

The Syllabus
Algebra.
(a) Groups, homomorphisms, cosets, Lagrange's Theorem, Sylow Theorems, symmetric group
Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, _elds,
algebraic extensions, _nite _elds
(b) Matrices, determinants, vector spaces, linear transformations, span, linear independence,
basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigenvectors,
upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rota-
tions, orthogonal matrices, GLn, SLn, On, SO2, SO3.

References:
(i) Algebra, M. Artin
(ii) Topics in Algebra, Herstein
(iii) Basic Algebra, Jacobson
(iv) Abstract Algebra, Dummit and Foote

Complex Analysis.
Holomorphic functions, Cauchy-Riemann equations, integration, zeroes of analytic functions, Cauchy
formulas, maximum modulus theorem, open mapping theorem, Louville's theorem, poles and sin-
gularities, residues and contour integration, conformal maps, Rouche's theorem, Morera's theorem
References:
(i) Functions of one complex variable, John Conway
(ii) Complex Analysis, L V Ahlfors
(iii) Complex Analysis, J Bak and D J Newman

Calculus and Real Analysis.
(a) Real Line: Limits, continuity, di_erentiablity, Reimann integration, sequences, series, lim-
sup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions,
(b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives,
total derivative, Jacobian, gradient, line integrals, surface integrals, vector _elds, curl, di-
vergence, Stoke's theorem
(c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass
approximation.


References:
(i) Principles of mathematical analysis, Rudin
(ii) Real Analysis, Royden
(iii) Calculus, Apostol
Topology. Topological spaces, base of open sets, product topology, accumulation points, bound-
ary, continuity, connectedness, path connectedness, compactness, Hausdor_ spaces, normal spaces,
Urysohn's lemma, Tietze extension, Tychono_'s theorem,
References: Topology, James Munkres