Organisation : Chennai Mathematical Institute
Announcement : Syllabus
Name Of The Exam : M.SC Maths Entrance Exam

Download Syllabus Here : http://www.indianjobtalks.com/upload...I-entrance.pdf
Home Page : http://www.cmi.ac.in//

CMI M.SC Entrance Exam Syllabus :
1. Algebra :
(a) Groups, homomorphisms, cosets, Lagrange’s Theorem, group actions, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields, algebraic extensions, finite fields

(b) Matrices, determinants, vector spaces, linear transformations, span, linear independence, basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigen vectors, upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rotations, 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

2. 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 singularities, 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

3. Calculus and Real Analysis :
(a) Real Line : Limits, continuity, differentiablity, Reimann integration, sequences, series, limsup, 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 fields, curl, divergence, 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

4. Topology :
Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness, Hausdorff spaces, normal spaces, Urysohn’s lemma, Tietze extension, Tychonoff’s theorem.

MSc/PhD Computer Science :
Topics covered in entrance examination
** Discrete Mathematics
Sets and relations, elementary counting techiniques, pigeon hole principle, partial orders,
** Elementary probability theory
** Automata Theory
Regular expressions, non deterministic and deterministic nite automata, subset construction, regular languages, non regularity (pumping lemma), context free grammars, basic ideas about computable and noncomputable functions.
** Algorithms
O notation, recurrence relations, time complexity of algorithms, sorting and searching (bubble sort, quick sort, merge sort, heap sort).
** Data structures
Lists, queues, stacks, binary search trees, heaps.
** Graphs
Basic denitions, trees, bipartite graphs, matchings in bipartite graphs, breadth rst search, depth rst search, minimum spanning trees, shortest paths.
** Algorithmic techniques
Dynamic programming, divide and conquer, greedy.
** Logic
Boolean logic, truth tables, boolean circuits | and, or, not, and, nand gates.

Suggested reading material :
1. Frank Harary: Graph Theory, Narosa.
2. John Hopcroft and Jerey D Ullman: Introduction to Automata, Languages and Computation, Narosa.
3. Jon Kleinberg and Eva Tardos: Algorithm Design, Pearson.
4. C. Liu: Elements of Discrete Mathematics, Tata McGraw-Hill.