Suggested syllabus for the First Year of the NBHM Nurture programme 2007-2011 :


I. Calculus and Real Analysis :-


Sequences of real and complex numbers, Series of real and complex numbers, 
absolute convergence and conditional convergence, tests for convergence, 
Cesaro summability and Cauchy products, Functions and their limits,
Continuity and differentiability of functions of one variable, 
Rolle's theorem, mean value theorem, Taylor's theorem, uniform continuity, 
chain rule, derivatives of inverse functions, successive differentiation, 
Leibniz' rule, L'Hospital's rule, maxima and minima.

References:

W.Rudin: Principles of Mathematical Analysis, Chapters 3-8

F.John and R.Courant: Introduction to Calculus and Analysis (Vol. 1), 
sections 1.2, 1.3, 1.4, 1.6, 1.7, S.1, S.2, chapters 2,3 and 5, sections 7.1 
and 7.2


---------------------------------------


II. Algebra and Number Theory :-


Sets and equivalence relations, mappings, integers, groups, subgroups, 
normal subgroups, quotient groups, group homomorphisms and automorphisms, 
Cayley's theorem, permutation groups, Sylow's theorems, direct products and 
finite abelian groups.

Matrices, vector spaces, quotients, homomorphisms, isomorphisms, 
linear independence, bases, dimensions and direct sums.

Divisibility, congruences, theorems of Wilson, Fermat and Euler, 
Chinese Remainder Theorem, quadratic reciprocity law, the Jacobi symbol, 
arithmetic functions, linear Diophantine equations, sums of 2,3 and 4 
squares, binary quadratic forms and equivalence of quadratic forms.


References :

I.N.Herstein: Topics in Algebra (Wiley Eastern), chapters 1 and 2,
sections 4.1 and 4.2.

K.Hoffman and R.Kunze: Linear Algebra (Prentice Hall of India), chapters 1 
and 2.

I. Niven and H. S. Zuckerman: An Introduction to the Theory of Numbers 
(Wiley Eastern), chapters 1, 2, 3, 4 and 5.

 
Additional references:

M.Artin: Algebra (Prentice Hall of India), chapters 1, 2 and 3.

N.Jacobson: Basic Algebra, Vol. 1 (Hindustan Publishing Co.), chapter 1.


----------------------------------------------


III. Combinatorics :-

Binomial coefficients, Generating functions, Partitions, Pigeon-hole principle, 
Inclusion-exclusion principle, Marriage theorem, Latin squares, 
Cayley's theorem on number of trees, conditions for Eulerian graphs, 
Hamiltonian paths to exist, knight tour.


References :

L.Comtet: Advanced combinatorics, D.Reidel Publ.Co.Inc., USA 1974, 
Chapters I,II.

V.Bryant: Aspects of combinatorics, Cambridge Univ.Press 1993, Sections 1 to 6 of chapter I.