Nurture Programme 2007-2011

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	Second year Algebra, Analysis, Combinatorics and Number theory, 
Syllabus :-

Algebra :

Sylow's Theorems and Applications. Permutation Groups, Simple Groups,
Solvable and Nilpotent Groups.  Direct Products, Finite Abelian Groups, Finitely
generated abelian groups, Elementary divisors.

Rings and Ideals, Homomorphisms, Quotient Rings, Algebras,
Ring Extensions. Integral Domains and Field of Fractions, Prime
and Maximal Ideals. Polynomial Rings, Polynomial Functions and
Roots of Polynomials. Factorisation of Integers and Polynomials.
Eucildean Domain, PID, UFD. Gauss' Theorem. Ring of Gaussian Integers.
Algebraic Integers.

Analysis: 

Applications of mean value theorem, L'Hopitals Rule.
Higher order derivatives, Taylors theorem .
The Riemann integral (follow Tao' book) 11.1 to 11.7.

Combinatorics :

Partitions, Latin squares, Graph theory (conditions for Eulerian graphs, 
Hamiltonian paths to exist, knight tour), Marriage theorem, Permutation 
group and Polya's theory of enumeration, Schutzenberger's jeu de taquin, 
RSK correspondence and generalizations.


Number Theory :

Quadratic reciprocity law, the Jacobi symbol, arithmetical functions, 
sums of 2, 3 and 4 squares, binary quadratic forms and equivalence of 
quadratic forms.


References :   

1. M.Artin : Algebra, Chapters 10,11.

2. V.Bryant :  Combinatorics.

3. W.Fulton, Young tableaux, LMS Student Texts, No.35., CUP 1997. 

4. N. Jacobson : Basic Algebra, Vol. 1.

5. V.Krishnamurthy : Introduction to combinatorics.

6. W. Rudin, Principles of Mathematical Analysis.

7. T. Tao, Analysis I.

8. Niven, Zuckerman & Montgomery : An Introduction to the Theory of 
Numbers, Chapters 2,3,4,10.