MT 701 Combinatorics
1. Counting principles, arrangements and selections, arrangements and selection with
repetition, distributions, binomial identities
2. Generating function : Generating function models, calculating coefficients of
generating functions, partitions, exponential generating functions, a summation method.
3. Recurrence Relations : Recurrence relation models, divide and conquer relations,
solution of linear and inhomogeneous recurrence relation, solution with generating functions.
4. Inclusion-exclusion: Counting with Venn diagrams, inclusion – exclusion formula,
restricted positions and Rook polynomials.
Prescribed Book :
1. Alan Tucker, Applied Combinatorics (fourth edition), John Wiley & sons , New York
(1995)
sections 5.1-5.6, 6.1-6.5, 7.1-7.5, 8.1-8.3.
Reference books :
1.V. Krishnamurthy, Combinatorial, Theory and Applications, East West Press, New
Delhi (1989) Scientific, (1996)
2.K.D. Joshi : Foundations of discrete mathematics,Wiley
3. Marshall Hall : Combinatorial theory ,Wiley.

MT 703 Functional Analysis

Hilbert spaces, operators on a Hilbert space, Banach spaces.

Prescribed book :

John B. Conway : A course in functional analysis. Springer (1997 ) Chapters 1,2,3.

MT 702 Field Theory

1. Field Extensions :

Basic Theory of Field Extensions

Algebraic Extensions

Classical Straightedge and Compass Constructions

Splitting Fields and Algebraic Closures

Separable and Inseparable Extensions

Cyclotomic Polynomials and Extensions

2. Galois Theory :

Basic Definitions

The Fundamental Theorem of Galois Theory

Finite Fields

Galois Groups of Polynomials

Solvable and Radical Extensions: Insolvability of the Quintic

Prescribed Book :

Dummit and Foote, Abstract Algebra, 2nd Edition, Wiley Eastern Ltd.

Chapters : 13.1 to 13.6

14.1 to 14.3, 14.6 , 14.7 (statements only)

Reference Books :

1. O. Zariski and P. Sammuel, Commutative Algebra, Vol. 1, Van Nostrand.

2. P. Bhattacharya and S. Jain, Basic Abstract Algebra, Second Edition,

Cambridge University Press.

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           MT 704 Graph Theory

Paths and cycles, trees, planarity, coloring, digraphs, matchings, marriage and Mengers theorem.

Prescribed Book :

R. J. Wilson, Introduction to graph theory, Pearson, (2003) Chapters 1 – 8.

MT – 706  Cryptography

 

Unit 1 :  Cryptography

Some simple cryptosystems,Enciphering matrices

 

Unit  2 :  Public Key

The idea of public key cryptography, RSA,Discrete log ,

              Knapsack, Zero- knowledge protocols and oblivious transfer

 

Unit 3:  Primality and Factoring

Pseudoprimes, The rho method, Fermat factorization and factor

bases, The continued fraction method,The quadratic sieve

method

 

Unit 4:  Elliptic curves

Basic facts, Elliptic curve cryptosystems, Elliptic curve

             primality test, Elliptic curve factorization

 

Unit 5:  Problem solving using ‘SAGE- Free Open Source Software’

 

Text Book: A course in Number Theory and cryptography, Neal Koblitz

Springer, second edition.

 

Chapters : 3 , 4 , 5, 6

 

Reference Books:1. Introduction to Modern Cryptography
Jonathan Katz and Yehuda Lindell
Publisher: Chapman & Hall/CRC
               2. Handbook of Applied Cryptography,

A. Menezes, P. van Oorschotand S. Vanstone,
 CRC Press
                3. Invitation to Cryptology,  Barr,  Prentice Hall


Extra Credit course on Scilab and Latex.