**Commutative Algebra):**- Regular Sequences and Depth : Regular Sequences, Grade and Depth, Depth and Projective Dimension, Some Linear Algebra, Graded rings and modules. The Koszul Complex. The Eagon-Northcottcompled.
- Cohen-Macaulay Rings : Cohen - Macaulay rings and modules, Regular rings and normal rings, Complete Intersections.
- DeterminantalRings : Graded Hodge Algebras, Starightening Laws on Posets of Monors, Properties of Determinantal Rings.

**UNIT I**

**Difference Equation and its Application **

Difference Calculus, Linear first - order difference equations, Nonlinear difference equations, Higher order linear difference equations, Systems of difference equations, Stability Theory, Applications.

**Computational Techniques using Mathematica and Matlab**

Mathematica basics. 2 D and 3 D Graphs .Basic Calculus.Ordinary Differential Equations.Partial Differential Equations and Boundary Value Problems.Mathematica Programming. Linear and Nonlinear Integral Equations.

Matrix Operations in MATLAB.Solution of Equations.Curve-fitting.Numeral Integration.MATLAB Programming.

**UNIT II**

**Operation Research**

Inventory control of style goods and perishable items.Production planning for unreliable production systems.Integrated production, quality and maintenance models. Production planning and inventory control in fuzzy environment. Supply chain - definition, decision phases, process view. Centralized supply network versus decentralized operation. Coordination. Bullwhip effect. Multi-echelon supply chains. Simple models of supply chain management. Solving inventory/supply chain management problem using Genetic Algorithms (GAs).

**Algebra**

Definition of normed and Banach Algebra and examples,Singular and Non-singular elements, the spectrum of an element, The spectral radius. Definition of C*-Algebras and examples, Self-adjoint, Unitary, Normal, Positive and Projection elements in C*-Algebras, Commutative C*-Algebras, C*-Homomorphisms, Representation of commutative C*-Algebras, sub algebras and the spectrum, The Specrtral Theorem, Positive linear functions in C*-algebras, States and the GNS construction.

**UNIT III**

**Topology**

Definition, Basic properties including translations in topological groups, neighbourhood system of identity, separation properties, uniform structure on topological groups. Locally compact groups, Lie groups, Measure and integration in locally compact spaces and then in locally compact groups, Haar measure, Haar integrals.

**Sequence Spaces**

Linear spaces, Linear metric spaces, paranorms, seminorms, norms, subspaces, dimensionality, factorspaces, basis, dimension, basic facts of normed linear spaces and Banach spaces (revision). Sequence spaces, Matrix and linear transformations, Algebras of matrices, summability, Tauberian Theorems.

**UNIT IV**

**Integral Equations **

Basic definitions, regular, singular, hypersingular integral equations.Occurrence of integral equations in classical mechanics, ordinary differential equations, partial differential equations.Occurrence in continuum mechanics (elasticy, fluid mechanics).Singular integral equations, Abel integral equations, solutions, Cauchy singular integral equations, solutions, applications.Hypersingular integral equations, solution of simple hypersingular integral equations, applications.Dual integral equations.Solution for trigonometric function kernels, applications.

Theory of Distribution
Good function, Fairly good function, Generalised function, Ordinary function as generalized function . Addition of generalized function, Derivatives of generalized functions, Fourier transform of generalized functions, Limits of generalized function, Powers of | x | as generalized functions, Even and odd gerneralised functions, Integration of generalized functions, Integration of generalized function, Multiplication of two generalized functions.

**Modules, Rings, Groups and Categories**
Tensor Product of Modules, Categories, Functions and Natural Transformations, Exact sequences, Projective, Injective and Flat Modules, Localization, Group Representation Theory.

**UNIT V**

This unit will be decided by the Ph.D. guide