The details of the topics to be covered under the courses mentioned above are given below.

**MATH801 ALGEBRA 3CU:**

Sylow theorems, direct products, fundamental theorem of finite Abelian groups, field of quotients,
Euclidean rings, Polynomial rings over commutative rings, inner product spaces, theory modules,
sub-modules, quotient modules, modules over principal ideal domains. Applications finitely generated
Abelian group fields extension fields elements of Galois theory, solvability radicals.

**MATH802 ALGEBRAIC TOPOLOGY 3CU:**

Review of categories and functors. Homology, fundamental group, covering transformation, simplical
complexes. Singular homology, Universal co-efficient theory for homology and cohomology. Spectral
sequence.

**MATH803 FUNCTIONAL ANALYSIS 3CU:**

Measures and integration. Outer measure. Lesbegue Measure. Basic properties of Banach and Hilbert spaces.
Operators, Duality. Basic theorems in functional analysis. Classical Banach spaces. Spectral theory of
Operators in Hilbert spaces. L2 space as a Hilbert space. Banach algebras. Gelfand theory, compact
operators. Examples and applications to classical analysis.

**MATH804 ADVANCED COMPLEX ANALYSIS 3CU:**

Periodic functions, Weierstrass functions, elliptic curves. Modular forms. Algebraic functions,
Reimann surfaces. Covering surfaces, covering transformations. Discontinuous groups of linear transforms,
automorphic forms.

**MATH805 PARTIAL DIFFERENTIAL EQUATIONS 3CU:**

Basic examples of linear partial differential equations and their fundamental equations and their
fundamental solutions. Existence and regularity of solutions (Local or Global) of the Cauchy problems;
boundary value problems and mixed boundary value problems. The fundamental solutions of their partial
differential equations.

**MATH806 ADVANCED SET THEORY 3CU:**

Review of Naive (Cantorian) Set Theory, Paradoxes, Type Structures, Culmination into axiomatic.
Introduction to first-order logic: Syntax and Semantics. Axioms of Set Theory (ZF - Axioms) Extensionality,
pairing, power set, comprehension, replacement, foundation, infinity, choice, arguments for these axioms.
Intuitive theory of cardinals: countable and uncountable sets, fundamental theorem for demonstrating
countability, Proof of Schroeder - Bernstein Theorem, cardinal arithmetic, continuum hypotheses.
Ordered Relations and Ordered sets properties of orderings, lattices and Boolean algebra, well-ordered
sets. The Axiom of choice, Well-ordering theorem, Maximal principles and Zorn's Lemma. Some consequences
of the axiom of choice, the axiom of determinacy, introductory concept of forcing.

**MATH807 SPACE DYNAMICS 3CU:**

The Solar System, The general three-body problem, Equations of motion, integrals of 3-body problem,
The restricted three-body problem, Equations of motion, Liberation points, Hill & limiting surface
Stability of liberation points, The N-body problem, Equations of motion, Integrals of the N-body problem.
Perturbation Theory, Equations of pertubated motion, Evolution of orbit due to atmospheric resistance,
Influence of the celestial bodies on the motion of the satellites. Artificial Satellites.

**MATH808 COMPUTATIONAL FLUID DYNAMICS 3CU:**

Basic concepts in flow through porous media, convection, numerical solution of fluid flow in a vertical
channel filled with porous matrix: Using fixed grid and variable grid finite difference technique, under
thermal boundary condition of first, second and third kind. Solution of fluid flow in a composite
vertical channel partially filled with porous matrix and partially filled with clear fluid: using finite
difference technique, flow behaviour in a vertical annulus filled with porous matrix having thermal
boundary condition of mixed kind, mathematical modeling of wall conduction effect on flow mechanism in a
composite vertical channel, some closed form solution of flow through porous media.

**MATH809 LINEAR AND NON-LINEAR PROGRAMMING 3CU:**

Optimization Problems – types, size, solution, algorithms and convergence of algorithms. Linear programming
problems–definitions, connected with LPPs, formulation of LPPs, examples, diet problems, manufacturing
problems, transportation problems, fundamental theorem of linear programming. The simplex algorithms –
computational procedure and optimality condition theorem, two phase and big M methods, procedures of
resolving cycling due to degeneracy. The revised simplex algorithms–Computational procedures and
algorithms.
Duality in LPPs – Interpretation and examples, interpretation of duality in diet, manufacturing and
transportation problems etc., duality theorem of LPP, simplex multipliers and sensitivity complimentary
slackness theorems. Dual simplex method – Computational procedure and examples. Primal dual algorithm -
Computational procedure and examples, primal dual optimality theorem Nonlinear programming – general NLPP
and Lagrange multiplier method for problems having equality constraints, Kuhn – Tucker conditions for
non-equal constraints. Quadratic programming – Wolfe’s modified simplex method and examples, Beale’s
method and examples. Separable Convex programming – Piecewise linear approximations and separable
programming algorithms, examples.

**MATH812 GROUP REPRESENTATION THEORY 3CU:**

Representations of groups by linear transformations; group algebras, character theory and modular
representations. Representation theory of algebraic groups; representation of finite groups;
representation of compact and locally compact groups; representation of Lie groups. Unitary representation
theory.

**MATH814 ALGEBRAIC THEORY OF SEMIGROUPS 3CU:**

Regular Semigroups 0-Simple Semigroups, completely 0-Simple Semigroups, Clifford Semigroups, Inverse
Semigroups, free Semigroups, free products Semigroups, locally inverse Semigroups, Bisimple inverse
Semigroups, Representation of Semigroups, amalgamation Semigroups, congruence on Semigroups, Transformation
Semigroups.

**MATH815 ADVANCED DIFFERENTIAL EQUATIONS 3CU:**

General First Order Differential Equation: Initial value problem. Existence of Approximate solution.
Ascoll’s Theorem and Cauchy-Peano Theorem on the existence of solutions of the initial value problem.
Lichchitz condition and uniqueness of solution. Picard iterates and Picard-Lindel Theorem. Higher Order
equations, Formulation as a system of first order equation. Existence and uniqueness of solution.
System of Linear Differential Equations. Different metrics on matrices. Consistent norms on matrices.
Exponential and logarithms of matrices. Fundamental matrices for linear homogeneous system of differential
equation. Non-homogeneous system and solution in terms of fundamental matrices. Linear Differential
Equations with constant coefficients. Stability Theory for Linear System. Concepts of stability,
uniform stability, asymptotic stability and uniform asymptotic stability and their relationship for
linear system.

**SCI801 MANAGEMENT AND ENTREPRENEURSHIP 2CU:**

The course will cover business environment, general management, financial management, entrepreneurship
development, feasibility studies, marketing and managerial problem solving

**SCI802 ICT AND RESEARCH METHODOLOGY 2CU:**

Essentials of Spreadsheet, Internet Technology, Statistical Packages, precision and accuracy of
estimates, principles of scientific research, concept of hypotheses formulation and testing, organization
of research and report writing.

**MATH881 Seminar (1 CU/Semester)**

**MATH891 Research/Thesis (3 CU/Semester)**

**MATH882 Seminar (1 CU/Semester)**

**MATH892 Research/Thesis (3 CU/Semester)**