Math. 611 Functional Analysis
Spectral notions – Introduction to Banach algebra – Adjoints and sesquilinear functionals – Some spectral results for normal and completely connection operators – Orthogonal projections and positive definite operators – Square roots and spectral decomposition theorem – Spectral theorem for completely continuous normal operators – Spectral theorem for bounded self-adjoint operators.

Math. 612 Modern Algebra
Group theory: Sylow’s theorems - Fundamental theorem of abelian groups, Jordan-Holder theorems and solvable groups – Modules: free modules and Zoru’ Lemma – Modules over PID – Categories: Products - Coproducts and free objects - Fields: algebraic and transcendental extensions - Algebraic closure - Galois theory of finite extensions and solvability by radicals – Linear algebra: canonical forms – Rings: Semi-simple rings and Jacobson radical.

Math. 613 Topology
The topics to be covered include topologies and bases – Subspaces - Product spaces - Quotient spaces - Metric spaces – Continuity – Connectedness – Compactness – Para-compactness - Separation axioms - Covering spaces and homotopy theory - Simplicial theory.

Math. 614+615
Function of one Complex Variable, Partial Differential Equations, Ordinary Differential Equations and Mathematical Logic, Analysis and Numerical Analysis: The student selects any two of the following:

(a) Function of one complex variable Conformal mapping – Application of conformal mapping – The Schwarz-Christoffel transformation – Analytic continuation and Riemann surface – Harmonic functions: Basic properties of harmonic functions – Harmonic functions on a disk – The Dirichlet problems – Green’s functions: Entire functions – Infinite products.
(b) Partial Differential Equations The single-first order equation – Second-order equations: hyperbolic equations for functions of two independent variables – Characteristic manifolds and Cauchy problem – Elliptic equations: Laplace’s equation - The classical maximum principle - Poisson’s equation and
Postgraduate Programs Guiding Regulations
Newtonian potential - Classical solutions - Schauder approach - Sobolev space - Generalized solutions and regularity, strong solutions – Hyperbolic equations in higher dimensions - Parabolic equations.
(c) Ordinary Differential Equations Existence and uniqueness of solution of initial value problems – Linear systems – Linearization – Plane systems – Stability.
(d) Mathematical Logic Formal systems (first order) – Proof of theoretic and model theoretic techniques and interconnections – Compactness and completeness theorems – Non-standard models with applications to analysis – Peano arithmetic and set theory as illustrations of important first order systems.
(e) Analysis Elementary set theory, Axiom of choice, elementary topology – Lebesque - measure and integration on the real line – Abstract measure and integration theory - Product measures and Fubini’s theorem - Absolute continuity and the Radon-Nikodym theorem - Signed measures and decomposition theorems - Integration on locally compact spaces - Lp-space and the Riesz representation theorem – Elementary theory of topological vector spaces - Normed space and Hilbert space - Elementary theory of continuous linear operators.
(f) Numerical analysis Second order partial differential equations – Classification - Properties - Initial and boundary conditions - Space of solutions – Finite difference approximations to derivatives – Parabolic equations and finite difference representation (explicit - implicit) weight average approximation properties of difference schemes – Stability - Consistency and convergence methods of improving accuracy – Hyperbolic equations and characteristics – Numerical integration along a characteristic – Courant, Friedrichs - Lewy condition – Simultaneous first order equations and stability – Introduction to the numerical solution of integral equation – Variational formulation PDE and the finite element method.