Real analysis: Metric Spaces, Completeness, Connectedness, Complete Review of Riemann Integral, Lebesgue measure, Lebesgue and convergence, LP spaces.

Complex analysis: complex-analytic functions, Cauchy’s integral formula, power series, Liouville’s theorem, maximum-modulus theorem, Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal mappings.

Introduction to Error: its sources, propagation and analysis. Root finding of non-linear equations.Systems of linear and non-linear equations.

Finite difference operators, Polynomial interpolation and error analysis, Hermite interpolation, Spline interpolation, B-Splines, Numerical differentiation, Numerical integration: Trapezoidal and Simpson’s rules, Newton-Cotes formula, Method of undermind coefficients and Gaussian Quadrature, Richardson Extrapolation.

Difference equations. Numerical solution of ordinary differential equations: Initial value problems: Numerical stability, Taylor series method, Euler and modified Euler methods and stability analysis, Runge-Kutta methods, Multistep methods, Predictor-Corrector method, convergence and stability. System of ordinary differential equations. Boundary Value Problems: Shooting and direct methods.

Kinematics of Fluids in Motion: Real Fluids and Ideal Fluids, Velocity of fluid at a point, Streamlines and Pathlines, Steady and Unsteady Flows, the Velocity Potential, Vorticity Vector, The Equation of continuity. Equation of motion of a fluid: Pressure at a point in fluid at rest, Pressure at a point in a moving fluid, Euler’s equation of motion, Bernoulli’s equation. Viscous and inviscid fluid, the Navier-Stokes equation of motion, rotational and irrational flows.

Theory of surface wave: Equation of Motion, Wave Terminology, Analytical solution of the wave problem, Dispersion relation of the wave motion, Classification of water waves, Particle motion and Pressure, Superposition of waves, Wave reflection and standing wave, Wave energy and group velocity, Wave Refraction, Wave Diffraction. Finite amplitude waves: Mathematical formulation, Perturbation method of solution. Linear and Nonlinear diffraction theory.

Nonlinear equations: autonomous and non-autonomous systems, phase portrait, stability of equilibrium points, Lyapunov exponents, periodic solutions, local and global bifurcations, Poincare-Bendixon theorem, Hartmann-G robmann theorem, Center Manifold theorem.

Nonlinear oscillations: perturbations and the Kolmogorov-Arnold-Moser theorem, limit cycles. Chaos: one-dimensional and two-dimensional Poincare maps, attractors, routes to chaos, intermittency, crisis and quasi periodicity. Synchronization in coupled chaotic oscillators. Applications: Examples from Biology, Chemistry, Physics and Engineering.

Galois Theory: Field Extensions, Automorphisms, self Normal Extensions, Separable and Inseparable Extensions, The Fundamental Theorem of Galois Theory; Some Galois Extensions: Finite Fields, Cyclotomic Extensions, Norms and Traces, Cyclic Extensions, Hilbert Theorem 90 and Group Cohomology, Kummer Extensions; Applications of Galois Theory: Discriminants, Polynomials of Degree 3 and 4, Ruler and Compass Constructions, Solvability by Radicals; Infinite Algebraic Extensions, Infinite Galois Extensions, Krull topology, The Fundamental Theorem of Infinite Galois theory; Transcendental Extensions: Transcendence Bases, Linear Disjointness, Affine Algebraic Varieties, Algebraic Function Fields, Derivations and Differentials.

Rings and Ideals ;Rings and ring homomorphism , Ideals, quotient rings, Zero divisors, Nil potent Element ,Units, Prime Ideals , and maximal ideals, Nilradical and Jacobson radical, Operation on ideals, Extension and contraction ; Modules ; modules and modules homomorphism ,sub module ,Direct sum product of modules, restriction and extension of scalars , Exactness properties of the tensor product ,algebras ,tensor product of algebra ;Rings and modules of fraction,L ocal Properties Extended and contracted ideals in the rings of fractions, Primary decomposition, Integral dependence, The going up theorem Integrally closed integral domains, Valuation Topologies and completion, Filtrations, Graded Rings and modules, The associated graded ring , Artin Ress Theorem , Dimension Theory, Hilbert function , Dimensions theory of Noetherian local rings, Regular local rings, Transcendental dimension, Depth :M-Regular Sequences, Cohen Macaulay Rings.

Operators on Hilbert spaces: Bounded linear operator on Hilbert spaces, spectrum of an operator, weak , norm and strong operator topologies, normal, self adjoint, unitary and compact operator and their spectra. Diagonalization, spectral theorem and applications: diagonalization for a compact self adjoint operator, spectral theorem for compact norm operator, spectral calculus, application to strum-Liouville problem. Positive operators : positive linear maps of finite dimensional space and their norms, Schur products, completely positive maps.

Introduction to Financial Derivatives: Futures and options Pricing with no-arbitrage principle, Black-Scholes equation for European style option, Quick introduction to stochastic processes, Basic facts about Brownian Motion and Martingales with applications in finance, Stochastic integration, Stochastic Calculus (Itô's formula, Girsanov theorem, Connection with PDEs) Black-Scholes formula, Greeks.

Prerequisite : Basic Knowledge in Single Variable Calculus

Sigma- rings, sigma algebra, measurable space, countability and sub – addivity of a measure, Borel measure, Lebesgue outer measure, measurable sets, construction of non-measureable set, the contor set. Measurable functions and their properties, almost everywhere property, approximation of measurable functions with the simple measurable functions and step functions. Egorov’s theorem and Lusin’s theorem. Lebesgue integral of functions, Lesbgue integral of integral functions and its linearity, Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem, Applications of convergence theorems. Singed measures, absolute continues functions and their properties, singular measures , Radon – Nikodym theorem with applications.

The Fundamental theorem of Arithmetic, Arithmetical Functions and Dirichlet Multiplication: The Mobius function, The Euler totient function, product formula, The Dirichlet product of arithmetical Functions, Dirichlet inverses and the Mobius inversion formula, The Von-Mangoldt function, Multiplicative functions, Dirichlet multiplication, The inverse of a completely multiplicative function, Liouville’s function, The divisor functions, Generalized convolutions, Formal power series, The Bell series of an arithmetical function, Bell series and Dirichlet multiplication, Derivatives of arithmetical functions, The Selberg identity; Averages of Arithmetical functions: Euler’s summation Formula, Some elementary asymptotic formulas, An application to the distribution of lattice points visible from the origin, The partial sums of a Dirichlet product, Another identity for the partial sums of a Dirichlet product; Some Elementary Theorems on the Distribution Numbers: Chebyshev’s functions, Prime number theorem, Some equivalent forms of the prime number theorem, Shapiro’s Tauberian theorem, An asymptotic formula for partial sums, The partial sums of the Mobius function, Brief sketch of an elementary proof of the prime number theorem, Selberg’s asymptotic formula.

Prerequisite : Nil

Weakly modular functions, modular functions, The modular group, modular forms, cusp forms, Eisenstein series, Congruence subgroups, The j-function, Dedekind eta function, The space of modular forms, dimension formula, expansions at infinity, elliptic points, parabolic points and hyperbolic points of congruence subgroups, fundamental domain, Hecke operators, Theta functions, Automorphic forms.

Calculus of Variation: Introduction, problem of barchistochrone, isoperimetric problem, concept of extrema of a functional, variation and it’s properties. Variational problems with fixed boundaries, The Euler equation, The fundamental lemma of calculus of variations. Variational problems with moving boundaries, Reflection and refraction extremals.

Transversality conditions, Sufficient conditions for an extremum, Field of extremals, Jacobi conditions, Legendre Condition. Second variations. Canonical equations and variational principles, Introduction to direct method for variational principle.

Integral Equations: Integral equations, Regular Integral equations: Voltera integral equations, Fredholm integral equations, Volterra and Fredholm equations with regular kernels. Degenerate kernel, Fredholm Thereom, Method of successive approximation.

Bernstein polynomials and its properties. Solving integral equations by using Bernstein polynomials and general polynomial. Numerical method: Quadrature method for Integral equations. Green's function in integral equations.

Mainfolds: Indentification (quotient) spaces and identification (quotient) maps; topology n-manifolds including surfaces, Sn, RPn, CPn. and lens spaces.

Triangulated manifolds: Representation of triangulated, closed 2-manifolds as connected sums of tori of projective planes.

Fundamental group and covering spaces: Fundamental group, functoriality, retract, deformation retract; Van Kampen’s Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, RPn, lens spaces via covering spaces.

Simplicial homology: Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed trangulated surfaces via gomology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.

Composition of knots, Reidemeister moves, links, Invariants of knots, Surfaces and Knots: Genus and Seifert surfaces, Torus knots knots and its properties, Setelite Knots, Hyperbolic Knots, Braid theory, Alexander polynomial, Bracket polynomial, HOMFLY polynomial, Jones polynomial, Vassiliev Invariants, Knot complements and 3-Manifolds.