Elementary Set Theory

• Sets and subsets

• Operations on sets: union, intersection, difference, complement

• Cartesian product

• Relations and functions

Finite, Countable, and Uncountable Sets

• Cardinality of sets

• Countable and uncountable sets

• Cantor's diagonal argument

Real Number System

• Construction of real numbers

• Properties of real numbers

• Completeness of the real number system

• Archimedean property

• Supremum and infimum

Sequences and Series

• Definition of sequences and series

• Convergence and divergence of sequences and series

• Cauchy sequences

• Limsup and liminf

• Tests for convergence of series

Bolzano-Weierstrass Theorem

• Statement and proof of the theorem

• Applications of the theorem

Heine-Borel Theorem

• Statement and proof of the theorem

• Applications of the theorem

Continuity

• Definition of continuity

• Properties of continuous functions

• Intermediate value theorem

• Uniform continuity

• Lipschitz continuity

Differentiability

• Definition of differentiability

• Properties of differentiable functions

• Mean value theorem

• Taylor's theorem

• Implicit function theorem

• Inverse function theorem

Sequences and Series of Functions

• Pointwise and uniform convergence

• Weierstrass M-test

• Uniform convergence and continuity

• Uniform convergence and integrability

Riemann Sums and Riemann Integral

• Definition of Riemann sums and Riemann integral

• Properties of Riemann integrable functions

• Improper integrals

Monotonic Functions and Discontinuities

• Definition of monotonic functions

• Types of discontinuities

• Functions of bounded variation

• Lebesgue measure and integral

Functions of Several Variables

• Partial derivatives

• Directional derivatives

• Chain rule

• Implicit function theorem

Metric Spaces

• Definition of metric spaces

• Open and closed sets

• Compactness

• Connectedness

Normed Linear Spaces

• Definition of normed linear spaces

• Examples of normed linear spaces

• Banach spaces and completeness

Spaces of Continuous Functions

• Definition of space of continuous functions

• Uniform convergence and continuity

• Arzela-Ascoli theorem

• Vector spaces:

  • Definition and examples

  • Subspaces

  • Linear dependence

  • Basis and dimension

  • Algebra of linear transformations

• Linear Transformations :

  • Definition Range and Null space of Linear Transformation

  • Sylvester's Law

  • Types of Linear Transformation

  • Properties of Linear transformation

  • Some Important Linear operators

  • Set of all linear transformation from V to V'

  • Properties

  • Isomorphism of Vector Spaces

  • Matrix Representation of Linear Transformation

  • Properties Important Matrices/Linear Operators

  • System of Linear Equations

• Algebra of matrices:

  • Operations on matrices

  • Rank and determinant of matrices

  • Solution of linear equations

• Eigenvalues and eigenvectors:

  • Definition and examples

  • Cayley-Hamilton theorem

  • Canonical forms

  • Diagonal forms

  • Triangular forms

  • Jordan forms

• Inner product spaces:

  • Definition and examples

  • Orthonormal basis

• Quadratic forms:

  • Definition and examples

  • Reduction of quadratic forms

  • Classification of quadratic forms

• Introduction and Geometrical Representation of Complex Numbers

  • Polar Forms of a Complex Number P(x,y)

  • Inverse Points with respect to a circle

  • Chordal Distance

  • Stereographic Projection and Point Set Topology

• Limit, Continuity and Differentiability

  • Topology in the Complex Plane

  • Limit of a function

  • Alternative Definition of limit of function

  • Limit of a function at z = ¥

  • Continuous functions

  • Uniform continuity

  • Differentiability

  • Cauchy-Riemann Equations

  • Polar Form of C-R Equations

  • Complex form of C-R Equation

  • Necessary condition for differentiability

  • Sufficient Condition of Differentiability

  • Assignment

• Singularities of Analytic Functions

  • Regular point and Analytic Functions

  • Singularity

  • Classifications of singular points

  • Entire functions

  • Result on Analyticity

  • Construction of Analytic function

• Complex Integration

  • Curves and Cauchy's Integral Formula

  • Extension of Cauchy's Integral formula to multiply connected Regions

  • Cauchy integral formula for the derivative of an Analytic function

  • Higher Order Derivatives

  • Morera's Theorem

  • Analytic functions on simplyh connected domains

  • Cauchy's inequality

  • Assignment

• Some Important Theorems and Their Applications

  • Liouville's Theorem

  • Fundamental Theorem of Algebra in C

  • Gauss's Theorem

  • Luca's Theorem

  • Generalized Version of Liouville's Theorem

• Power Series

  • Power Series

  • Result on the Radius of convergence

• Taylor and Laurent Expansion

  • Taylor Series Expansion

  • Laurent Series Expansion

  • Analysis of singularities through Laurent series

  • Picard's little Theorem

  • Picard's Great Theorem

• Some Special functions related to the Exponential

  • The exponential function

  • The logarithm function

  • The square Root function

• Meromorphic functions

  • Argument Theorem

  • Rouche's Theorem

• Calculus of Residues

  • Residue at a finite point

  • Some result on poles

  • Residue at infinity

  • Cauchy residue theorem

  • Extended Residue formula

  • Assignment

• Conformal Mapping

  • Defination

  • Conformal Mapping/Conformality

  • Magnifications factor and scale factor

  • Linear fractional/Bilinear/Mobius Transformation

  • Matrix Interpretation of a Mobius transformation

  • Fixed points

  • Normal form or canonical form of a bilinear transform

  • Classification of bilinear transformation on the basis of normal form

  • Cross Ratio

  • Automorphisms of Disks and Half-plane

  • Automorphism of the unit disk

• Maximum and Minimum Modulus Principle and Schwarz Lemma

  • Mean value property

  • The open mapping theorem

  • maximum modulus principle

  • minimum modulus theorem

  • Schwarz pick lemma

• Permutations

• Combinations

• Pigeon-hole principle

• Inclusion-exclusion principle

• Derangements

• Fundamental theorem of arithmetic

• Divisibility in Z

• Congruences

• Chinese Remainder Theorem

• Euler’s Ø- function

• Primitive roots

• Groups

• Subgroups

• Normal subgroups

• Quotient groups

• Homomorphisms

• Cyclic groups

• Permutation groups

• Cayley’s theorem

• Class equations

• Sylow theorems

• Rings

• Ideals

• Prime and maximal ideals

• Quotient rings

• Unique factorization domain

• Principal ideal domain

• Euclidean domain

• Polynomial rings

• Irreducibility criteria

• Fields

• Finite fields

• Field extensions

• Galois Theory

• Introduction to Topology

• Topological Spaces and Continuous Functions

• Basis and Subbasis for a Topology

• Dense Sets and Closure

• Separation Axioms

• T1 Separation Axiom

• T2 Separation Axiom

• T3 Separation Axiom

• T4 Separation Axiom

• Regular and Normal Spaces

• Urysohn's Lemma and Tietze's Extension Theorem

• Compact Spaces

• Compactness and Finite Intersection Property

• Heine-Borel Theorem and Bolzano-Weierstrass Theorem

• Connected Spaces

• Path Connected Spaces

• Product Spaces and Tychonoff's Theorem

Chapter 1: First Order Ordinary Differential Equations

• Existence and uniqueness of solutions of initial value problems for first order ODEs

• Separable and linear equations

• Exact equations and integrating factors

• Singular solutions of first order ODEs

• System of first order ODEs

Chapter 2: Linear Second Order Ordinary Differential Equations

• Homogeneous linear second order ODEs

• Nonhomogeneous linear second order ODEs

• Method of undetermined coefficients

• Variation of parameters

• Cauchy-Euler equations

Chapter 3: Higher Order Linear Ordinary Differential Equations

• Homogeneous linear higher order ODEs with constant coefficients

• Nonhomogeneous linear higher order ODEs with constant coefficients

• Method of undetermined coefficients

• Variation of parameters

• Cauchy-Euler equations

Chapter 4: Special Topics in Ordinary Differential Equations

• Power series solutions of differential equations

• Laplace transform method for solving differential equations

• Systems of first order linear differential equations

• Sturm-Liouville boundary value problem

• Green's function method for solving differential equations.

• Lagrange method for solving first order PDEs

• Charpit method for solving first order PDEs

• Cauchy problem for first order PDEs

• Classification of second order PDEs

• General solution of higher order PDEs with constant coefficients

• Method of separation of variables for Laplace equation

• Method of separation of variables for Heat equation

• Method of separation of variables for Wave equation

Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods. 

Calculus of Variations: Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations. 

Linear Integral Equations: Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel. 

Classical Mechanics: Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s principle and principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical equations for the motion of a rigid body about an axis, theory of small oscillations. 

• Probability:

  • Sample space

  • Discrete probability

  • Independent events

  • Bayes theorem

  • Random variables and distribution functions (univariate and multivariate)

  • Expectation and moments

  • Independent random variables, marginal and conditional distributions

  • Characteristic functions

  • Probability inequalities (Tchebyshef, Markov, Jensen)

  • Modes of convergence

  • Weak and strong laws of large numbers

  • Central Limit theorems (i.i.d. case)

• Markov Chains:

  • Finite and countable state space

  • Classification of states

  • Limiting behaviour of n-step transition probabilities

  • Stationary distribution

  • Poisson and birth-and-death processes

• Standard Distributions:

  • Discrete and continuous univariate distributions

  • Sampling distributions

  • Standard errors and asymptotic distributions

  • Distribution of order statistics and range

• Estimation:

  • Methods of estimation

  • Properties of estimators

  • Confidence intervals

• Hypothesis Testing:

  • Tests of hypotheses: most powerful and uniformly most powerful tests, likelihood ratio tests

  • Analysis of discrete data and chi-square test of goodness of fit

  • Large sample tests

  • Simple nonparametric tests for one and two sample problems

  • Rank correlation and test for independence

  • Elementary Bayesian inference

• Linear Models:

  • Gauss-Markov models

  • Estimability of parameters

  • Best linear unbiased estimators

  • Confidence intervals

  • Tests for linear hypotheses

  • Analysis of variance and covariance

  • Fixed, random and mixed effects models

  • Simple and multiple linear regression

  • Elementary regression diagnostics

  • Logistic regression

• Multivariate Analysis:

  • Multivariate normal distribution

  • Wishart distribution and their properties

  • Distribution of quadratic forms

  • Inference for parameters, partial and multiple correlation coefficients and related tests

  • Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation