• 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

1. Numerical Methods
– Numerical solutions of algebraic equations
– Method of iteration and Newton-Raphson method
– Rate of convergence

2. Linear Algebra
– Solution of systems of linear algebraic equations using Gauss elimination method
– Solution of systems of linear algebraic equations using Gauss-Seidel methods

3. Interpolation and Approximation
– Finite differences
– Lagrange interpolation
– Hermite interpolation
– Spline interpolation

4. Differentiation and Integration
– Numerical differentiation
– Numerical integration

5. Ordinary Differential Equations (ODEs)
– Numerical solutions using Picard’s method
– Euler’s method
– Modified Euler’s method
Runge-Kutta methods

1. Variation of a Functional
2. Euler-Lagrange Equation
3. Necessary and Sufficient Conditions for Extrema
4. Variational Methods for Boundary Value Problems in:
– Ordinary Differential Equations (ODEs)
– Partial Differential Equations (PDEs)

1. Linear Integral Equations
– Linear integral equation of the first kind
– Linear integral equation of the second kind
– Fredholm type equations
– Volterra type equations

2. Solutions with Separable Kernels

3. Characteristic Numbers and Eigenfunctions

4. 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