Doctor of Philosophy in Mathematics

The Department of Mathematics started its PhD programme in July 2013. A wide range of the following research areas are offered in which the enrolled students can pursue their PhD work:

• Numerical Analysis
• Differential Equations and Boundary Value Problems
• Fourier Analysis
• Analysis, Function Spaces
• Integral Operators and weighted Norm Inequalities
• Graph Theory, Discrete Mathematics
• Finite Elements Methods
• Parallel Computations
• Statistical Approximation, Stochastic Processes
• Mathematical Biology, Non-linear Dynamical Systems
• Optimization, Swarm Intelligence

Candidates must have completed a minimum of 17 years of formal education, i.e. 12 years of regular schooling, followed by either a 3-year Bachelor’s degree and a 2-year Master’s degree, or a 4-year Bachelor’s degree and a 1-year Master’s degree in Mathematics, Computer Science, Statistics, Operations Research, Physics, or an equivalent field, with a minimum aggregate of 55% marks or an equivalent grade.

Candidates with 4-year Bachelor degree in the subject are also eligible for admission to PhD programme provided they have secured a minimum of 80% marks or equivalent grade.

Modes of Admission

Admission to various PhD programmes is offered through two modes:

1. SAU Entrance Test Mode:

• For Indian Candidates: Center-based online tests conducted by the University on stipulated dates.
• For other SAARC Candidates (excluding India): Proctored online tests on stipulated dates.

2. Direct Admission Mode (Without SAU Entrance Test):

• In India: Based on scores from national-level tests.
• In Other SAARC Countries: Based on national-level test scores or qualifying examination results from recognized institutions/boards.
• For Non-SAARC Candidates: Based on scores obtained in their qualifying examination (from recognized institutions).

Under the Entrance Test mode, the admission procedure for the PhD programme consists of an Entrance Test followed by an interview.
Under the Direct and Executive modes, candidates will be shortlisted and called directly for the interview.
For further details, please refer to the General Guidelines for the PhD Programmes.

• The duration of the Entrance Test will be 2 hours.
• The question paper will consist of 70 multiple choice questions.
• There will be no negative marking.
• Calculators will not be allowed. However, Log Tables may be used.

The questions may be asked in the Entrance Test in the following areas:

Analysis

• Real functions; limit, continuity, differentiability
• Sequences; series; uniform convergence
• Functions of complex variables; analytic functions; complex integration
• Singularities; power and Laurent series
• Metric spaces; stereographic projection
• Topology, compactness, connectedness
• Normed linear spaces; inner product spaces
• Dual spaces; linear operators
• Lebesgue measure and integration; convergence theorems

Algebra

• Basic theory of matrices and determinants; eigenvalues and eigenvectors
• Groups and their elementary properties; subgroups; normal subgroups; cyclic groups; permutation groups
• Lagrange's theorem; quotient groups; homomorphism of groups
• Cauchy Theorem and p-groups; structure of groups; Sylow's theorems and applications
• Rings, integral domains and fields; ring homomorphism and ideals
• Polynomial rings and irreducibility criteria
• Vector spaces, subspaces, linear independence, basis, dimension
• Inner product spaces; orthonormal basis; Gram–Schmidt process
• Linear transformations

Differential Equations

• First order ODEs; initial value problems; singular solutions
• System of linear first order ODEs
• Method of solution of dx/P = dy/Q = dz/R
• Orthogonal trajectory
• Pfaffian differential equations in three variables
• Linear second order ODEs; Sturm–Liouville problems
• Laplace transformation of ODEs; series solutions
• Cauchy problem for first order PDEs; method of characteristics
• Second order linear PDEs and classification
• Separation of variables
• Solutions of Laplace, wave and diffusion equations
• Fourier and Laplace transforms of PDEs

Numerical Analysis

• Numerical solution of algebraic and transcendental equations
• Direct and iterative methods for systems of linear equations
• Matrix eigenvalue problems
• Interpolation and approximations
• Numerical differentiation and integration
• Composite and double numerical integration
• Initial value problems (numerical solution)
• Finite difference and finite element methods for boundary value problems

Probability and Statistics

• Axiomatic approach of probability
• Random variables
• Expectation; moment generating functions
• Density and distribution functions
• Conditional expectation

Linear Programming

• Linear programming problem and formulation
• Graphical method; simplex method
• Artificial starting solution
• Sensitivity analysis
• Duality and post-optimality analysis