## PhD in Mathematics

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**Minimum Eligibility**

A minimum of 17 years of education with a Master's degree in Mathematics/Computer Science/Statistics/Operations Research/Physics or an equivalent degree with 55% marks in the aggregate or equivalent grade.

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**Admission Procedure**: Entrance Test and Interview. A minimum of 50% marks will have to be secured separately, in both the Entrance Test and the Interview, in order to be eligible for admission. The weightage of the Entrance Test will be 50% and the weightage of the Interview will be 50%.

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**Format of the Entrance Test Paper**

The duration of the Entrance Test will be 2 hours and the question paper will consist of 50 multiple choice questions.

Preference will be given to those opting for the research in the following areas:

1. Numerical Methods

2. PDE

3. Computational Fluid Dynamics

4. Graph Theory

5. Analysis

6. Function Spaces

7. Fourier Analysis

8. Parallel Algorithms

9. Scientific Computing

10. Optimization Techniques

11. Swarm Intelligence

12. Mathematical Biology

13. Nonlinear Dynamical System

14. Bio-informatics

15. Optimization

16. Graph theory

17. Differential Equations

18. Numerical Analysis

19. Mathematical Modeling

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**Negative Marks for Wrong Answers: **If the answer given to any of the Multiple Choice Questions is wrong, 1/4^{th} of the marks assigned to that question will be deducted.

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**Interview:** Candidates up to five times the number of seats available will be short-listed for interview on the basis of their performance in the Entrance Test, subject to a minimum cut off.

A final merit list will be drawn by adding together the marks of the Entrance Test and the interview. Separate merit lists will be made for (a) candidates from all SAARC countries other than India, and (b) candidates from India. Equal number of candidates will be offered admission from these two lists, according to merit and availability of seats.

Calculators are not allowed. However, Log Tables may be used.

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**The areas from which questions may be asked in the Entrance Test will include the following:**

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**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; eigen values and eigen vectors; 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; the structure of groups; Sylow's theorems and their applications; rings, integral domains and fields; ring homomorphism and ideals; polynomial rings and irreducibility criteria; vector space, vector subspace, linear independence of vectors, basis and dimensions of a vector space, inner product spaces, orthonormal basis; Gram-Schmidt process, linear transformations.

**Differential Equations:** First order ordinary differential equations (ODEs); solution of first order initial value problems; singular solution of first order ODEs; system of linear first order ODEs; method of solution of dx/P=dy/Q=dz/R; orthogonal trajectory; solution of Pfaffian differential equations in three variables; linear second order ODEs; Sturm-Liouville problems; Laplace transformation of ODEs; series solutions; Cauchy problem for first order partial differential equations (PDEs); method of characteristics; second order linear PDEs in two variables and their classification; separation of variables; solution of Laplace, wave and diffusion equations; Fourier transform and Laplace transform of PDEs.

**Numerical Analysis: **Numerical solution of algebraic and transcendental equations; direct and iterative methods for system of linear equations; matrix eigenvalue problems; interpolation and approximations; numerical differentiation and integration; composite numerical integration; double numerical integration; numerical solution for initial value problems; finite difference and finite element methods for boundary value problems.

**Probability and Statistics: **Axiomatic approach of probability; random variables; expectation, moments generating functions, density and distribution functions; conditional expectation.

**Linear Programming:** Linear programming problem and its formulation; graphical method, simplex method; artificial starting solution; sensitivity analysis; duality and post-optimality analysis.