PhD scholars need to complete sufficient course work with at least 15 credits (relaxation applies to students from central institutes upon the recommendation of the Doctoral committee). Students also need to qualify the Comprehensive Viva within the prescribed period.

Comprehensive Examination Schedule: Winter 2024-2025

The schedule for the next comprehensive examination (Written part) is decided as follows;

Date of Examination : 23.01.2025

Last date for submitting applications : 10.01.2025, 5 pm

Guidelines regarding the Comprehensive Viva

Syllabus for Written section of PhD Comprehensive Examination (2022 Admission onwards)

Core Section - Module I :

Real Analysis: Review of limit continuity and differentiation,  Uniform continuity,  Mean value theorems, Taylor’s theorem, Functions of bounded variation, Riemann Integration, Fundamental theorem of calculus,  Uniform convergence of sequence of functions, Derivative of functions of several variables,  Jacobian,  Directional derivative,  Inverse function theorem, Implicit function theorem.

Measure Theory: Sigma algebra, Measure, Caratheodory’s extension theorem, Lebesgue measure, Measurable functions, Lebesgue integral, Fatou’s lemma, Monotone convergence theorem, Dominated convergence theorem, Fubini’s theorem.

Complex Analysis: Analytic functions, Contour integral,  Cauchy’s theorem,  Cauchy’s integral formula, Taylor and Laurent’s series, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem, Calculus of residues.

Functional Analysis: Banach spaces, LP spaces, Bounded linear transformation, Hahn- Banach extension theorem, Open mapping theorem, Closed graph theorem, Uniform boundedness principle, Hilbert spaces, Orthonormal bases, Riesz representation theorem, Self- adjoint operators, Normal and unitary operators, Projections.

Differential Equation: Introduction and motivation, Geometrical interpretation of solution. Solution methods for first-order and second-order equations, Existence and uniqueness of solution of initial value problems, Picard’s and  Peano’s theorems,  Euler and improved Euler’s methods, Higher order linear equations and Linear systems. Fundamental solutions, Wronskian, Matrix exponential solution, Behaviour of solutions. Boundary value problems for second-order equations: Green’s function, Sturm comparison theorems and oscillations, and eigenvalue problems.

Topology: Metric spaces, Topological spaces, Subspaces, Continuous functions, The Product and Quotient spaces, Separation axioms,  Countability properties,  Compactness, Connectedness and metrizability.

Linear Algebra: Vector spaces, Dimension, Linear transformations and their matrix representations, rank-nullity theorem, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton theorem, Diagonal form, Triangular form,  Jordan form,  Inner product spaces, Self-adjoint operators.

Abstract Algebra: Groups, Subgroups, Cosets, Normal subgroups, Quotient groups, Rings, Integral domains, Fields, Quotient fields of an Integral domain, Ideals, Maximal ideals, Quotient rings, Ring of polynomials, Prime and irreducible elements.

Specialization Section:

Module II: Fuzzy Graph Theory

Fuzzy sets- membership functions – methods of generating membership functions – defuzzification methods, Operations on fuzzy set, Fuzzy relations, Operations on fuzzy relations, similarity relations – compatibility or tolerance relations.  Fuzzy  numbers- arithmetic operations on intervals- arithmetic operations on fuzzy numbers

Fuzzy graphs- Path and connectedness – strongest path – strong path – types of arcs in fuzzy graphs - complete fuzzy graph – complement of a fuzzy graph.  Connectivity  in Fuzzy Graphs- Bridges and cut nodes – bonds -Trees and cycles -  end nodes  –  blocks  – node connectivity and arc connectivity – fuzzy analogue of Whitney’s theorem

Metric in fuzzy graph – -distance - g distance - - distance – ss – distance – eccentricity  – radius– diameter – self-centred fuzzy graph- operations on fuzzy graph - -Fuzzy line graphs, Fuzzy intersection graphs, Fuzzy interval graphs.

Module III: Graph Theory

Graphs: review of basics in graphs - -Trees- Blocks- Matrices-Operations on graphs. Connectivity: Vertex Connectivity and edge connectivity – n- connected graphs-Menger’s Theorem.

Traversability: Euler graphs-Hamiltonian Graphs-Planar and Nonplanar graphs.
Metric in graph: Centre, Median, eccentric vertex, Eccentric graph, boundary vertex, complete vertex, interior vertex.

Distance Sequences: Degree sequence, Eccentric Sequence - Distance Sequences - The Distance Distribution, Mean distance.

Matchings: Maximum matching-Perfect matching-Matching in bipartite graphs

Factorization: Coverings and independence-1-factorization-2-factorization-Arboricity Domination: Dominating set-Domination number-total dominating set –total domination number.

Module IV: Numerical Analysis and Computational Methods for PDE

Preliminaries: Round-off error, Truncation error, Absolute error, Relative error,  Percentage error;    Solving system of linear equations:    Direct methods Gauss elimination,    LUdecomposition, Iterative methods: Gauss-Seidel, Gauss Jacobi,  Relaxation methods; Vector norms, matrix norms and condition number. Eigenvalues, eigenvectors, Gerschgorin circle theorem. Newton’s method to solve nonlinear systems of equations.  Numerical methods to solve ODEs: Euler method, modified Euler method, Runge-Kutta methods, Stiff and nonstiff ODEs, finite difference methods, Backward difference formulas,  Stability  Theory,  A- stability, L-stability, B-stability.

Numerical methods for PDEs; Heat equation, advection equation, elliptic equation; types of boundary conditions; finite difference approximations; FTCS, BTCS and Crank-Nicolson schemes; ADI schemes; Naïve, Upwind, Downwind, Lax Wendroff and Lax  Friedrichs schemes, Beam-Warming Method; Numerical methods for Laplace Equations; Stability, Von- Neumann stability, CFL condition, consistency and convergence analysis,  Lax theorem; Method of lines; Numerical solution of convection-diffusion, diffusion-reaction and convection-diffusion-reaction problems; Implementation of numerical schemes,  Numerical error, Computational order.

Module V: Fractal Theory and Applications

The completeness of space of fractals, Transformations on the real line,  Affine transformations in the Euclidean plane, Mobius transformations on the Riemann sphere, Analytic transformations, The contraction mapping theorem,  Condensation sets,  Addresses of points on fractals, Continuous transformations from code space to fractals, Dynamical systems, Dynamics on fractals, Equivalent dynamical systems, Shadowing theorem, Chaotic dynamics on fractals, Fractal dimension,  Theoretical and experimental determination of fractal dimension, Hausdorff- Besicovitch dimension. Fractal interpolation functions, Fractal dimension of fractal interpolation functions, Hidden  variable  fractal  interpolation,  Space-filling curves, Escape time algorithm, Julia sets, IFS for Julia sets, Map  of  fractals, Mandelbrot’s sets,

Module VI : Sobolev Spaces and its Applications to PDEs

Sobolev Spaces and its properties: Fundamental solutions, non-existence of classical solutions, weak convergence,  test functions,  distributions,  weak and distributional derivative, compact support, mollifiers, partition of unity.  Sobolev spaces:  definition and basic properties,  approximation by smooth functions.  extension theorem,  Poincaré inequality, Sobolev inequality, embedding theorems,  compactness theorems,  traces,  dual space, fractional Sobolev spaces, Hardy’s inequality.

Variational Methods: First variation, second variation, Euler-Lagrange equation, existence of minimizers: classical  method  and  direct  method,  Dirichlet  principle,  Dirichlet  integral and p- Dirichlet Integral, existence and nonexistence of minimizers: examples and counterexamples, Wierstrass function, weak  form  of  the  Euler-Lagrange  equations, convexity, coercivity, lower semicontinuity,

Applications to PDEs: Second-order elliptic equations: Laplace and Poisson equation, weak solutions, the existence of weak solutions, Lax-Milgram lemma, weak formulations of elliptic boundary value problems, weak convergence method,  Schauder’s fixed point theorem, method of subsolutions and supersolutions, comparison principle, nonlinear eigenvalue problems, Pohozaev identity for the nonexistence of solutions,  critical points and saddle points of energy functional, Palais-Smale sequence, mountain pass lemma, interior and boundary regularity, Harnack inequality, Hopf lemma, weak and strong maximum principles.

Module VII: Fuzzy Set Theory

Basic concepts of fuzzy sets, Membership functions, Methods of generating membership functions, Defuzzification methods, Extension principle, Operations on fuzzy sets, Fuzzy complement, Fuzzy union, Fuzzy intersection, combinations of operations, and General aggregation operations. Fuzzy numbers, Arithmetic operations on intervals, Arithmetic operations on fuzzy numbers, Fuzzy equations, Fuzzy relations, Projections and Extensions, Binary Fuzzy relations, Similarity relations, Compatibility relations.  Fuzzy measures, Evidence Theory, Belief and  Plausibility measures,  Joint  Basic  Assignment,  Dempester’s rule of Combination, Marginal bodies of Evidence, Possibility and Necessity measures, Possibility distribution, Basic distribution, Probability measures.

Module VIII: Generalized Set Theory

An overview of basic operations on Fuzzy sets, Intuitionistic fuzzy sets, Hesitant fuzzy sets, Multisets, Multiset relations, Compositions, Equivalence multiset relations and partitions of multisets, Multiset functions, Fuzzy Multisets.Rough sets, Knowledge representation, Information systems, Exact sets, rough sets, approximations, Set-algebraic structures,Topological structures, Decision  systems,  Knowledge  reduction,Reducts  via boolean reasoning,discernibility approach . Reducts in decision systems, Rough membership functions Soft sets, Tabular representation of a  soft  set,  Operations  with  Soft  sets:  soft subset, complement of a soft set, null  and  absolute  soft  sets,  AND  and  OR  operations, Union and intersection of soft sets, DeMorgan laws, Fuzzy soft sets and soft fuzzy sets, Intuitionistic Fuzzy Soft Sets and Soft Intuitionistic Fuzzy Sets, Hesitant Fuzzy Soft Sets, Soft Rough Sets and Rough Soft Sets, Fuzzy rough sets and rough fuzzy sets.

Module IX: Algebraic Topology

Geometric Complexes and Polyhedra, Orientation of Geometric Complexes, Simplicial Homology Groups - Chains, cycles, Boundaries, Homology groups, examples of Homology Groups, The structure of  Homology  Groups,  The  Euler  Poincare  Theorem,  Pseudo manifolds and the Homology Groups of Sn.Simplicial Approximation,  induced Homomorphisms on the Homology groups, the Browder Fixed point theorem and  related results. The Fundamental  group  -  Introduction,  Homotopic  paths  and  the  fundamental group, the covering homotopy  property  for  S1,  Examples  of  Fundamental  group,  the relation between H1(K) and π1([K]). Covering Spaces – Definition and examples, basic properties of Covering spaces, Classification  of  covering  spaces, Universal covering spaces and applications.

Module X: Probability statistics and system reliability

Basics of probability statistics: Random variables, Discrete and  continuous  random variables, Moments, Moment generating function  and  Characteristic  function,  Random vectors, Jointly distributed random variables, Joint probability distributions, Conditional expectation.
Distributions of random variables and limit theorems: Bi-variate normal distribution, Transformations of random variables, Transformations of random vectors, Order statistics, Chebyshev’s theorem, Limit theorems in probability, Modes of convergence, Weak law of large numbers, Strong law of large numbers, Limiting moment generating  function, Central limit theorem.

Statistical inference: Introduction to population and samples, Sampling distribution of the mean and variance, Point estimation, Maximum Likelihood Estimation (MLE), Method of moments, Properties of estimators, Tests of hypothesis, Uniformly Most Powerful (UMP) Tests, Newman-Pearson lemma,  Inference  concerning  single  mean  and  two  means, Inference concerning one variance and two variances, Inference concerning one proportion and several proportions, Chi -square test for goodness of fit.

System reliability: Basic concepts, Cut sets, Path sets, Minimal cut and path sets, Bounds for reliability, Reliability and Quality, Maintainability and Availability, Reliability analysis, Causes of failures, Catastrophic and Degradation failures, Useful life of  components, Component reliability and hazard models, Mean time to failure, system reliability models, System with components in series, parallel, k/n systems, System with mixed mode failures. Redundancy techniques: Basics of redundancy techniques, Component v/s unit redundancy, Weakest link techniques, Mixed  redundancy,  Stand  by  redundancy,  Redundancy optimization, Double failure and redundancy, Maintainability and availability concepts, Two-unit parallel system with repair, Signal redundancy, Time redundancy, Software redundancy

Reliability evaluation and allocation: Hierarchical systems, Path determination method, Boolean Algebra method,  Cut  set  approach,  Logic  diagram  approach,  Conditional probability approach, System cost and reliability approximations, Reliability  allocation problems

Module XI: Reliability, stochastic process, acceptance sampling plans and optimization

Reliability estimation: Life testing: Introduction, hazard rate functions, Exponential distribution in life testing, Simultaneous testing-stopping at r-th failure,  stopping by fixed time, sequential testing, Accelerated testing, Equipment acceptance testing.

Stochastic process: Elements of stochastic processes, Classification of general stochastic processes. Markov Chains: Definition, examples, transition probability matrix, classification of states, basic limit theorem, limiting distribution of  Markov  Chains, Continuous-time Markov Chains: General pure birth processes and Poisson processes, more about Poisson processes, A counter model, Birth and Death processes with absorbing states, Finite state continuous time Markov Chains.

Acceptance sampling plans: Sampling plans by attributes and variables, Single, double, multiple, and sequential sampling plans- acceptable quality level, LTPD-producer’s risk, consumers risk.

Optimization techniques: Linear programming problems (LPP)-Formulation of LPP, Simplex method-Simplex algorithm-Charles M Method, Two phase method, Duality in LPP, Dual simplex method, Advanced linear LPP, Sensitivity  analysis-  Parametric  programming, Bounded variable problem, Dynamic programming-Bellman’s optimality principle.

Module XII: Time Series Analysis

Fundamental Concepts: Stochastic Processes, The Autocovariance and Autocorrelation functions, The Partial auto-correlation function, the White  Noise  Process,  Estimation of the Mean, Autocovariances and Autocorrelation.

Stationary Time Series Models: Auto-Regressive Processes, Moving Average processes, The Dual Relationship between AR(p) and MA(q) processes, Auto Regressive Moving Average (ARMA(p,q)) processes.

ARIMA Models and their Identification : Nonstationary Time Series Models - Nonstationarity in the Mean, Auto Regressive  Integrated  Moving  Average  (ARIMA) Models, Nonstationarity in the Variance and Auto-covariance.

Time-Series Forecasting: Minimum Mean Square Error Forecasts, Computation of Forecasts, The ARIMA Forecast as a weighted average of previous observations.

Time Series Model Identification:  Steps  for  Model  Identification,  Inverse Autocorrelation Function.

Parameter Estimation: The Method of Moments, Maximum Likelihood Method, Nonlinear Estimation, Ordinary Least  Squares  (OLS)  Estimation  in  Time  Series Analysis.

Module XIII: Wavelet Theory

Inner Product Spaces, Orthonormal Sets, Hilbert Spaces, Bessels Inequality, Fourier transforms, Parseval identity and Plancherel theorem, Basic Properties  of  Discrete Fourier Transforms, Translation invariant Linear Transforms, The Fast Fourier Transforms.

Construction of wavelets on Z N: The Haar system, Shannon Wavelets, Real Shannon wavelets  Wavelets on  Z:  l2 ( Z ),  Complete orthonormal sets in  Hilbert spaces,  L2 (−π, π ) and   Fourier series, The   Fourier   Transform and convolution on   l2( Z ),   First stage Wavelets on Z.

Wavelets on  R:  L2 (R ) and approximate identities,  The  Fourier transform on  R, Multiresolution analysis.

Module XIV: Matrix Theory for Spectral Graph Theory

The Minimax Principle for Eigenvalues, Cauchy’s Interlacing Theorem, Weyl’s Inequalities, Wielandt’s Minmax Principle, Hermitian and Skew- Hermitian Ma- trices, Optimal matching distance, Hausdorff distance, Estimates in the Frobenius Norm, Singular Value Decomposition(SVD) of matrices

Graphs: review of basics in graphs - Trees- Blocks- Matrices-Operations on graphs. Connectivity: Vertex Connectivity and edge connectivity – n- connected graphs-Menger’s Theorem. Traversability: Euler Graphs-Hamiltonian Graphs-Planar and Nonplanar graphs.

Module XV: Number Theory

Elementary Number Theory: Euclid’s algorithm, Chinese remainder theorem, primitive roots, Quadratic reciprocity law, applications, Binary quadratic forms, Sums of two squares, sums of four squares, Diophantine approximation, theorems of Dirichlet and Liouville, Continued fractions, quadratic irrationalsQuadratic extensions of rationales.

Galois Theory: Algebraic extensions and algebraic closures, Splitting fields,  Normal  extensions, Perfect fields, Separable extensions, Galois extensions,  Fundamental  theorem  of Galois theory.
Algebraic Number Theory The Gaussian integers, Integrality, Ideals, Noetherian and Principal Ideal Domains, Dedekind domain,  Lattices,  Algebraic  numbers  and  number  fields.  Ideal  class group, Finiteness of the ideal class group, Units in number rings, Dirichlet’s unit theorem. Theory of Valuations The p-adic numbers, the p-adic absolute value, Valuations, Completions, Local fields, Hensel's lemma.

Module XVI: Univalent Function Theory And Bohr’s Inequality

Univalent functions, Elementary properties of univalent functions, the Area theorem, Coefficient estimates for univalent functions, The maximum modulus of univalent functions, Bieberbach Conjecture, The Koebe One-Quarter theorem, Growth and Distortion theorems, Subclasses of Univalent functions: Starlike Functions, Convex Functions, Spirallike Functions, Close-to-Convex Functions, Concave Univalent Functions, Ma-Minda Class of Starlike and  Convex  Functions. Odd Univalent Functions, Robertson Conjecture.

Bohr and the Dirichlet series, Classical Bohr Radius Problem, Majorant series, Subordinations, Bohr Phenomenon, Bohr-Rogosinski Radius, Bohr radius for Concave-Wedge domain, Bohr Phenomenon for various subclasses of analytic function, Bohr radius in the study of Banach spaces.

Module XVII: Combinatorial commutative algebra

Fundamental of algebra: group and subgroups, order and involution, direct product of
abelian groups, cyclic group and normal subgroup, homomorphism, group action, rings, subrings, integral domains, ring homomorphism, ideals, quotient rings, zero divisors, nilpotent elements, units, prime ideals, maximal ideals, operations on ideals, polynomial rings, factorisation of polynomials, irreducible and reducible elements, Euclidean domains.

Combinatorics: magic square and magic rectangle, Kozig array, graph, subgraph, and induced subgraphs, trees, blocks, matrices, and operations on graphs. Connectivity: cut-vertex,cut-edge, vertex, and edge connectivity, n-connected, Menger’s theorem. Traversability: Euler graphs, Hamiltonian graphs, minimum spanning tree problems, planar graphs and matching.

Module XVIII: Operator Theory

Spectral properties of bounded linear operators: Parts of Spectrum, resolvent and spectrum, resolvent operator, spectral theory, Complex analysis in spectral theory, Gelfand-Mazour theory.

Theory of compact operators, spectral theory of compact self-adjoint operators, Spectral properties of bounded self-adjoint linear operators, normal operators, unitary operators, Positive operators, Square root of an operator, Polar decomposition, Numerical range of operators.

Projection operators, Orthogonal Projections, Invariant Subspaces, Reducing Subspaces, Shifts, Toeplitz operators.

Module XIX: Water waves

Mathematical techniques: Laplace transforms, Fourier series, Fourier transform, Z-Transform, Green’s function technique, Integral equation.
Linear wave theory: Conservation of mass, moment, and Energy-Euler Equation, Bernoulli’s Equation, Governing differential equation, Separable solutions of the equation, Boundary conditions for water waves, Progressive Waves, Dispersion relation-Non dimensionalization and scaling.
Formulation of wave equations: Surface gravity waves of uniform finite depth, Surface gravity waves of deep water, Flexural gravity waves of uniform finite depth and deep water, Capillary-gravity waves, Waves over Smooth, rigid, porous bottoms.

Module XX: Lie Algebra/Super algebra

Lie algebra: Classical algebras, ideals, homomorphisms and representation (Lie modules), derived series, lower central series and classification of nilpotent and solvable Lie algebras and its representations (Engel, Lie’s theorems), simple and semisimple Lie algebras, Killing form, Cartan’s criterion and it’s applications, Weyl’s theorem and complete reducibility of representations, classification of irreducible sl(2,C)-representation, Cartan subalgebra (CSA), root space decomposition and properties, abstract root system, Weyl group, classification of irreducible root system and Dynkin diagrams.

Lie superalgebra: general linear and special linear Lie superalgebras, orthosymplectic, queer and periplectic Lie superalgebras, solvable, nilpotent, semi-simple Lie superalgebras.

Module XXI: Schwarz-Pick Type Inequalities

Subordinate functions-Bieberbach’s conjecture by de Branges-Theorems of Jenkins and Sheil-Inverse coefficients-Domains with bounded boundary rotation-The Schwarz-Pick inequality-Estimates using the Euclidean distance- An application of Teichmuller theorem-Domains with uniformly perfect boundary-Derivatives of the conformal radius Theorems of Ruscheweyh and Yamashita-Pairs of simply connected domains-Holomorphic mappings into convex domains-Punishing factors for convex pairs.

Module XXII: Linear and Non-linear Optimization

Theorems on extremas, Linear programs, basic feasible solutions, simplex method, Duality theorem, complementary slackness, Sensitivity analysis.
Convex sets, convexity preserving operations, Separation Theorems, Theorems of alternatives: Farkas’ theorem, Gordan’s theorem, Convex functions and their continuity, Lipschitz continuity and directional derivative.
Local optima and its variants, saddle point, coercive functions, existence of optima: Weierstrass theorem, Theorem for coercive functions, Descent direction and its first order and second order characterizations for minima.
Lagrange multipliers method, Cones for constraint optimization: cone, dual cone, cone of feasible directions, linearizing cone, cone of descent directions, tangent cone, Geometric optimality conditions,
First order Fritz Jhon and Karush-Kuhn-Tucker(KKT) necessary optimality conditions, First order KKT sufficient optimality condition, Second order KKT necessary and sufficient optimality conditions.

Module XXIII Zero Product Determined Algebras:
 

The Definition of a zpd Nonassociative Algebra, Symmetrically zpd Nonassociative Algebras, Stability Under Algebraic Constructions, zpd Rings, Examples and Non-examples of zpd Algebras, The Finite - Dimensional Case, Lie Algebras and Jordan Algebras, Zero Lie/Jordan Product Determined Algebras. Rings of Central Quotients, Classical Rings of Quotients, Ore Domains,  artindale Rings of Quotients, The Extended Centroid, Linear (In)dependence in Prime Rings, Prime GPI-Rings, Primitive PI-Rings, Prime PI-Rings and Central Polynomials.