# Instructions for PhD. written test 2019-2020 odd Semester

This for the students who have applied for the PhD position for odd semester of 2019-20 academic year. The details syllabus  of the PhD written test is the following:

Depending on the number of students appearing in the written test, a cut off mark may be decided by the committee to short list the candidates to appear in the interview.

## Syllabus for Written Test

### Algebra:

Vector spaces, bases and dimension, Linear transformations and their matrix representations, Dual vector spaces, Determinants and their properties, Eigenvalues and eigenvectors, Characteristic polynomial and minimal polynomial, Diagonalization.

Groups, normal subgroups, quotient groups, isomorphism theorems, automorphisms, permutation groups, group actions, Sylows theorem, classification of finite abelian groups.

Rings, ideals, quotient rings, isomorphism theorems, prime ideals, maximal ideals, Field of fractions, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains, Polynomial rings, irreducibility criteria.

Field extensions, algebraic extensions, splitting fields, algebraic closures, separable extensions, cyclotomic polynomials and extensions, automorphism groups and fixed fields, Galois extensions, Fundamental theorem of Galois theory, Finite fields.

### Analysis:

Topology of Rn, Heine-Borel Theorem, Connectedness, Completeness. Series and sequences, Continuous and differentiable functions, Mean value theorem, Maxima and minima, Riemann integration. Sequence and series of functions, Uniform convergence, Space of continuous functions C[0; 1], Arzela-Ascoli theorem, Weirstrass approximation theorem, Contraction mapping principle, Existence and uniqueness of ODE, Picard Lindelof’s theorem, Fourier series.

Lebesgue measure, Measurable functions, Lebesgue integral, Basic properties of Lebesgue integral, Differentiation and Lebesgue measure, LSpaces, Holder and Minkowski inequalities, convergence in measure, Monotone and Dominated convergence theorems, Fatou’s lemma,

Normed linear spaces, Banach spaces, Hilbert spaces, Dual spaces, Compact operators, Function spaces like C([0; 1]) and Lp([0; 1]), Linear operators. Hahn-Banach Theorem, Open mapping theorem, Closed graph theorem and Uniform boundedness principle.

Holomorphic functions, Cauchy’s theorem and Cauchy integral formula, Cauchy estimates and Liouvilles theorem, Maximum modulus principle, Singularities, Laurent series, Theory of residues, Zeros of holomorphic functions, Argument principle, Rouches theorem, Contour integration, The Open Mapping theorem, Schwarz lemma.

### Discrete Mathematics:

Pigeonhole principle, Counting principles, Binomial coefficients, Principles of inclusion and exclusion, recurrence relations, generating functions, Partition numbers, Partially ordered sets, Lattices, Boolean algebra.

Graphs, graph isomorphisms, degree sequence, trees, bipartite graphs, Hamilton cycles, Euler tours, directed graphs, matching, Tuttes theorem, connectivity, Mengers theorem, planar graphs, vertex and edge colouring of graphs, matrices associated with graphs.

Divisibility, Primes, Fundamental theorem of arithmetic, Congruences, Chinese remainder theorem, Linear congruences, Fermats little theorem, Wilsons theorem, Euler function and its applications, Group of units, primitive roots, Quadratic residues, Jacobi symbol, Arithmetic functions, Mobius Inversion formula.

### Probability:

Combinatorial probability and urn models; Conditional probability and independence; Random variables discrete and continuous; Expectations, variance and moments of random variables; Transformations of univariate random variables; Jointly distributed random variables; Conditional expectation; Generating functions; Limit theorems.

### Statistics:

Standard univariate and multivariate distributions, Multivariate normal distribution, Types of convergence of random variables, Descriptive statistical measures, Standard sampling distributions, Order statistics, Theory of point estimation: unbiasedness, consistency, sufficiency, minimum variance unbiased estimation, Methods of estimation: method of moments, maximum likelihood estimation and its properties, Bayes estimation, Confidence set estimation, Bayesian intervals, Tests of hypothesis: uniformly most powerful tests for simple and composite hypotheses, p-value, Likelihood ratio tests, Large sample tests, Correlation, Multiple linear regression and related inference, Logistic regression, Analysis of variance, Standard Nonparametric tests.

### Theoretical computer Sciences:

Design and analysis of algorithms: Basic concepts, asymptotic notations, recurrence relation, sorting and searching, divide and conquer, greedy algorithm, dynamic programming, approximation algorithms (basics), randomized algorithm (basics)

Graph algorithms: graphs representation, BFS, DFS, shortest path, connectivity, cycles, trees, spanning tree, Eulerian cycle and Hamiltonian paths, independent set, coloring, chromatic number, dominating sets

Theory of Computation: finite state automata; DFA,NFA, regular expressions, regular languages, pumping lemma, context free languages and grammars, pushdown automata (PDA), Turing machine, decidability, recognizability, notions of P, NP, co-NP, reduction, NP complete problems.

Logic: propositional logic, equivalence and implications. truth tables, De Morgans law, quantifiers, inference and proofs, first order logic.

Basics of cryptology

### Topology:

Topological spaces, Product topology, Subspace topology, Compact spaces, Locally compact spaces, Sequentially compact space, Limit point-compact spaces, Continuous functions, Open map, Homeomorphisms, Separation Axioms: T1, Hausdorff, regular, normal spaces; Uryshons lemma, Tietze Extension Theorem, One point compactification.

Complete metric spaces, Uniform continuity, Connected spaces, Path connected spaces, Quotient topology, Homotopy, Fundamental group: Basic definitions, Covering spaces, Fundamental group of the circle.

PDF file is available here