**Syllabus:**

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The course work for the Ph.D. program in SMS is consist of 8 courses out of which 4 be compulsory. There is a pool of compulsory courses consists of seven groups of courses, each consisting of two courses. A student has to choose atleast two groups from the pool based on the recommendation of PGCS, SMS. Moreover besides the existing pool of elective courses, remaining courses in the pool of compulsory courses will be also treated as elective courses. Each course is of 4 credits. Therefore, the total of 32 credit will be for course-work. Comprehensive examination will be based on 4 compulsory courses.

Pool of compulsory courses is the following:

(a) Analysis – M603:Analysis I & M604:Analysis II

(b) Algebra – M601: Algebra I & M602: Algebra II

(c) Topology – M659: Topology and Complex Analysis & M660: Advanced Topology

(d) Probability – M472: Advanced Probability & M455: Introduction to Stochastic Processes

(e) Statistics – M481: Statistical Inference I & M567: Statistical Inference II

(f) Theoretical Computer Sciences – M469: Theory of Computation & M462: Cryptology

(g) Discrete Mathematics – M661: Combinatorics and Graph Theory & M566: Designs and Codes

Discrete Markov chains with countable state space; Classification of states: recurrences, transience, periodicity. Stationary distributions, reversible chains, Several illustrations including the Gambler’s Ruin problem, queuing chains, birth and death chains etc. Poisson process, continuous time Markov chain with countable state space, continuous time birth and death chains.

- P. G. Hoel, S. C. Port, C. J. Stone, “Introduction to Stochastic Processes”, Houghton Mifflin Co., 1972.
- R. Durrett, “Essentials of Stochastic Processes”, Springer Texts in Statistics, Springer, 2012.
- G. R. Grimmett, D. R. Stirzaker, “Probability and Random Processes”, Oxford University Press, 2001.
- S. M. Ross, “Stochastic Processes”, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, 1996

Overview: Cryptography and cryptanalysis, some simple cryptosystems (e.g.,shift, substitution, affine, knapsack) and their cryptanalysis, classification of cryptosystems, classification of attacks; Information Theoretic Ideas: Perfect secrecy, entropy;

Secret key cryptosystem: stream cipher, LFSR based stream ciphers, cryptanalysis of stream cipher (e.g., correlation attack, algebraic attacks), block cipher, DES, linear and differential cryptanalysis, AES;

Public-key cryptosystem: Implementation and cryptanalysis of RSA, ElGamal public key cryptosystem, Discrete logarithm problem, elliptic curve cryptography; Data integrity and authentication: Hash functions, message authentication code, digital signature scheme, ElGamal signature scheme; Secret sharing: Shamir’s threshold scheme, general access structure and secret sharing.

- D. R. Stinson, “Cryptography: Theory And Practice”, Chapman & Hall/CRC, 2006.
- A. J. Menezes, P. C. van Oorschot, S. A. Vanstone, “Handbook of Applied Cryptography”, CRC Press, 1997.

Automata and Language Theory: Finite automata, regular expression, pumping lemma, context free grammar, context free languages, Chomsky normal form, push down automata, pumping lemma for CFL;

Computability: Turing machines, Churh-Turing thesis, decidability, halting problem, reducibility, recursion theorem;

Complexity: Time complexity of Turing machines, Classes P and NP, NP completeness, other time classes, the time hierarchy.

- J. E. Hopcroft, R. Motwani, J. D. Ullman, “Introduction to Automata Theory, Languages, and Computation”, Addison-Wesley, 2006.
- H. Lewis, C. H. Papadimitriou, “Elements of the Theory of Computation”, Prentice- Hall, 1997.
- M. Sipser, “Introduction to the Theory of Computation”, PWS Publishing, 1997.

Probability spaces, Random Variables, Independence, Zero-One Laws, Expectation, Product spaces and Fubini’s theorem, Convergence concepts, Law of large numbers, Kolmogorov three-series theorem, Levy-Cramer Continuity theorem, CLT for i.i.d. components, Infinite Products of probability measures, Kolmogorov’s Consistency theorem, Conditional expectation, Discrete parameter martingales with applications.

- A. Gut, “Probability: A Graduate Course”, Springer Texts in Statistics, Springer, 2013.
- K. L. Chung, “A Course in Probability Theory”, Academic Press, 2001.
- S. I. Resnick, “A Probability Path”, Birkh ̈auser, 1999.
- P. Billingsley, “Probability and Measure”, Wiley Series in Probability and Statistics, John Wiley & Sons, 2012.
- J. Jacod, P. Protter, “Probability Essentials”, Universitext, Springer-Verlag, 2003

Review: joint and conditional distributions, order statistics, group family, exponential family (3 hrs)

Introduction to parametric inference, sufficiency principle and data reduction, factorization theorem, minimal sufficient statistics, Fisher information, ancillary statistics, complete statistics, Basu’s theorem (9 hrs)

Unbiasedness, best unbiased and linear unbiased estimator, Rao-Blackwell theorem, Lehmann-Scheffe theorem and UMVUE, Cramer-Rao lower bound and UMVUE, multi-parameter cases (8 hrs)

Location and scale invariance, principle of equivariance (4 hrs)

Methods of estimation: method of moments, likelihood principle and maximum likelihood estimation, properties of MLE: invariance, consistency, asymptotic normality (5 hrs)

Hypothesis testing: error probabilities and power, most powerful tests, Neyman-Pearson lemma and its applications, p-value, uniformly most powerful (UMP) test via Neyman-Pearson lemma, UMP test via monotone likelihood ratio property, existence and nonexistence of UMP test for two sided alternative, unbiased and UMP unbiased tests (13 hrs)

Likelihood (generalized) ratio tests and its properties, invariance and most powerful invariant tests (5 hrs)

Introduction to confidence interval estimation, methods of fining confidence intervals: pivotal quantity, inversion of a test, examples such as confidence interval for mean, variance, difference in means, optimal interval estimators, uniformly most accurate confidence bound, large sample confidence intervals (7 hrs)

- Lehmann, E.L. and Casella, G.(1998), “Theory of Point Estimation”, 2nd edition, New York: Springer.
- Lehmann, E.L. and Romano, J. P. (2005), “Testing Statistical Hypotheses”, 3rd edition, Springer.
- Nitis Mukhopadhyay (2000), “Probability and Statistical Inference”, New York: Marcel Dekker.
- George Casella and Roger L. Berger, “Statistical Inference”, 2nd edition, Cengage Learning, 2001.
- A.M. Mood, F.A. Graybill and D.C. Boes (1974), “Introduction to the Theory of Statistics”, 3rd edition, McGraw Hill.

Incidence structures, affine planes, translation plane, projective planes, conics and ovals, blocking sets. [10 lectures]

Introduction to Balanced Incomplete Block Designs (BIBD), Symmetric BIBDs, Difference sets, Hadamard matrices and designs, Resolvable BIBDs, Latin squares. [20 lectures]

Basic concepts of Linear Codes, Hamming codes, Golay codes, Reed-Muller codes, Bounds on the size of codes, Cyclic codes, BCH codes, Reed-Solomon codes. [20 lectures]

G. Eric Moorhouse, Incidence Geometry, 2007 (available online).Douglas R. Stinson, Combinatorial Designs, Springer-Verlag, New York, 2004.W. Cary Huffman, V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.

General decision problem, loss and risk function, minimax estimation, minimaxity and admissibility in exponential family (5 hrs)

Introduction to Bayesian estimation, Bayes rule as average risk optimality, prior and posterior, conjugate families, generalized Bayes rules (8 hrs)

Bayesian intervals and construction of credible sets, Bayesian hypothesis testing (8 hrs)

Empirical and nonparametric empirical Bayes analysis, admissibility of Bayes and generalized Bayes rules, discussion on Bayes versus non-Bayes approaches (7 hrs)

Large sample theory: review of modes of convergences, Slutsky’s theorem, Berry-Essen bound, delta method, CLT for iid and non iid cases, multivariate extensions (10 hrs)

Asymptotic level tests, asymptotic equivalence, comparison of tests: relative efficiency, asymptotic comparison of estimators, efficient estimators and tests, local asymptotic optimality (11 hrs)

Bootstrap sampling: estimation and testing (5 hrs)

- Lehmann, E.L. and Romano, J. P. (2005), “Testing Statistical Hypotheses”, 3rd edition, Springer.
- Lehmann, E.L. (1999), “Elements of Large-Sample Theory”, Springer-Verlag.
- James O Berger (1985), “Statistical Decision Theory and Bayesian Analysis”, 2nd Edition, New York: Springer.
- Lehmann, E.L. and Casella, G.(1998), “Theory of Point Estimation”, 2nd edition, New York: Springer

Group Theory: Dihedral groups, Permutation groups, Group actions, Sylow’s theorems, Simplicity of the alternating groups, Direct and semidirect products, Solvable groups, Nilpotent groups, Jordan Holder Theorem, free groups.

Ring Theory: Properties of Ideals, Chinese remainder theorem, Field of fractions, Euclidean domains, Principal ideal domains, Unique factorization domains, Polynomial Rings, Irreducibility criteria, Matrix rings.

Module Theory: Examples, quotient modules, isomorphism theorems, Generation of modules, free modules, tensor products of modules, Exact sequences - Projective, Injective and Flat modules.

- D. S. Dummit and R. M. Foote, Abstract Algebra. John Wiley & Sons, 2004.
- T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer, 1980.
- M. Artin, Algebra, Prentice Hall, 1991.

Linear Algebra: Matrix of a Linear transformation, dual vector spaces, determinants, Tensor algebras, Symmetric algebras, Exterior algebras,

Modules over PIDs: Basic theory, Structure theorem for finitely generated abelian groups, Rational and Jordan canonical forms.

Field Theory: Algebraic extensions, Splitting fields, Algebraic closures, Separable and Inseparable extensions, Cyclotomic polynomials and extensions, Galois extensions, Fundamental Theorem of Galois theory, Finite fields, Composite extensions, Simple extensions, Cyclotomic extensions and Abelian extensions over rational field, Galois groups of polynomials, Fundamental theorem of algebra, Solvable and Radical extensions, Computation of Galois groups over rational field.

- D. S. Dummit and R. M. Foote, Abstract Algebra. John Wiley & Sons, 2004.
- T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer, 1980.
- M. Artin, Algebra, Prentice Hall, 1991.

Spaces of functions: Continuous functions on locally compact spaces, Stone-Weierstrass theorems, Ascoli-Arzela Theorem.

Review of Measure theory: Sigma-algebras, measures, construction and properties of the Lebesgue measure, non-measurable sets, measurable functions and their properties.

Integration: Lebesgue Integration, various limit theorems, comparison with the Riemann Integral, Functions of bounded variation and absolute continuity.

Measure spaces: Signed-measures, Radon-Nikodym theorem, Product spaces, Fubini's thoerem (without proof) and its applications.

$L^p$-spaces: Holder and Minkowski inequalities, completeness, Convolutions, Approximation by smooth functions.

Fourier analysis: Fourier Transform, Inverse Fourier transform, Plancherel Theorem for $\mathbb R$.

Review of Measure theory: Sigma-algebras, measures, construction and properties of the Lebesgue measure, non-measurable sets, measurable functions and their properties.

Integration: Lebesgue Integration, various limit theorems, comparison with the Riemann Integral, Functions of bounded variation and absolute continuity.

Measure spaces: Signed-measures, Radon-Nikodym theorem, Product spaces, Fubini's thoerem (without proof) and its applications.

$L^p$-spaces: Holder and Minkowski inequalities, completeness, Convolutions, Approximation by smooth functions.

Fourier analysis: Fourier Transform, Inverse Fourier transform, Plancherel Theorem for $\mathbb R$.

- D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
- G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
- P. R. Halmos, Measure Theory, Springer-Verlag, 1974.
- H. L. Royden, Real Analysis, Macmillan 1988.
- W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.

Banach spaces: Review of Banach spaces, Hahn-Banach Theorem and its applications, Baire Category theorem and its applications like Closed graph theorem, Open mapping theorem.

Topological Vector spaces: Weak and weak* topologies, locally convex topological vector spaces.

Hilbert spaces: Review of Hilbert spaces and operator Theory, Compact operators, Schauder's theorem on the spectral theory of compact operators.

Banach algebras: Elementary properties, Resolvent and spectrum, Spectral radius formula, Ideals and homomorphisms, Gelfand transforms, Gelfand theorem for commutative Banach algebras.

- D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
- G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
- G. K. Pederson, Analysis NOW, GTM series, Springer-Verlag, 1991.
- W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.
- W. Rudin, Functional Analysis, TMH Edition, 1974.
- K. Yosida, Functional Analysis, Springer-Verlag 1968.

**Topology:** Topological spaces, Continuous maps between topological spaces, product topology, Quotient spaces, Connectedness, Compactness,Winding Numbers of Closed Curves, Brouwer Fixed Point Theorem (statement only), Borsuk-Ulam Theorem (Statement Only). [20 lectures]**Complex Analysis:** Complex line integrals, Goursat’s theorem; Local existence of primitives and Cauchy’s theorem in a disc , Cauchy’s integral formula , Applications of Cauchy’s integral, Singularities and their classifications, zeros, poles and residue theorem. Applications of residue theorem. Argument principle and applications. Maximum Modulus principle, Schwarz lemma. Biholomorphic between between complex plane, Disc to itself, Statement of Riemann Mapping theorem. [30 lectures]

- Armstrong, Basic Topology, Springer, 1983
- Munkres, Topology, Pearson Education, 2005.
- Greene and Krantz, Function Theory of One Complex Variable, gsm 40, University Press, 2006
- Stein and Shakarchi, Complex Analysis (Princeton Lectures in Analysis, No. 2), Princeton University Press, 2003.
- Gamelin, Complex Analysis (Undergraduate Texts in Mathematics), Springer, 2003.
- Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.

**Homotopy Theory:**Fundamental groups and its functorial properties, examples, Van-Kampen Theorem, [8 lectures]**Covering spaces:**Covering spaces, Computation of fundamental groups using coverings. The classification of covering spaces. Deck transformations. [8 lectures]**Simply connected spaces:**Simply connected spaces-Universal covering spaces of locally simply connected and pathwise connected spaces. - Universal covering group of connected subgroups of General Linear groups. [16 lectures]**Homology groups:**Affine spaces, simplexes and chains - Homology groups - Properties of Homology groups. - Chain Complexes, Relation Between one dimensional Homotopy and Homology groups. - (As in sections 8 - 12 of Part II of Greenberg and Harper.) [16 lectures]

- Armstrong, Basic Topology, Springer, 1983
- Greenberg & Harper, Algebraic Topology: A First Course, Addition Wesley, 1984.
- Munkres, Topology, Pearson Education, 2005. 1974

Pigeonhole principle, Counting principles, Binomial coefficients, Principles of inclusion and exclusion, recurrence relations, generating functions, Catalan numbers, Stirling numbers, Partition numbers, Schroder numbers. [25 lectures]

Graphs, subgraphs, graph isomorphisms, Hamilton cycles, Euler tours, directed graphs, matching, Tutte’s theorem, Menger’s theorem, planar graphs, Kuratowski’s theorem, graph colourings, network flows, max-flow min-cut theorem, Ramsey theory for graphs, Matrices associated with graphs: Incidence matrix, Adjacency matrix, Laplacian matrix. [25 lectures]

- R. A. Brualdi, Introductory Combinatorics, Pearson Prentice Hall, 2010.
- J. H. van Lint, R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001.
- R. P. Stanley, Enumerative Combinatorics Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2012.
- R. Diestel, Graph Theory, Graduate Texts in Mathematics, 173, Springer, 2010.
- B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, 184, Springer-Verlag, 1998.
- J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244, Springer, 2008.

**School of Mathematical Sciences**

NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050

Tel: +91-674-249-4081

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