Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Leraning some important theorems in complex analysis such as Riemann mapping theorem, Weirstrass factorization theorem, Runges theorem, Hardamard factorization theorem, Little Picards theorem and Great Picards theorem.
• Learning some basic techniques of harmonic functions and characterization of Dirichlet Region.

Prerequisite:
MAT306

Syllabus:

1. Review of basic Complex Analysis: Cauchy-Riemann equations, Cauchy’s theorem and estimates, power series expansions, maximum modulus principle, Classification of singularities and calculus of residues.
2. Space of continuous functions, Arzela’s theorem, Spaces of analytic functions, Spaces of meromorphic functions, Riemann mapping theorem, Weierstrass Factorization theorem, Runge’s theorem, Simple connectedness, Mittag-Leffler’s theorem, Analytic continuation, Schwarz reflection principle, Mondromy theorem, Jensen’s formula, Genus and order of an entire function, Hadamard factorization theorem, Little Picard theorem, Great Picard theorem, Harmonic functions.

References

1. L. V. Ahlfors, "Complex Analysis", Tata McGraw-Hill,
2. J. B. Conway, "Functions of One Complex Variable II", Graduate Texts in Mathematics 159, Springer-Verlag,
3. W. Rudin, "Real and Complex Analysis", Tata McGraw-Hill,
4. R. Remmert, "Theory of Complex Functions", Graduate Texts in Mathematics 122, Springer, 2008.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the concept of topological vector space, as a generalisation of normed linear spaces, and various properties of operators defined on them.

Prerequisite:
MAT401

Syllabus:

1. Definition and examples of topological vector spaces (TVS) and locally convex spaces (LCS); Linear operators; Hahn-Banach Theorems for TVS/ LCS (analytic and geometric forms); Uniform boundedness principle; Open mapping theorem; Closed graph thoerem; Weak and weak* vector topologies; Bipolar theorem; dual of LCS spaces; Krein-Milman theorem for TVS; Krien-Smulyan theorem for Banach spaces; Inductive and projective limit of LCS.

References

1. W. Rudin, "Functional Analysis", Tata McGraw-Hill,
2. A. P. Robertson, W. Robertson, "Topological Vector Spaces", Cambridge Tracts in Mathematics 53, Cambridge University Press,
3. J. B. Conway, "A Course in Functional Analysis", Graduates Texts in Mathematics 96, Springer, 2006.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning various decomposition results of matrices and their applications.

Prerequisite:
MAT205

Syllabus:

1. Rational and Jordan canonical forms, Inner product spaces, Unitary and Normal operators, Forms on inner product spaces, Spectral theorems, Bilinear forms, Matrix decomposition theorems, Courant- Fischer minimax and related theorems, Nonnegative matrices, Perron-Frobenius theory, Generalized inverse, Matrix Norm, Perturbation of eigenvalues.

References

1. R. A. Horn, C. R. Johnson, "Matrix Analysis", Cambridge University Press,
2. K. Hoffman, R. Kunze, "Linear Algebra", Prentice-Hall of India,
3. S. Roman, "Advanced Linear Algebra", Graduate Texts in Mathematics 135, Springer, 2008.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the explicit representations of solutions of four important classes of PDEs, namely, Transport equations, Heat equation, Laplace equation and wave equation for initial value problems.
• Learning the properties of solutions of these equations such as mean value property, maximum principles and regularity.
• Understanding Cauchy-Kowalevski Theorem and uniqueness theorem of Holmgreen for quasilinear equations.

Prerequisite:
MAT302, MAT303, MAT305

Syllabus:

1. Classification of Partial Differential Equations, Cauchy Problem, CauchyKowalevski Theorem, Lagrange-Green identity, The uniquness theorem of Holmgren.
2. Transport equation: Initial value problem, nonhomogeneous problem.
3. Laplace equation: Fundamental solution, Mean Value formula, properties of Harmonic functions, Green’s function, Energy methods, Harnack’s inequality.
4. Heat equation: Fundamental solution, Mean value formula, properties of solutions.
5. Wave equation: Solution by spherical means, Nonhomogeneous problem, properties of solutions.

References

1. L. C. Evans, "Partial Differential Equations", Graduate Studies in Mathematics 19, American Mathematical Society,
2. F. John, "Partial Differential Equations", Springer International Edition,
3. G. B. Folland, "Introduction to Partial Differential Equations", Princeton University Press,
4. S. Kesavan, "Topics in Functional Analysis and Applications", John Wiley & Sons, 1989.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the theory of discrete time and continuous time Markov chains.

Prerequisite:
MAT206

Syllabus:

1. 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.

References

1. P. G. Hoel, S. C. Port, C. J. Stone, "Introduction to Stochastic Processes", Houghton Mifflin Co.,
2. R. Durrett, "Essentials of Stochastic Processes", Springer Texts in Statistics, Springer,
3. G. R. Grimmett, D. R. Stirzaker, "Probability and Random Processes", Oxford University Press,
4. S. M. Ross, "Stochastic Processes", Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, 1996

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the fundamentals of classical algebraic geometry.
• Learning about the theory of Riemann surfaces, divisors, line bundles, Chern Classes and the Riemann Roch Theorem.

Prerequisite:
MAT205, MAT301

Syllabus:

1. Prime ideals and primary decompositions, Ideals in polynomial rings, Hilbert Basis theorem, Noether normalisation lemma, Hilbert’s Nullstellensatz, Affine and Projective varieties, Zariski Topology, Rational functions and morphisms, Elementary dimension theory, Smoothness, Curves, Divisors on curves, Bezout’s theorem, Riemann-Roch for curves, Line bundles on Projective spaces.

References

1. K. Hulek, "Elementary Algebraic Geometry", Student Mathematical Library 20, American Mathematical Society,
2. I. R. Shafarevich, "Basic Algebraic Geometry 1: Varieties in Projective Space", Springer,
3. J. Harris, "Algebraic geometry", Graduate Texts in Mathematics 133, SpringerVerlag,
4. M. Reid, "Undergraduate Algebraic Geometry", London Mathematical Society Student Texts 12, Cambridge University Press,
5. K. E. Smith et. al., "An Invitation to Algebraic Geometry", Universitext, SpringerVerlag,
6. R. Hartshorne, "Algebraic Geometry", Graduate Texts in Mathematics 52, SpringerVerlag, 1977.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the different algebraic techniques used in the study of the graphs

Prerequisite:
MAT205, MAT208

Syllabus:

1. Adjacency matrix of a graph and its eigenvalues, Spectral radius of graphs, Regular graphs and Line graphs, Strongly regular graphs, Cycles and Cuts, Laplacian matrix of a graph, Algebraic connectivity, Laplacian spectral radius of graphs, Distance matrix of a graph, General properties of graph automorphisms, Transitive and Arc-tranisitve graphs, Symmetric graphs.

References

1. N. Biggs, "Algebraic Graph Theory", Cambridge University Press,
2. C. Godsil, G. Royle, "Algebraic Graph Theory", Graduate Texts in Mathematics 207, Springer-Verlag,
3. R. B. Bapat, "Graphs and Matrices", Universitext, Springer, Hindustan Book Agency, New Delhi, 2010.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basic properties of number fields, computation of class numbers and zeta functions.

Prerequisite:
MAT207, MAT304

Syllabus:

1. Number Fields and Number rings, prime decomposition in number rings, Dedekind domains, Ideal class group, Galois theory applied to prime decomposition, Gauss reciprocity law, Cyclotomic fields and their ring of integers, finiteness of ideal class group, Dirichlet unit theorem, valuations and completions of number fields, Dedekind zeta function and distribution of ideal in a number ring.

References

1. D. A. Marcus, "Number Fields", Universitext, Springer-Verlag,
2. G. J. Janusz, "Algebraic Number Fields", Graduate Studies in Mathematics 7, American Mathematical Society,
3. S. Alaca, K. S. Williams, "Introductory Algebraic Number Theory", Cambridge University Press,
4. S. Lang, "Algebraic Number Theory", Graduate Texts in Mathematics 110, SpringerVerlag,
5. A. Frohlich, M. J. Taylor, "Algebraic Number Theory", Cambridge Studies in Advanced Mathematics 27, Cambridge University Press,
6. J. Neukirch, "Algebraic Number Theory", Springer-Verlag, 1999.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about the descriptive statistics of data sets including graphical representation using some statistical software.
• Understanding the basic theory of point estimation, interval estimation, hypothesis testing and linear regression.

Prerequisite:
MAT206

Syllabus:

1. Descriptive Statistics, Graphical representation of data, Curve fittings, Simple correlation and regression, Multiple and partial correlations and regressions, Sampling, Sampling distributions, Standard error. Normal distribution and its properties, The distribution of X and S2 in sampling from a normal distribution, Exact sampling distributions: χ2, t, F. Theory and Methods of Estimation: Point estimation, Criteria for a good estimator, Properties of estimators: Unbiasedness, Efficiency, Consistency, Sufficiency, Robustness. A lower bound for a variance of an estimate, Method of estimation: The method of moment, Least square method, Maximum likelihood estimation and its properties, UMVU Estimator, Interval estimation. Test of Hypothesis: Elements of hypothesis testing, Unbiased test, Neyman-Pearson Theory, MP and UMP tests, Likelihood ratio and related tests, Large sample tests, Test based on χ2, t, F.

Text Books

1. H. J. Larson, "Introduction to Probability Theory and Statistical Inference", John Wiley & Sons,
2. V. K. Rohatgi, "Introduction to Probability Theory and Mathematical Statistics", John Wiley & Sons, 1976.

References

1. I. Miller, M. Miller, "John E. Freund’s Mathematical Statistics with Applications", Pearson, 2013.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning data structure, design and analysis algorithms.
• Understanding some important algorithms like sortings, graph theoretics, polynomial related and optimization related.

Prerequisite:
MAT208

Syllabus:

1. Algorithm analysis, asymptotic notation, probabilistic analysis; Data Structure: stack, queues, linked list, hash table, binary search tree, red-black tree; Sorting: heap sort, quick sort, sorting in linear time; Algorithm design: divide and conquer, greedy algorithms, dynamic programming; Algebraic algorithms: Winograd’s and Strassen’s matrix multiplication algorithm, evaluation of polynomials, DFT, FFT, efficient FFT implementation; Graph algorithms: breadth-first and depth-first search, minimum spanning trees, single-source shortest paths, all-pair shortest paths, maximum flow; NP-completeness and approximation algorithms.

References

1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, "The Design and Analysis of Computer Algorithms", Addison-Wesley Publishing Co.,
2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to Algorithms", MIT Press, Cambridge,
3. E. Horowitz, S. Sahni, "Fundamental of Computer Algorithms", Galgotia Publication,
4. D. E. Knuth, "The Art of Computer Programming Vol. 1, Vol. 2, Vol 3", AddisonWesley Publishing Co., 1997, 1998, 1998.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Llearning the practical use of some important results from real analysis and linear algebra.

Prerequisite:
MAT201, MAT303

Syllabus:

1. Errors in computation: Representation and arithmetic of numbers, source of errors, error propagation, error estimation.
2. Numerical solution of non-linear equations: Bisection method, Secant method, Newton-Raphson method, Fixed point methods, Muller’s method.
3. Interpolations: Lagrange interpolation, Newton divided differences, Hermite interpolation, Piecewise polynomial interpolation.
4. Approximation of functions: Weierstrass and Taylor expansion, Least square approximation.
5. Numerical Integration: Trapezoidal rule, Simpson’s rule, Newton-Cotes rule, Guassian quadrature.
6. Numerical solution of ODE: Euler’s method, multi-step methods, Runge-Kutta methods, Predictor-Corrector methods.
7. Solutions of systems of linear equations: Gauss elimination, pivoting, matrix factorization, Iterative methods - Jacobi and Gauss-Siedel methods.
8. Matrix eigenvalue problems: power method.

Text Books

1. K. E. Atkinson, "An Introduction to Numerical Analysis" Wiley-India Edition, 2013.

References

1. S. D. Conte, C. De Boor, "Elementary Numerical Analysis, Tata McGraw-Hill,
2. W. H. Press et. al., "Numerical Recipes - The Art of Scientific Computing", Cambridge University Press, 2007.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of cryptography and cryptanalysis.
• Understanding the theory and design of cryptographic schemes like stream ciphers, block ciphers and public key ciphers like RSA, ElGamal, elliptic curve cryptosystem.
• Learning about data authentication, integrity and secret sharing.

Prerequisite:
MAT202, MAT207

Syllabus:

1. Overview of Cryptography and cryptanalysis, some simple cryptosystems (e.g., shift, substitution, affine, knapsack) and their cryptanalysis, classification of cryptosystems, classification of attacks;
2. Information Theoretic Ideas: Perfect secrecy, entropy;
3. 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;
4. Public-key cryptosystem: Implementation and cryptanalysis of RSA, ElGamal public-key cryptosystem, Discrete logarithm problem, elliptic curve cryptography;
5. Data integrity and authentication: Hash functions, message authentication code, digital signature scheme, ElGamal signature scheme;
6. Secret sharing: Shamir’s threshold scheme, general access structure and secret sharing.

References

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

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the structures of finite fields, factorization of polynomials, some applications towards cryptography, coding theory and combinatorics.

Prerequisite:
MAT304

Syllabus:

1. Structure of finite fields: characterization, roots of irreducible polynomials, traces, norms and bases, roots of unity, cyclotomic polynomial, representation of elements of finite fields, Wedderburn’s theorem;
2. Polynomials over finite field: order of polynomials, primitive polynomials, construction of irreducible polynomials, binomials and trinomials, factorization of polynomials over small and large finite fields, calculation of roots of polynomials;
3. Linear recurring sequences: LFSR, characteristic polynomial, minimal polynomial, characterization of linear recurring sequences, Berlekamp-Massey algorithm;
4. Applications of finite fields: Applications in cryptography, coding theory, finite geometry, combinatorics.

References

1. R. Lidl, H. Neiderreiter, "Finite Fields", Cambridge university press,
2. G. L. Mullen, C. Mummert, "Finite Fields and Applications", American Mathematical Society,
3. A. J. Menezes et. al., "Applications of Finite Fields", Kluwer Academic Publishers,
4. Z-X. Wan, "Finite Fields and Galois Rings", World Scientific Publishing Co., 2012.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning how to measure information and encoding of information.
• Understanding the theory and techniques of error correcting codes like Reed-Muller codes, BCH codes, Reed-Solomon codes, Algebraic codes.

Prerequisite:
MAT205, MAT304

Syllabus:

1. Information Theory: Entropy, Huffman coding, Shannon-Fano coding, entropy of Markov process, channel and mutual information, channel capacity;
2. Error correcting codes: Maximum likelihood decoding, nearest neighbour decoding, linear codes, generator matrix and parity-check matrix, Hamming bound, Gilbert-Varshamov bound, binary Hamming codes, Plotkin bound, nonlinear codes, Reed-Muller codes, Cyclic codes, BCH codes, ReedSolomon codes, Algebraic codes.

References

1. R. W. Hamming, "Coding and Information Theory", Prentice-Hall,
2. N. J. A. Sloane, F. J. MacWilliams, "Theory of Error Correcting Codes", NorthHolland Mathematical Library 16, North-Holland,
3. S. Ling, C. Xing, "Coding Theory: A First Course", Cambridge University Press,
4. V. Pless, "Introduction to the Theory of Error-Correcting Codes", Wiley-Interscience Publication, John Wiley & Sons,
5. S. Lin, "An Introduction to Error-Correcting Codes", Prentice-Hall, 1970.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the propositional logic and first order theory.
• Understanding the completeness and compactness theorems with Godels incompleteness theorem.

Prerequisite:
MAT101

Syllabus:

1. Propositional Logic, Tautologies and Theorems of propositional Logic, Tautology Theorem.
2. First Order Logic: First order languages and their structures, Proofs in a first order theory, Model of a first order theory, validity theorems, Metatheorems of a first order theory, e. g., theorems on constants, equivalence theorem, deduction and variant theorems etc.
3. Completeness theorem, Compactness theorem, Extensions by definition of first order theories, Interpretations theorem, Recursive functions, Arithmatization of first order theories, Godels first Incompleteness theorem, Rudiments of model theory including Lowenheim-Skolem theorem and categoricity.

References

1. J. R. Shoenfield, "Mathematical logic", Addison-Wesley Publishing Co.,
2. E. Mendelson, "Introduction to Mathematical Logic", Chapman & Hall, 1997.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the concept of measures and measurable functions.
• Understanding integration and their various properties

Prerequisite:
MAT302

Syllabus:

1. σ−algebras of sets, measurable sets and measures, extension of measures, construction of Lebesgue measure, integration, convergence theorems, RadonNikodym theorem, product measures, Fubini’s theorem, differentiation of integrals, absolutely continuous functions, Lp-spaces, Riesz representation theorem for the space C[0; 1].

References

1. G. De Barra, "Measure theory and integration".
2. J. Nevue, "Mathematical foundations of the calculus of probability", Holden-Day, Inc.,
3. I. K. Rana, "An introduction to measure and integration", Narosa Publishing House.
4. P. Billingsley, "Probability and measure", John Wiley & Sons, Inc.,
5. W. Rudin, "Real and complex analysis", McGraw-Hill Book Co.,
6. K. R. Parthasarathy, "Introduction to probability and measure", The Macmillan Co. of India, Ltd., 1977.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning calculus in Banach Spaces, degree theory and it’s application for fixed point theorems of Brouwer and Schauder.
• Learning homotopy, homotopy extension and invariance theorems and its applications.

Prerequisite:
MAT304, MAT401

Syllabus:

1. Calculus in Banach spaces, inverse and multiplicit function theorems, fixed point theorems of Brouwer, Schauder and Tychonoff, fixed point theorems for nonexpansive and set-valued maps, predegree results, compact vector fields, homotopy, homotopy extension, invariance theorems and applications.

References

1. S. Kesavan, "Nonlinear Functional Analysis", Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the concepts of C*-algebra, von-Neuman algebra and toeplitz operators and the notion of index for Fredholm operators.

Prerequisite:
MAT401

Syllabus:

1. Compact operators on Hilbert Spaces. (a) Fredholm Theory (b) Index, C*-algebras - noncommutative states and representations, Gelfand-Neumark representation theorem, Von-Neumann Algebras;
2. Projections, Double Commutant theorem, L1 functionalCalculus, Toeplitz operators.

References

1. W. Arveson, "An invitation to C*-algebras", Graduate Texts in Mathematics, No.
2. Springer-Verlag,
3. N. Dunford and J. T. Schwartz, "Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space", Interscience Publishers John Wiley i& Sons
4. R. V. Kadison and J. R. Ringrose, "Fundamentals of the theory of operator algebras. Vol. I. Elementary theory", Pure and Applied Mathematics, 100, Academic Press, Inc.,
5. V. S. Sunder, "An invitation to von Neumann algebras", Universitext, SpringerVerlag, 1987.

Course: UG-Elective

Duration: 56 Hours

Credit:

Outcome:

• Learning Automata and Language theory by studying automata and context free language.
• Learning Computability theory by studying Turing machine and halting problem.
• Learning Complexity theory by studying P and NP class problems.

Prerequisite:
MAT101

Syllabus:

1. 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;
2. Computability: Turing machines, Churh-Turing thesis, decidability, halting problem, reducibility, recursion theorem;
3. Complexity: Time complexity of Turing machines, Classes P and NP, NP completeness, other time classes, the time hierarchy.

References

1. J. E. Hopcroft, R. Motwani, J. D. Ullman, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley,
2. H. Lewis, C. H. Papadimitriou, "Elements of the Theory of Computation", PrenticeHall,
3. M. Sipser, "Introduction to the Theory of Computation", PWS Publishing, 1997.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge on Haar measure, convolution structure on Lie group with emphasize to harmonic analysis on the groups Circle and real line.

Prerequisite:
MAT302, MAT306, MAT401

Syllabus:

1. Topological Groups: Basic properties of topological groups, subgroups, quotient groups. Examples of various matrix groups. Connected groups.
2. Haar measure: Discussion of Haar measure without proof on R, T, Z and simple matrix groups, Convolution, the Banach algebra L1(G) and convolution with special emphasis on L1(R), L1(T) and L1(Z).
3. Basic Representation Theory: Unitary representation of groups, Examples and General properties, The representations of Group and Group algebras, C∗-algebra of a group, GNS construction, Positive definite functions, Schur’s Lemma.
4. Abelian Groups: Fourier transform and its properties, Approximate identities in L1(G), Classical Kernels on R, The Fourier inversion Theorem, Plancherel theorem on R, Plancherel measure on R; T; Z.
5. Dual Group of an Abelian Group: The Dual group of a locally compact abelian group, Computation of dual groups for R; T; Z, Pontryagin’s Duality theorem.

References

1. G. B. Folland, "A Course in Abstract Harmonic Analysis", CRC Press,
2. H. Helson, "Harmonic Analysis", Texts and Readings in Mathematics, Hindustan Book Agency,
3. Y. Katznelson, "An Introduction to Harmonic Analysis", Cambridge University Press,
4. L. H. Loomis, "An Introduction to Abstract Harmonic Analysis", Dover Publication,
5. E. Hewitt, K. A. Ross, "Abstract Harmonic Analysis Vol. I", Springer-Verlag,
6. W. Rudin, "Real and Complex Analysis", Tata McGraw-Hill, 2013.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning p-adic numbers, quadratic forms, Dirichlet series and modular forms.

Prerequisite:
MAT207, MAT304, MAT306

Syllabus:

1. Review of Finite fields, Gauss Sums and Jacobi Sums, Cubic and biquadratic reciprocity, Polynomial equations over finite fields, Theorems of Chevally and Warning, Quadratic forms over prime fields.
2. Ring of p-adic integers, Field of p-adic numbers, completion, p-adic equations, Hensel’s lemma, Hilbert symbol, Quadratic forms with p-adic coefficients.
3. Dirichlet series: Abscissa of convergence and absolute convergence, Riemann Zeta function and Dirichlet L-functions. Dirichlet’s theorem on primes in arithmetic progression.
4. Functional equation and Euler product for L-functions.
5. Modular Forms and the Modular Group, Eisenstein series, Zeros and poles of modular functions, Dimensions of the spaces of modular forms, The j-invariant L-function associated to modular forms, Ramanujan τ function.

References

1. J.-P. Serre, "A Course in Arithmetic", Graduate Texts in Mathematics 7, SpringerVerlag,
2. K. Ireland, M. Rosen, "A Classical Introduction to Modern Number Theory", Graduate Texts in Mathematics 84, Springer-Verlag,
3. H. Hasse, "Number Theory", Classics in Mathematics, Springer-Verlag,
4. W. Narkiewicz, "Elementary and Analytic Theory of Algebraic Numbers", Springer Monographs in Mathematics, Springer-Verlag,
5. F. Q. Gouv^ea, "p-adic Numbers", Universitext, Springer-Verlag, 1997.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about measure theoretic probability starting from probability spaces to theory of martingales.

Prerequisite:
MAT206, MAT302

Syllabus:

1. 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.

References

1. A. Gut, "Probability: A Graduate Course", Springer Texts in Statistics, Springer,
2. K. L. Chung, "A Course in Probability Theory", Academic Press,
3. S. I. Resnick, "A Probability Path", Birkh¨auser,
4. P. Billingsley, "Probability and Measure", Wiley Series in Probability and Statistics, John Wiley & Sons,
5. J. Jacod, P. Protter, "Probability Essentials", Universitext, Springer-Verlag, 2003.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the use of different algebraic technique to study the combinatorial problems.

Prerequisite:
MAT202, MAT203

Syllabus:

1. Catalan Matrices and Orthogonal Polynomials, Catalan Numbers and Lattice Paths, Combinatorial Interpretation of Catalan Numbers, Symmetric Polynomials and Functions, Schur Functions, Jacobi-Trudi identity, RSK Algorithm, Standard Tableaux, Young diagrams and q-binomial coefficients, Plane Partitions, Group actions on boolean algebras, Enumeration under group action, Walks in graphs, Cubes and the Radon transform, Sperner property, Matrix-Tree Theorem.

References

1. R. P. Stanley, "Algebraic Combinatorics", Undergraduate Texts in Mathematics, Springer,
2. M. Aigner, "A Course in Enumeration", Graduate Texts in Mathematics 238, Springer,
3. R. P. Stanley, "Enumerative Combinatorics Vol. 2", Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 1999.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the theoretical study of cryptography which puts foundation for the study and design of real-life cryptography.

Prerequisite:
MAT102, MAT206

Syllabus:

1. Introduction to cryptography and computational model, computational difficulty, pseudorandom generators, zero-knowledge proofs, encryption schemes, digital signature and message authentication schemes, cryptographic protocol.

References

1. O. Goldreich, "Foundations of Cryptography - Vol. I and Vol. II", Cambridge University Press, 2001,
2. S. Goldwasser, Mihir Bellare, "Lecture Notes on Cryptography", 2008, available online from http://cseweb.ucsd.edu/mihir/papers/gb.html

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding different kinds of incidence structures such as projective spaces, affine spaces, generalized quadrangles, polar spaces and quadratic sets.

Prerequisite:
MAT205

Syllabus:

1. Definitions and Exampleas, projective planes, affine planes, projective spaces, affine spaces, collineations of projective and affine spaces, fundamental theorem of projective and affine spaces, polar spaces, generalized quadrangles, quadrics and quadratic sets.

References

1. J. Ueberberg, "Foundations of Incidence Geometry", Springer Monographs in Mathematics, Springer,
2. L. M. Batten, "Combinatorics of Finite Geometries", Cambridge University Press,
3. E. E. Shult, "Points and Lines", Universitext, Springer,
4. L. M. Batten, A. Beutelspacher, "The Theory of Finite Linear Spaces: Combinatorics of points and lines", Cambridge University Press,
5. G. E. Moorhouse, "Incidence Geometry", 2007, available online from http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basics of Lie Algebra

Prerequisite:
MAT202, MAT205, MAT304

Syllabus:

1. Definitions and Examples, Derivations, Ideals, Homomorphisms, Nilpotent Lie Algebras and Engel’s theorem, Solvable Lie Algebras and Lie’s theorem, Jordan decomposition and Cartan’s criterion, Semisimple Lie algebras, Casimir operator and Weyl’s theorem, Representations of sl(2; F), Root space decomposition, Abstract root systems, Weyl group and Weyl chambers, Classification of irreducible root systems, Abstract theory of weights, Isomorphism and conjugacy theorems, Universal enveloping algebras and PBW theorem, Representation theory of semi-simple Lie algebras, Verma modules and Weyl character formula.

References

1. J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Graduate Texts in Mathematics 9, Springer-Verlag,
2. K. Erdmann, M. J. Wildon, "Introduction to Lie Algebras", Springer Undergraduate Mathematics Series, Springer-Verlag,
3. J.-P. Serre, "Complex Semisimple Lie Algebras", Springer Monographs in Mathematics, Springer-Verlag,
4. N. Jacobson, "Lie Algebras", Dover Publications, 1979.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the different techniques used to solve the linear and nonlinear programming problem

Prerequisite:
MAT102, MAT205

Syllabus:

1. Linear programming problem and its formulation, convex sets and their properties, Graphical method, Simplex method, Duality in linear programming, Revised simplex method, Integer programming, Transportation problems, Assignment problems, Games and strategies, Two-person (non) zerosum games, Introduction to non-linear programming and techniques.

References

1. J. K. Strayer, "Linear Programming and its Applications", Undergraduate Texts in Mathematics, Springer-Verlag,
2. P. R. Thie, G. E. Keough, "An Introduction to Linear Programming and Game Theory", John Wiley & Sons,
3. L. Brickman, "Mathematical Introduction to Linear Programming and Game Theory", Undergraduate Texts in Mathematics, Springer-Verlag,
4. D. G. Luenberger, Y. Ye, "Linear and Nonlinear Programming", International Series in Operations Research & Management Science 116, Springer, 2008.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of distribution Theory, Sobolev Spaces and their properties.

Prerequisite:
MAT401, MAT454

Syllabus:

1. Distribution Theory, Sobolev Spaces, Embedding theorems, Trace theorem.
2. Dirichlet, Neumann and Oblique derivative problem, Weak formulation, Lax-Milgram, Maximum Principles - Weak and Strong Maximum Principles, Hopf Maximum Principle, Alexandroff-Bakelmann-Pucci Estimate.

References

1. L. C. Evans, "Partial Differential Equations", Graduate Studies in Mathematics 19, American Mathematical Society,
2. H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations", Universitext, Springer,
3. R. A. Adams, J. J. F. Fournier, "Sobolev Spces", Pure and Applied Mathematics 140, Elsevier/Academic Press,
4. S. Kesavan, "Topics in Functional Analysis and Applications", John Wiley & Sons,
5. M. Renardy, R. C. Rogers, "An Introduction to Partial Differential Equations", Springer, 2008.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning random graphs and their applications.

Prerequisite:
MAT206, MAT208

Syllabus:

1. Models of random graphs and of random graph processes; illustrative examples; random regular graphs, configuration model; appearance of the giant component small subgraphs; long paths and Hamiltonicity; coloring problems; eigenvalues of random graphs and their algorithmic applications; pseudo-random graphs.

References

1. N. Alon, J. H. Spencer, "The Probabilistic Method", John Wiley & Sons, 2008
2. B. Bollob´as, "Random Graphs", Cambridge Studies in Advanced Mathematics 73, Cambridge University Press,
3. S. Janson, T. Luczak, A. Rucinski, "Random Graphs", Wiley-Interscience,
4. R. Durrett, "Random Graph Dynamics", Cambridge University Press,
5. J. H. Spencer, "The Strange Logic of Random Graphs", Springer-Verlag, 2001.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning how to use probabilistic techniques to different areas of mathematics and computer science.

Prerequisite:
MAT206

Syllabus:

1. Inequalities of Markov and Chebyshev (median algorithm), first and second moment method (balanced allocation), inequalities of Chernoff (permutation routing) and Azuma (chromatic number), rapidly mixing Markov chains (random walk in hypercubes, card shuffling), probabilistic generating functions (random walk in d-dimensional lattice)

References

1. R. Motwani, P. Raghavan, "Randomized Algorithms", Cambridge University Press,
2. M. Mitzenmacher, E. Upfal, "Probability and Computing: Randomized algorithms and probabilistic analysis", Cambridge University Press, 2005.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding about parametric statistical inference to be applicable to almost all branches of statistics.
• Learning various methods of estimation and hypothesis testing and their large sample and small sample properties.

Prerequisite:
MAT206, MAT459

Syllabus:

1. Review of joint and conditional distributions, order statistics, group family, exponential family. Introduction to parametric inference, sufficiency principle and data reduction, factorization theorem, minimal sufficient statistics, Fisher information, ancillary statistics, complete statistics, Basu’s theorem. Unbiasedness, best unbiased and linear unbiased estimator, Rao-Blackwell theorem, Lehmann- Scheffe theorem and UMVUE, Cramer-Rao lower bound and UMVUE, multi-parameter cases. Location and scale invariance, principle of equivarience. Methods of estimation: method of moments, likelihood principle and maximum likelihood estimation, properties of MLE: invariance, consistency, asymptotic normality. Hypothesis testing: error probabilities and power, most powerful tests, Neyman-Pearson lemma and its applications, p-value, uniformly most powerful (UMP) test via NeymanPearson lemma, UMP test via monotone likelihood ratio property, existence and nonexistence of UMP test for two sided alternative, unbiased and UMP unbiased tests. Likelihood (generalized) ratio tests and its properties, invariance and most powerful invariant tests. 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.

References

1. E. L. Lehmann and G. Casella, "Theory of Point Estimation", 2nd edition, Springer, New York,
2. E. L. Lehmann and J. P. Romano, "Testing Statistical Hypothesis", 3rd edition, Springer,
3. N. Mukhopadhyay, "Probability and Statistical Inference", Marcel Dekker, New York.
4. G. Casella and R. L. Berger, "Statistical Inference", 2nd edition, Cengage Learning,
5. A. M. Mood, F. A. Graybill and D. C. Boes, "Introduction to the theory of Statistics", 3rd edition, McGraw Hill, 1974.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about various modern statistical tools to analyze and draw inference from multivariate data sets
• Learning inference about multivariate sample mean and variance, techniques of dimension reduction, introductory factor analysis, cluster analysis and statistical pattern recognition.

Prerequisite:
MAT205, MAT305, MAT459

Syllabus:

1. Review of matrix algebra (optional), data matrix, summary statistics, graphical representations. Distribution of random vectors, moments and characteristic functions, transformations, some multivariate distributions: multivariate normal, multinomial, Dirichlet distribution, limit theorems. Multivariate normal distribution: properties, geometry, characteristics function, moments, distributions of linear combinations, conditional distribution and multiple correlation. Estimation of mean and variance of multivariate normal, theoretical properties, James-Stein estimator (optional), distribution of sample mean and variance, the Wishart distribution, large sample behavior of sample mean and variance, assessing normality. Inference about mean vector: testing for normal mean, Hotelling T 2 and likelihood ratio test, confidence regions and simultaneous comparisons of component means, paired comparisons and a repeated measures design, comparing mean vectors from two populations, MANOVA. Techniques of dimension reduction, principle component analysis: definition of principle components and their estimation, introductory factor analysis, multidimensional scaling. Classification problem: linear and quadratic discriminant analysis, logistic regression, support vector machine. Cluster analysis: non-hierarchical and hierarchical methods of clustering.

References

1. K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis", Academic Press,
2. T. W. Anderson, "An introduction to Multivariate Statistical Analysis", Wiley,
3. C. Chatfield and A. J. Collins, "Introduction to Multivariate Analysis", Chapman & Hall,
4. R. A. Johnson and D. W. Wichern, "Applied Multivariate Statistical Analysis", 6th edition, Pearson,
5. Brian Everitt and Torsten Hothorn, "An Introduction to Applied Multivariate Analysis with R", Springer,
6. M. L. Eaton, "Multivariate Statistics", John Wiley, 1983.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge of smooth manifolds, tangent and cotangent spaces, vector bundles, (co)tangent bundles, vector fields, differential forms, exterior differentiation, De-Rham cohomology, integration on manifolds, homotopy invariance of De-Rham cohomology and the statement of Poincare Duality.

Prerequisite:
MAT307

Syllabus:

1. Differentiable manifolds and maps: Definition and examples, Inverse and implicit function theorem, Submanifolds, immersions and submersions.
2. The tangent and cotangent bundle: Vector bundles, (co)tangent bundle as a vector bundle, Vector fields, flows, Lie derivative.
3. Differential forms and Integration: Exterior differential, closed and exact forms, Poincar´e lemma, Integration on manifolds, Stokes theorem, De Rham cohomology.

References

1. Michael Spivak, "A comprehensive introduction to differential geometry", Vol. 1, 3rd edition,
2. Frank Warner, "Foundations of differentiable manifolds and Lie groups", SpringerVerlag, 2nd edition,
3. John Lee, "Introduction to smooth manifolds", Springer Verlag, 2nd edition,
4. Louis Auslander and Robert E. MacKenzie, "Introduction to differentiable manifolds", Dover, 2nd edition, 2009.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the various properties of commutative rings, various class of commutative rings, and dimension theory.

Prerequisite:
MAT301

Syllabus:

1. Commutative rings, ideals, operations on ideals, prime and maximal ideals, nilradicals, Jacobson radicals, extension and contraction of ideals, Modules, free modules, projective modules, exact sequences, tensor product of modules, Restriction and extension of scalars, localization and local rings, extended and contracted ideals in rings of fractions, Noetherian modules, Artinian modules, Primary decompositions and associate primes, Integral extensions, Valuation rings, Discrete valuation rings, Dedekind domains, Fractional ideals, Completion, Dimension theory.

Text Books

1. M. F. Atiyah, I. G. Macdonald, "Introduction to Commutative Algebra", AddisonWesley Publishing Co., 1969.

References

1. R. Y. Sharp, "Steps in Commutative Algebra", London Mathematical Society Student Texts,
2. Cambridge University Press,
3. D. S. Dummit, R. M. Foote, "Abstract Algebra", Wiley-India edition, 2013.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basics of fundamental group (π1) and singular homology.
• Learning different techniques to compute the fundamental group such as homotopy invariance and Van-Kampen Theorem.
• Learning different techniques to compute singular homology of a space, including homotopy invariance, Mayer-Vietoris, excision, long exact sequence.

Prerequisite:
MAT301, MAT307

Syllabus:

1. Homotopy Theory: Simply Connected Spaces, Covering Spaces, Universal Covering Spaces, Deck Transformations, Path lifting lemma, Homotopy lifting lemma, Group Actions, Properly discontinuous action, free groups, free product with amalgamation, Seifert-Van Kampen Theorem, BorsukUlam Theorem for sphere, Jordan Separation Theorem.
2. Homology Theory: Simplexes, Simplicial Complexes, Triangulation of spaces, Simplicial Chain Complexes, Simplicial Homology, Singular Chain Complexes, Cycles and Boundary, Singular Homology, Relative Homology, Short Exact Sequences, Long Exact Sequences, Mayer-Vietoris sequence, Excision Theorem, Invariance of Domain.

Text Books

1. J. R. Munkres, "Topology", Prentice-Hall of India,
2. A. Hatcher, "Algebraic Topology", Cambridge University Press, 2009.

References

1. G. E. Bredon, "Topology and Geometry", Graduates Texts in Mathematics 139, Springer, 2009.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• It is a unique style of course where the mathematics students having interest in computation can learn to compute different algebraic problems in computer. Here students will learn the computation of the problems related (i) linear algebra, (ii) non-linear system of equations like Grobner bases, (iii) polynomial, (iv) algebraic number theory and (v) elliptic curve.

Prerequisite:
MAT205, MAT304

Syllabus:

1. Linear algebra and lattices: Asymptotically fast matrix multiplication algorithms, linear algebra algorithms, normal forms over fields, Lattice reduction; Solving system of non-linear equations: Gr¨obner basis, Buchberger’s algorithms, Complexity of Gr¨obner basis computation; Algorithms on polynomials: GCD, Barlekamp-Massey algorithm, factorization of polynomials over finite field, factorization of polynomials over Z and Q; Algorithms for algebraic number theory: Representation and operations on algebraic numbers, trace, norm, characteristic polynomial, discriminant, integral bases, polynomial reduction, computing maximal order, algorithms for quadratic fields; Elliptic curves: Implementation of elliptic curve, algorithms for elliptic curves.

References

1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, "The Design and Analysis of Computer Algorithms", Addison-Wesley Publishing Co.,
2. H. Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics 138, Springer-Verlag,
3. D. Cox, J. Little, D. O’shea, "Ideals, Varieties and Algorithms: An introduction to computational algebraic geometry and commutative algebra", Undergraduate Texts in Mathematics, Springer-verlag, 2007.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the elementary properties of Dirichlet series and distribution of primes.

Prerequisite:
MAT201, MAT207, MAT306

Syllabus:

1. Arithmetic functions, Averages of arithmetical functions, Distribution of primes, finite abelian groups and characters, Gauss sums, Dirichlet series and Euler products, Reimann Zeta function, Dirichlet L-functions, Analytic proof of the prime number theorem, Dirichlet Theorem on primes in arithmetic progression.

References

1. T. M. Apostol, "Introduction to Analytic Number Theory", Springer International Student Edition,
2. K. Chandrasekharan, "Introduction to Analytic Number Theory", Springer-Verlag,
3. H. Iwaniec, E. Kowalski, "Analytic Number Theory", American Mathematical Society Colloquium Publications 53, American Mathematical Society, 2004.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basic facts about classical groups defined over fields such as General Linear groups, Special Linear groups, Symplectic groups, Orthogonal groups and Unitary groups.

Prerequisite:
MAT202, MAT205, MAT304

Syllabus:

1. General and special linear groups, bilinear forms, Symplectic groups, symmetric forms, quadratic forms, Orthogonal geometry, orthogonal groups, Clifford algebras, Hermitian forms, Unitary spaces, Unitary groups.

References

1. L. C. Grove, "Classical Groups and Geometric Algebra", Graduate Studies in Mathematics 39, American Mathematical Society,
2. E. Artin, "Geometric Algebra", John Wiley & sons, 1988.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of Ergodic Theory.

Prerequisite:
MAT201

Syllabus:

1. Measure preserving systems; examples: Hamiltonian dynamics and Liouvilles theorem, Bernoulli shifts, Markov shifts, Rotations of the circle, Rotations of the torus, Automorphisms of the Torus, Gauss transformations, Skew-product, Poincare Recurrence lemma: Induced transformation: Kakutani towers: Rokhlins lemma. Recurrence in Topological Dynamics, Birkhoffs Recurrence theorem, Ergodicity, Weak-mixing and strong-mixing and their characterizations, Ergodic Theorems of Birkhoff and Von Neumann. Consequences of the Ergodic theorem. Invariant measures on compact systems, Unique ergodicity and equidistribution. Weyls theorem, The Isomorphism problem; conjugacy, spectral equivalence, Transformations with discrete spectrum, Halmosvon Neumann theorem, Entropy. The Kolmogorov-Sinai theorem. Calculation of Entropy. The Shannon Mc-MillanBreiman Theorem, Flows. Birkhoffs ergodic Theorem and Wieners ergodic theorem for flows. Flows built under a function.

References

1. Peter Walters, "An introduction to ergodic theory", Graduate Texts in Mathematics, Springer-Verlag,
2. Patrick Billingsley, "Ergodic theory and information", Robert E. Krieger Publishing Co.,
3. M. G. Nadkarni, "Basic ergodic theory", Texts and Readings in Mathematics, Hindustan Book Agency,
4. H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory", Princeton University Press,
5. K. Petersen, "Ergodic theory", Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1989.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowldege on Fourier Series, Fourier transforms and celebrated differentiation theorem and important operators like Hilbert transform and Maximal function.

Prerequisite:
MAT302

Syllabus:

1. Fourier series and its convergences, Dirichlet kernel, Frejer kernel, Parseval formula and its applications. Fourier transforms,the Schwartz space, Distribution and tempered distribution, Fourier Inversion and Plancherel theorem. Fourier analysis on Lp-spaces. Maximal functions and boundedness of Hilbert transform. Paley-Wiener Theorem for distribution. Poisson summation formula, Heisenberg uncertainty Principle, Wiener’s Tauberian theorem.

References

1. Y. Katznelson, "An Introduction to Harmonic Analysis", Cambridge University Press,
2. E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton Mathematical Series 32, Princeton University Press,
3. G. B. Folland, "Fourier Analysis and its Applications", Pure and Applied Undergraduate Texts 4, America Mathematical Society, 2010.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the rudiments of Lie groups and irreducible representations of compact Lie groups parametrised by Weyl Character formula.

Prerequisite:
MAT205, MAT305, MAT307

Syllabus:

1. General Properties: Definition of Lie groups, subgroups, cosets, group actions on manifolds, homogeneous spaces, classical groups. Exponential and logarithmic maps, Adjoint representation, Lie bracket, Lie algebras, subalgebras, ideals, stabilizers, center Baker-Campbell-Hausdorff formula, Lie’s Theorems.
2. Structure Theory of Lie Algebras: Solvable and nilpotent Lie algebras (with Lie/Engel theorems), semisimple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria, Jordan decomposition.
3. Complex semisimple Lie algebras, Toral subalgebras, Cartan subalgebras, Root decomposition and root systems.
4. Weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, BGG resolution, Weyl character formula.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. J. Faraut, "Analysis on Lie Groups", Cambridge Studies in Advanced Mathematics 110, Cambridge University Press,
3. B. C. Hall, "Lie Groups, Lie algebras and Representations", Graduate Texts in Mathematics 222, Springer-Verlag,
4. W. Fulton, J. Harris, "Representation Theory: A first course", Springer-Verlag,
5. J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Graduate Texts in Mathematics 9, Springer-Verlag,
6. A. Kirillov, "Introduction to Lie Groups and Lie Algebras", Cambridge Studies in Advanced Mathematics 113, Cambridge University Press,
7. V. S. Varadharajan, "Lie Groups, Lie Algebras and their Representations", SpringerVerlag, 1984.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the concepts and various structure theorems of C*-algebra and von-Neuman algebra.

Prerequisite:
MAT401

Syllabus:

1. Banach algebras/C*-algebras: Definition and examples; Spectrum of a Banach algebra; Gelfand transform; Gelfand-Naimark theorem for commutative Banach algebras/ C*-algebras; Functional calculus for C*-algebras; Positive cone in a C*-algebra; Existance of an approximate identity in a C*-algebra; Ideals and Quotients of a C*-algebra; Positive linear functionals on a C*-algebra; GNS construction. Locally convex topologies on the algebras of bounded operators on a Hilbert space, von-Neumann’s bi-commutant theorem; Kaplansky’s density theorem. Ruan’s characterization of Operator Spaces (if time permites).

References

1. R. V. Kadison, J. R. Ringrose, "Fundamentals of the Theory of Operator Algebras Vol. I", Graduate Studies in Mathematics 15, American Mathematical Society,
2. G. K. Pedersen, "C*-algebras and their Automorphism Groups", London Mathematical Society Monographs 14, Academic Press,
3. V. S. Sunder, "An Invitation to von Neumann Algebras", Universitext, SpringerVerlag,
4. M. Takesaki, "Theory of Operator Algebras Vol. I", Springer-Verlag, 2002.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the first principles of representations and understanding the important examples of 3 different types of groups, viz., compact, nilpotent and solvable groups.

Prerequisite:
MAT205, MAT305, MAT307

Syllabus:

1. Introduction to topological group, Haar measure on locally compact group, Representation theory of compact groups, Peter Weyl theorem, Linear Lie groups, Exponential map, Lie algebra, Invariant Differentail operators, Representation of the group and its Lie algebra. Fourier analysis on SU(2) and SU(3). Representation theory of Heisenberg group . Representation of Euclidean motion group.

References

1. J. E. Humphreys, "Introduction to Lie algebras and representation theory", SpringerVerlag,
2. S. C. Bagchi, S. Madan, A. Sitaram, U. B. Tiwari, "A first course on representation theory and linear Lie groups", University Press,
3. Mitsou Sugiura, "Unitary Representations and Harmonic Analysis", John Wiley & Sons,
4. Sundaram Thangavelu, "Harmonic Analysis on the Heisenberg Group", Birkhauser,
5. Sundaram Thangavelu, "An Introduction to the Uncertainty Principle", Birkhauser, 2003.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge on representaion on compact lie groups with examples SU(2), SO(n).

Prerequisite:
MAT205, MAT401, MAT307

Syllabus:

1. Review of General Theory: Locally compact groups, Computation of Haar measure on R; T, SU(2), SO(3) and some simple matrix groups, Convolution, the Banach algebra L1(G).
2. Representation Theory: General properties of representations of a locally compact group, Complete reducibility, Basic operations on representations, Irreducible representations.
3. Representations of Compact groups: Unitarilzibality of representations, Matrix coefficients, Schur’s orthogonality relations, Finite dimensionality of irreducible representations of compact groups.
4. Various forms of Peter-Weyl theorem, Fourier analysis on Compact groups, Character of a representation. Schur’s orthogonality relations among characters.
5. Weyl’s Chracter formula, Computing the Unitary dual of SU(2); SO(3); Fourier analysis on SO(n).

References

1. T. Brocker, T. Dieck, "Representations of Compact Lie Groups", Springer-Verlag,
2. J. L. Clerc, "Les Repr´esentatios des Groupes Compacts, Analyse Harmonique" (J. L. Clerc et. al., ed.), C.I.M.P.A.,
3. G. B. Folland, "A Course in Abstract Harmonic Analysis", CRC Press,
4. M. Sugiura, "Unitary Representations and Harmonic Analysis", John Wiley & Sons,
5. E. B. Vinberg, "Linear Representations of Groups", Birkh¨auser/Springer,
6. A. Wawrzy´nczyk, "Group Representations and Special Functions", PWN-Polish Scientific Publishers, 1984.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning modular forms over 2 and their congruence subgroups, and their Hecke theory.

Prerequisite:
MAT202, MAT205, MAT207, MAT306

Syllabus:

1. SL2(Z) and its congruence subgroups, Modular forms for SL2(Z), Modular forms for congruence subgroups, Modular forms and differential operators, Hecke theory, L-series, Theta functions and transformation formula.

References

1. J.-P. Serre, "A Course in Arithmetic", Graduate Texts in Mathematics 7, SpringerVerlag,
2. N. Koblitz, "Introduction to Elliptic Curves and Modular Forms", Graduate Texts in Mathematics 97, Springer-Verlag,
3. J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier, "The 1-2-3 of Modular Forms", Universitext, Springer-Verlag,
4. F. Diamond, J. Shurman, "A First Course in Modular Forms", Graduate Texts in Mathematics 228, Springer-Verlag,
5. S. Lang, "Introduction to Modular Forms", Springer-Verlag,
6. G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Forms", Princeton University Press, 1994.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning elliptic curves and the structure of their rational points.

Prerequisite:
MAT202, MAT207, MAT306

Syllabus:

1. Congruent numbers, Elliptic curves, Elliptic curves in Weierstrass form, Addition law, Mordell-Weil Theorem, Points of finite order, Points over finite fields, Hasse-Weil L-function and its functional equation, Complex multiplication.

References

1. J. H. Silverman, J. Tate, "Rational Points on Elliptic Curves", Undergraduate Texts in Mathematics, Springer-Verlag,
2. N. Koblitz, "Introduction to Elliptic Curves and Modular Forms", Graduate Texts in Mathematics 97, Springer-Verlag,
3. J. H. Silverman, "The Arithmetic of Elliptic Curves", Graduate Texts in Mathematics 106, Springer,
4. A. W. Knapp, "Elliptic Curves", Mathematical Notes 40, Princeton University Press,
5. J. H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Graduate Texts in Mathematics 151, Springer-Verlag, 1994.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about the theory of Brownian motion and it applications to stochastic differential equations.

Prerequisite:
MAT472

Syllabus:

1. Brownian Motion, Martingale, Stochastic integrals, extension of stochastic integrals, stochastic integrals for martingales, It^o’s formula, Application of It^o’s formula, stochastic differential equations.

References

1. H. H. Kuo, "Introduction to Stochastic Integration", Springer,
2. J. M Steele, "Stochastic Calculus and Financial Applications", Springer-Verlag,
3. F. C. Klebaner, "Introduction to Stochastic Calculus with Applications", Imperial College, 2005.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the fundamentals of Lie groups and Lie Algebras.
• Learning about (bi)invariant vector fields, integration on Lie Groups, Cartan’s Theorem.

Prerequisite:
MAT305, MAT307

Syllabus:

1. Review of Several variable Calculus: Directional Derivatives, Inverse Function Theorem, Implicit function Theorem, Level sets in Rn, Taylor’s theorem, Smooth function with compact support. Manifolds: Differentiable manifold, Partition of Unity, Tangent vectors, Derivative, Lie groups, Immersions and submersions, Submanifolds. Vector Fields: Left invariant vector fields of Lie groups, Lie algebra of a Lie group, Computing the Lie algebra of various classical Lie groups. Flows: Flows of a vector field, Taylor’s formula, Complete vector fields. Exponential Map: Exponential map of a Lie group, One parameter subgroups, Frobenius theorem (without proof). Lie Groups and Lie Algebras: Properties of Exponential function, product formula, Cartan’s Theorem, Adjoint representation, Uniqueness of differential structure on Lie groups. Homogeneous Spaces: Various examples and Properties. Coverings: Covering spaces, Simply connected Lie groups, Universal covering group of a connected Lie group. Finite dimensional representations of Lie groups and Lie algebras.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces", Graduate Studies in Mathematics 34, American Mathematical Society,
3. S. Kumaresan, "A Course in Differential Geometry and Lie Groups", Texts and Readings in Mathematics 22, Hindustan Book agency,
4. F. W. Warner, "Foundations of Differentiable Manifolds and Lie Groups", Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the representation theory of compact Lie groups and the group SL(2,C).
• Learning classifications of all simple Lie algebras through root system.

Prerequisite:
MAT556

Syllabus:

1. General theory of representations, operations on representations, irreducible representations, Schur’s lemma, Unitary representations and complete reducibility. Compact Lie groups, Haar measure on compact Lie groups, Schur’s Theorem, characters, Peter-Weyl theorem, universal enveloping algebra, Poincare-Birkoff-Witt theorem, Representations of Lie(SL(2; C)). Abstract root systems, Weyl group, rank 2 root systems, Positive roots, simple roots, weight lattice, root lattice, Weyl chambers, simple reflections, Dynkin diagrams, classification of root systems, Classification of semisimple Lie algebras. Representations of Semisimple Lie algebras, weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, Weyl Character formula, The representation theory of SU(3), Frobenius Reciprocity theorem, Spherical Harmonics.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. J. Faraut, "Analysis on Lie Groups", Cambridge Studies in Advanced Mathematics 110, Cambridge University Press,
3. B. C. Hall, "Lie Groups, Lie algebras and Representations", Graduate Texts in Mathematics 222, Springer-Verlag,
4. W. Fulton, J. Harris, "Representation Theory: A first course", Springer-Verlag,
5. A. Kirillov, "Introduction to Lie Groups and Lie Algebras", Cambridge Studies in Advanced Mathematics 113, Cambridge University Press,
6. A. W. Knapp, "Lie Groups: Beyond an introduction", Birk¨auser,
7. B. Simon, "Representations of Finite and Compact Groups", Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about the mathematical modeling of simple stock markets and techniques to analyze them.

Prerequisite:
MAT472

Syllabus:

1. Financial market models in finite discrete time, Absence of arbitrage and martingale measures, Valuation and hedging in complete markets, Basic facts about Brownian motion, Stochastic integration, Stochastic calculus: It^o’s formula, Girsanov transformation, It^o’s representation theorem, BlackScholes formula

References

1. J. Jacod, P. Protter, "Probability Essentials", Universitext, Springer-Verlag,
2. D. Lamberton, B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance", Chapman-Hall,
3. H. F¨ollmer, A. Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter, 2011.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the technique used for constructing combinatorial designs and its relation with linear codes.

Prerequisite:
MAT205, MAT304

Syllabus:

1. Incidence structures, affine planes, translation plane, projective planes, conics and ovals, blocking sets. Introduction to Balanced Incomplete Block Designs (BIBD), Symmetric BIBDs, Difference sets, Hadamard matrices and designs, Resolvable BIBDs, Latin squares. 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.

References

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

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning decision theory and Bayesian estimation and testing.
• Learning about large sample theory including asymptotic tests, confidence intervals, asymptotic efficiency and optimality of estimators and tests.

Prerequisite:
MAT481

Syllabus:

1. General decision problem, loss and risk function, minimax estimation, minimaxity and admissibility in exponential family. Introduction to Bayesian estimation, Bayes rule as average risk optimality, prior and posterior, conjugate families, generalized Bayes rules. Bayesian intervals and construction of credible sets, Bayesian hypothesis testing. Empirical and nonparametric empirical Bayes analysis, admissibility of Bayes and generalized Bayes rules, discussion on Bayes versus non-Bayes approaches. 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. Asymptotic level α tests, asymptotic equivalence, comparison of tests: relative efficiency, asymptotic comparison of estimators, efficient estimators and tests, local asymptotic optimality. Bootstrap sampling: estimation and testing.

References

1. E. L. Lehmann and G. Casella, "Theory of Point Estimation", 2nd edition, Springer, New York,
2. E. L. Lehmann, "Elements of Large-Sample Theory", Springer-Verlag,
3. E. L. Lehmann and J. P. Romano, "Testing Statistical Hypothesis", 3rd edition, Springer,
4. James O Berger, "Statistical Decision Theory and Bayesian Analysis", 2nd edition, Springer, New York, 1985.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the vector order structure and its relation with Functional Analysis.

Prerequisite:
MAT401

Syllabus:

1. Cones and orderings; order convexity; order units; approximate order units; bases. Positive linear mappings and functionals; extension and separation theorems; decomposition of linear functionals into positive linear functionals. Vector lattices; basic theory. Norms and orderings; duality of ordered spaces; (approximate) order unit spaces; base normed spaces. Normed and Banach lattices; AM-spaces, AL-spaces; Kakutani theorems for AM-spaces and ALspaces. Matrix ordered spaces: matricially normed spaces; matricial ordered normed spaces; matrix order unit spaces; Arveson-Hahn-Banach extension theorem. Recommended books:
2. G.J.O. Jameson, "Lecture Notes in Mathematics" 141 Springer-Verlag,
3. N.C. Wong and K.F. Ng, "(2) Partially ordered topological vector spaces", Oxford University Press,
4. C.D. Aliprantis and O. Burkinshaw, "Positive operators", Academic Press,
5. H.H. Schaefer, "Banach lattices and positive operators", Berlin: Springer, 1974.

References

1. W.A. J. Luxemburg and A.C. Zaanen, "Riesz Spaces", Elsevier,
2. A.C. Zaanen, "Introduction to operator theory in Riesz spces (Vol 1 & Vol 2)", Springer, 1997

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding analytic and harmonic functions on the unit disc.
• Understanding properties Hp spaces, for 1 ≤ p < 1.
• Understanding invarint subspaces for the shift operator on H2 space.

Prerequisite:
MAT302, MAT306

Syllabus:

1. Fourier Series: Cesaro Means, Characterization of Types of Fourier Series;
2. Analytic and Harmonic Functions in the Unit Disc: The Cauchy and Poisson Kernels, Boundary Values, Fatou’s Theorem, Hp Spaces;
3. The Space H1: The Helson-Lowdenslager Approach, Szego’s Theorem, Completion of the Discussion of H1;
4. Factorization for Hp functions: Inner and Outer Functions, Blaschke Products and Singular Functions, The Factorization Theorem, Absolute Convergence of Taylor Series, Functions of Bounded Characteristic;
5. Analytic Functions with Continuous Boundary Values: Conjugate Harmonic Functions, Theorems of Fatou and Rudin;
6. The Shift Operator: The Shift Operator on H2, Invariant Subspaces for H2 of the Half-plane, Isometries, The Shift Operator on L2.

References

1. Kenneth Hoffman, Banach spaces of analytic functions, Reprint of the 1962 original, Dover Publications, Inc., New York,
2. Walter Rudin, Real and complex analysis, Third edition. McGraw-Hill Book Co., New York,
3. Duren, Peter L., Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 1970.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding contractive operators by exhibiting as a compression of unitary operators.
• Understanding Hardy classes and H1 space.
• Understanding dilations of commuting and non-commuting contractions.

Prerequisite:
MAT401

Syllabus:

1. Contractions and Their Dilations: Unilateral shifts, Wold decomposition, Bilateral shifts, Contractions, Canonical decomposition, Isometric and unitary dilations, Matrix construction of the unitary dilation, a discussion on rational dilation;
2. Geometrical and Spectral Properties of Dilations: Structure of the minimal unitary dilations, Isometric dilations, Dilation of commutants;
3. Functional Calculus: Hardy classes, Inner and outer functions, The classes H1 and H1 T , The role of outer functions, Contractions of class C0;
4. Operator-Valued Analytic Functions: The spaces L2(U) and H2(U), Inner and outer functions, Lemmas on Fourier representation, Factorizations, Analytic kernels;
5. Functional Models: Characteristic functions, Functional models for a given contraction, Functional models for analytic functions; A discussion on Commuting and non-commuting contractions and their dilations.

References

1. B´ela Sz.-Nagy, Ciprian Foias, Hari Bercovici and L´aszl´o K´erchy, Harmonic analysis of operators on Hilbert space; Second edition. Revised and enlarged edition. Universitext. Springer, New York,
2. Vern Ival Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics,
3. Cambridge University Press, Cambridge,
4. Jim Agler, John Harland and Benjamin J. Raphael, Classical function theory, operator dilation theory, and machine computation on multiply-connected domains. (English summary) Mem. Amer. Math. Soc. 191 (2008), no.
5. Nikolai K. Nikolskii, Operators, functions, and systems: an easy reading. Vol. 1 and Vol
6. Hardy, Hankel, and Toeplitz, American Mathematical Society, Providence, RI, 2002.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• This first part of the course shall give an overview of classical theory of Binary quadratic forms, computation of class numbers, understanding of composition rule and correspondence between binary quadratic forms and classes of ideals in quadratic fields. The second part will give a more general theory of quadratic forms over various fields and local-global principle.

Prerequisite:
MAT201, MAT205, MAT207, MAT304

Syllabus:

1. Binary Quadratic forms-Elementary concepts, Modular groups and its fundamental domain, Positive definite forms and reduction, Indefinite forms and reduction, Automorphs and Pells equation; The class group: Representation and Genera, Composition of forms, Class number computations; Quadratic number fields and Ideals, Binary quadratic forms and Classes of ideals.
2. Quadratic forms over any field K, with char(K) 6= 2, Isotropic elements and hyperbolic planes, Witts extension Theorem; p-adic fields and Hilbert symbol; Quadratic forms over Qp, Hasse-Minkowski Theory of Quadratic forms over Q, Quadratic forms with prescribed invariants. Some applications: Legendres Theorem (on representation of zero by aX2 +bY 2 +cZ2), Gausss Theorem (Sum of three squares), Lagranges Theorem (Sum of Four Squares).

References

1. Binary Quadratic forms by D. A. Buell (Springer)(Chapter 17)
2. A Course on Arithmetic, J.-P. Serre (Springer) (Part-I)
3. Algebra: Volume 3 by Luther, Passi (Narosa) (Chapter 5)
4. The Arithmetic Theory of Quadratic Forms by Buton. W. Jones (The Carus Mathematical Monographs)
5. Rational Quadratic Forms by J.W.S. Cassels (Dover Publications)
6. Number Theory, Vol. I by Henri Cohen (Springer)
7. An Introduction to the Theory of Numbers by I. Niven, H. S. Zuckerman and H. L. Montgomery (Wiley)
8. Number Theory by Z. I Borevich and I. Shafarevich (Academic Press)
9. Elementary Number Theory by E. Landau (Chelsea Publishing Company)

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• The main aim of this course is to familiarize the students with fundamentals of variational calculus in function spaces and optimal control of ordinary differential equations. By the end of this course, students will have an in-depth knowledge of these topics and will be ready for more advanced courses in calculus of variations.

Prerequisite:
MAT303, MAT204, MAT205

Syllabus:

1. Calculus in normed linear spaces, convexity and fundamental theorems of optimization, Euler Lagrange equations, examples (the brachistochrone problem, minimal surface area of revolution).
2. Controllability: linear case and nonlinear autonomous systems, Bang-Bang principle, Existence theorems for Optimal Control problems.
3. Necessary conditions for Optimal Controls: The Pontryagin Maximum Principle.

References

1. J. Macki and A. Strauss, Introduction to Optimal Control Theory, New York: Springer;
2. J.L. Troutman, Variational Calculus with Elementary Convexity, New York: Springer;
3. A. Sasane, Optimization in function spaces, Dover Publications,
4. D.G. Luenberger, Optimization by Vector Space methods. Wiley,
5. I.M. Gelfand and S.V. Fomin, Calculus of Variations. Dover,
6. J. Zabczyk, Mathematical control theoryan introduction. Second edition, Systems Control Found. Appl., Birkhuser/Springer, Cham,
7. H.O. Fattorini, Infinite-dimensional optimization and control theory. Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge,
8. W. Liu, Elementary Feedback Stabilization of the Linear ReactionConvection-Diffusion Equation and the Wave Equation, Mathmatiques & Applications (Berlin) [Mathematics & Applications], SpringerVerlag, Berlin,
9. A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and control of infinite dimensional systems. Second edition. Systems & Control: Foundations & Applications. Birkhauser Boston, Inc., Boston, MA,
10. L.C. Evans, Mathematical Methods for Optimization: Finite Dimensional Optimization, Lecture notes.
11. L.C. Evans, Mathematical Methods for Optimization: Dynamic Optimization, Lecture notes.
12. L.C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture notes.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Coxeter groups find applications in many areas of mathematics. Studying these groups involves the interplay of geometry, algebra, and combinatorics. This course aims to familiarize students with the general theory of Coxeter groups (mostly from a combinatorial approach). Upon successful completion of the course, students will be familiar with various examples of finite and affine Coxeter groups and finite reflection groups. They will also learn the linearity of Coxeter groups, Exchange properties, Root system, Hecke algebra, Classification of finite and affine Coxeter groups, and Automorphisms of Coxeter groups.

Prerequisite:
MAT202, MAT205

Syllabus:

1. Coxeter systems and Coxeter groups, Coxeter graphs, matrices and corresponding Artin groups, Irreducible Coxeter systems, Types of Coxeter groups and Artin groups: Spherical, 2-Spherical, Crystallographic, Rightangled, Universal, Large, Simply laced, Even, Odd, etc., Various examples of finite and affine Coxeter groups, and Artin groups, Parabolic subgroups, Length function. Linearity of Coxeter groups, Root system, Permutation representation, Exchange properties, and Deletion conditions. Bruhat order. Finite reflection groups, Polynomial Invariants of finite reflection groups, Hecke Algebras, Kazhdan- Lusztig, and R-polynomials. Classification of finite and Affine Coxeter groups, Automorphisms of Coxeter groups. Additional Topics: (If time permits, then some of the following topics may be covered.) Isomorphism problem in Coxeter groups, K(Π; 1) conjecture for Artin groups, Enumeration in Coxeter groups, Stanley symmetric function.

References

1. A. Bjorner, and F. Brenti, Combinatorics of Coxeter Groups, Springer,
2. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press,
3. Richard Kane, Reflection Groups and Invariant theory, CMS Books in Mathematics (Springer-Verlag),
4. P. Bahls, The isomorphism problem in Coxeter Groups, Imperial College Press,
5. Nicolas Bourbaki, Elements Of Mathematics: Lie Groups and Lie Algebras: Chapters 4-6, Springer
6. Michael W. Davis, The Geometry and Topology of Coxeter Groups, Princeton University Press, 2008.

Course: UG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• The first part of this course shall give an overview of Category theory and functors. Different types of covariant and contravariant functors, such as Hom, tensor, Tor, and Ext, will be described. In the second part, the homology and cohomology of groups will be introduced, and spectral sequence will be discussed. This course aims to familiarize the students with categories, functors, group cohomology, spectral sequences and commutative diagrams. By the end of these courses, students will have a working knowledge of this topic.

Prerequisite:
MAT202, MAT301

Syllabus:

1. Categories, Functors, Abelian categories, Category of modules, Hom and Tensor functors. Localization, Projective and Injective modules, Derived functors, Tor and Ext functors.
2. Group rings, G-modules, Bar resolution, Homology and cohomology of groups, low dimensional homologies, Universal coefficient theorem, Second cohomology group and Extensions, Kunneth Formula, The Schur Multiplier and its properties, Transgression, Restriction and Inflation homomorphisms, Hochschild-Serre spectral sequence and applications.

References

1. Joseph J. Rotman, An introduction to Homological algebra, New York: Springer;
2. Weibel, Charles A. An introduction to homological algebra. No.
3. Cambridge University Press,
4. Cartan, Henry, and Samuel Eilenberg. Homological algebra. Vol.
5. Princeton university Press,
6. Maclane, Saunders. Categories for the working mathematician. Vol.
7. Springer Science & Business Media,
8. MacLane, Saunders. Homology. Springer Science & Business Media,
9. Hilton, Peter J., and Urs Stammbach. A course in homological algebra. Vol.
10. Springer Science & Business Media,
11. L.R. Vermani, An elementary approach to homological algebra. CRC Press,
12. Grothendieck, A. (1957), Sur quelques points d’algbre homologique, Thoku Mathematical Journal, (2), 9: 119-221. (Grothendieck Tohoku paper)

Course: PG-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Students will learn basic properties of groups, rings, and modules and will be able to use these algebraic structures to solve research problems.

Prerequisite:

Syllabus:

1. 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.
2. Ring Theory: Properties of Ideals, Chinese remainder theorem, Field of fractions, Euclidean do- mains, Principal ideal domains, Unique factorization domains, Polynomial Rings, Irreducibility criteria, Matrix rings.
3. Module Theory: Examples, Quotient modules, Isomorphism theorems, Generation of modules, Free modules, Tensor products of modules, Exact sequences - Projective, Injective and Flat modules.

References

1. D. S. Dummit and R. M. Foote, Abstract Algebra, John Wiley & Sons,
2. T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer,
3. M. Artin, Algebra, Prentice Hall,
4. N. Bourbaki, Algebra, Springer,
5. C Musili, Introduction to Rings and Modules, Narosa Publishing House.
6. N. S. Gopalakrishnan, University Algebra, New Age International.

Course: PG-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will be familiar with various advanced concepts and techniques from functional analysis, measure theory and harmonic analysis (on the real line).

Prerequisite:

Syllabus:

1. Spaces of functions: Continuous functions on locally compact spaces, Stone-Weierstrass theorems, Ascoli-Arzela Theorem.
2. Review of Measure theory: Sigma-algebras, measures, construction and properties of the Lebesgue measure, non-measurable sets, measurable functions and their properties.
3. Integration: Lebesgue Integration, various limit theorems, comparison with the Riemann Integral, Functions of bounded variation and absolute continuity.
4. Measure spaces: Signed-measures, Radon-Nikodym theorem, Product spaces, Fubini’s theorem (without proof) and its applications.
5. Lp-spaces: Holder and Minkowski inequalities, completeness, Convolutions, Approximation by smooth functions.
6. Fourier analysis: Fourier Transform, Inverse Fourier transform, Plancherel Theorem for Real numbers.

References

1. D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag
2. G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc.,
3. P. R. Halmos, Measure Theory, Springer-Verlag,
4. H. L. Royden, Real Analysis, Macmillan
5. W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York,
6. Elliott H. Lieb and Michael Loss , Analysis, American Mathematical Society, 2001.

Course: PG-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will aware of various properties of topological space and various properties of functions on topological spaces. The students also learn continuous maps between topological spaces, product topology, Quotient spaces, Connectedness, Compactness, Path connected spaces, separation axioms, Tychonoff spaces, Urysohn's lemma and metrization theorem. Furthermore, the student will learn various prop-erties of functions for several variables.

Prerequisite:

Syllabus:

1. Topological spaces, Continuous maps between topological spaces, Product topology, Quotient spaces, Connectedness, Compactness, Path connected spaces, Separation axioms, Tychonoff spaces, Urysohn’s lemma and Metrization theorem.
2. Differentiable functions on Rn, Jacobian criteria, Taylor’s theorem, Inverse function theorem, Implicit function theorem, Maxima-minima, Lagrange multiplier.

References

1. Armstrong, Basic Topology, Springer,
2. Munkres, Topology, Pearson Education,
3. J. Dugundji, Topology.
4. J. J. Duistermaat, J. A. C. Kolk: Multidimensional Real Analysis I: Differentiation.
5. K. Janich, Topology, Springer.
6. John L Kelley: General Topology (free download: https://archiveorg/details/GeneralTopology)
7. F. Simmons: Introduction to Topology and Modern Analysis.
8. S.Kumaresan: A Course in Differential Geometry and Lie Groups, TRIM series.
9. T. M. Apostol: Calculus: Multi-Variable Calculus and Linear Algebra With Applications to Differential Equations And Probability- Vol 2.

Course: PG-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will learn some important theorems in complex analysis such as Riemann mapping theorem, Weierstrass factorization theorem, Runges theorem, Hardamard factorization theorem, Little Picards theorem and Great Picards theorem. They will also learn some basic techniques of harmonic functions and characterization of Dirichlet Region. These results are very useful in many branches of mathematics such as Number Theory, Differential Geometry, Operator theory, Partial Differential Equations etc.

Prerequisite:

Syllabus:

1. Review of basic Complex Analysis: Cauchy-Riemann equations, Cauchy’s theorem and estimates, power series expansions, maximum modulus principle, Classification of singularities and calculus of residues; Normal families, Arzela-Ascoli theorem, Riemann mapping theorem; Weierstrass factorization theorem, Runges theorem, Mittag-Lefflers theorem; Hadamard factorization theorem, Analytic Continuation, Gamma and Zeta functions.

References

1. L. V. Ahlfors, Complex Analysis, Tata McGraw-Hill,
2. J. B. Conway, Functions of one complex variable, Second edition. Graduate Texts in Mathematics,
3. Springer-Verlag, New York-Berlin,
4. R. Narasimhan and Y. Nievergelt, Complex analysis in one variable, Second edition. Birkhuser Boston, Inc., Boston, MA,
5. W. Rudin, Real and Complex Analysis, Tata McGraw-Hill,
6. Wolfgang Fischer, Ingo Lieb, A Course in Complex Analysis: From Basic Results to Advanced Topics, Springer, 2012
7. Eberhard Freitag, Rolf Busam, Complex Analysis, Springer,
8. Stein and Shakarchi, Complex Analysis, Princeton University Press,
9. T. Gamelin, Complex Analysis, Springer, 2000.

Course: PG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Students will learn basic properties of fields and Galois theory and will be able to use these results to solve other mathematical problems.

Prerequisite:

Syllabus:

1. Linear Algebra: Matrix of a Linear transformation, dual vector spaces, determinants, Tensor algebras, Symmetric algebras, Exterior algebras,
2. Modules over PIDs: Basic theory, Structure theorem for finitely generated abelian groups, Rational and Jordan canonical forms.
3. 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.

References

1. D. S. Dummit and R. M. Foote, Abstract Algebra, John Wiley & Sons,
2. T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer,
3. M. Artin, Algebra, Prentice Hall,
4. T. T. Moh: Algebra, World Scientific, 1992
5. N. Bourbaki, Algebra, Springer, 1989.

Course: PG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will aware of various properties of norm linear vector spaces and topological vector spaces. They will also learn various properties of linear transformations defined on these norm liner spaces and topological vector spaces.

Prerequisite:

Syllabus:

1. 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.
2. Topological Vector spaces: Weak and weak* topologies, locally convex topological vector spaces.
3. Hilbert spaces: Review of Hilbert spaces and operator Theory, Compact operators, Schauder’s theorem on the spectral theory of compact operators.
4. Banach algebras: Elementary properties, Resolvent and spectrum, Spectral radius formula, Ideals and homo- morphisms, Gelfand transforms, Gelfand theorem for commutative Banach algebras.

References

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

Course: PG-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will learn covering spaces, homotopy theory and several concepts of topological invariants, namely, fundamental groups, homology groups.

Prerequisite:

Syllabus:

1. Homotopy Theory: Fundamental groups and its functorial properties, examples, Van-Kampen Theorem, Computation of fundamental group of S
2. Covering spaces: Covering spaces, Computation of fundamental groups using coverings. The classification of covering spaces. Deck transformations.
3. 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.
4. Homology groups: Affine spaces, simplexes and chains - Homology groups - Properties of Homology groups. Chain Complexes, Relation Between one dimensional Homotopy and Homology groups. Computation of Homology groups Sn, Brouwer’s fixed point theorem.

References

1. M. A. Armstrong, Basic Topology, Springer, 1983
2. M. J. Greenberg & J. R. Harper, Algebraic Topology: A First Course, Addition Wesley,
3. J. R. Munkres, Topology, Pearson Education,
4. 1974

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning different fundamental results of linear transformations and matrices, e.g. eigenvalues and eigenvectors, diagonalization, triangulation, rational and Jordan canonical forms.

Prerequisite:

Syllabus:

1. System of linear equations, matrices, Gauss elimination, Basis, dimension of a vector space, Linear Transformations and its representations by Matrices, rank-nullity theorem, Transpose of a Linear Transformation, Determinants, Characteristic Values, Annihilating Polynomials, Diagonalization and Triangulation, Primary Decomposition Theorem, Rational and Jordan canonical forms, Inner product spaces, GramSchmidt orthonormalization, linear functionals and adjoint, Hermitian, self-adjoint, unitary and normal operators, spectral theorem for normal operators, Bilinear forms, symmetric and skew-symmetric bilinear forms, groups preserving bilinear forms.

Text Books

1. Hoffman, K.; Kunze, R.; Linear Algebra, Prentice Hall.

References

1. Artin, M.; Algebra, Prentice Hall,
2. Lax, P., Linear ALgebra and its applications, John Wiley & Sons, Second edition,
3. Rose, H.E.; Linear Algebra, Birkhauser, 2002.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will have a thorough understanding of the basic concepts of metric spaces. They will also be familiar with the concepts of sequences and series of functions and approximation theorems.

Prerequisite:

Syllabus:

1. Metric spaces, open balls and open sets, limit and cluster points, closed sets, dense sets, complete metric spaces, completion of a metric space, Continuity, uniform continuity, Banach contraction principle, Compactness, Connectedness.
2. Sequences of functions, Pointwise convergence and uniform convergence, ArzelaAscoli Theorem, Weierstrass Approximation Theorem, power series, radius of convergence, uniform convergence and Riemann integration, uniform convergence and differentiation.
3. Functions of bounded variation. Riemann-Stieltjes Integrals.

Text Books

1. N. L. Carothers, Real Analysis, Cambridge University Press,
2. W. Rudin, Principles of mathematical analysis, McGraw-Hill Book Co., 1976.

References

1. S. Kumaresan, Topology of Metric Spaces, Narosa Publishing House,
2. G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill,
3. Tom M. Apostol, Mathematical analysis, Addison-Wesley,
4. R. P. Boas, A Primer of Real Functions, MAA/AMS, Carus Monographs, Volume 13, Fourth Edition, 1960.Course Title : Number Theory

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the elementary properties of rings of integers including divisibility, congruences, continued fractions and Gauss reciprocity laws.

Prerequisite:

Syllabus:

1. Divisibility, Primes, Fundamental theorem of arithmetic, Congruences, Chinese remainder theorem, Linear congruences, Congruences with prime-power modulus, Fermat’s little theorem, Wilson’s theorem, Euler function and its applications, Group of units, primitive roots, Quadratic residues, Jacobi symbol, Binary quadratic form, Arithmetic functions, M¨obius Inversion formula, Dirichlet product, Sum of squares, Continued fractions and rational approximations, Riemann zeta function.

Text Books

1. I. Niven, H. S. Zuckerman, H. L. Montgomery, "An Introduction to the Theory of Numbers", Wiley-India Edition, 2008.

References

1. T. M. Apostol, "Introduction to Analytic Number Theory", Springer International Student Edition,
2. G. A. Jones, J. M. Jones, "Elementary Number Theory", Springer Undergraduate Mathematics Series. Springer-Verlag, 1998.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning different techniques to obtain explicit solutions of 1st order and second order ODE and its applications.
• learning general theory existence, uniqueness and continuous dependence of general ODE.
• Understanding propeties of solutions as maximum principle, asymptotic behaviour and phase portrait analysis of 2nd order equations.
• Leraning characteristics method for solving 1st order partial Differential Equations.

Prerequisite:

Syllabus:

1. Classifications of Differential Equations: origin and applications, family of curves, isoclines.
2. First order equations: separation of variable, exact equation, integrating factor, Bernoulli equation, separable equation, homogeneous equations, orthogonal trajectories, Picard’s existence and uniqueness theorems.
3. Second order equations: variation of parameter, annihilator methods.
4. Series solution: power series solutions about regular and singular points. Method of Frobenius, Bessel’s equation and Legendre equations. Wronskian determinant, Phase portrait analysis for 2nd order system, comparison and maximum principles for 2nd order equations.
5. Linear system: general properties, fundamental matrix solution, constant coefficient system, asymptotic behavior, exact and adjoint equation, oscillatory equations, Green’s function. SturmLiouville theory.
6. Partial Differential Equations: Classifications of PDE, method of separation of variables, characteristic method.

Text Books

1. S. L. Ross, "Differential Equations", Wiley-India Edition,
2. E. A. Coddington, "An Introduction to Ordinary Differential Equations", Prentice-Hall of India, 2012.

References

1. G. F. Simmons, S. G. Krantz, "Differential Equations", Tata Mcgraw-Hill Edition,
2. B. Rai, D. P. Choudhury, "A Course in Ordinary Differential Equation", Narosa Publishing House, New Delhi,
3. R. P. Agarwal, D. ORegan, "Ordinary and Partial Differential Equations", Universitext. Springer,
4. C. M. Bender, S. A. Orszag, "Advanced mathematical methods for Scientists and Engineers", Springer Verlag, 1999.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Upon successful completion of the course, students will be familiar with various advanced concepts and techniques from measure theory.

Prerequisite:

Syllabus:

1. Abstract measure. Construction of Lebesgue measure. Measurable functions. Integration. Comparison of Riemann and Lebesgue integration. Convergence in measure. Monotone convergence theorem. Dominated convergence theorem. Fatou’s lemma. Product measures (including infinite product). Fubini’s theorem. Convolutions. Change of variables. Integration in polar co-ordinates. Signed measures and Radon-Nikodym theorem. Lp spaces. Dual of Lp spaces. Complex measures. Riesz representation theorem.

Text Books

1. G. B. Folland, Real Analysis, Wiley-Interscience Publication, John Wiley & Sons,
2. S. Kesavan, Measure and Integration, Texts and Readings in Mathematics 77, Hindustan Book Agency, 2019.

References

1. W. Rudin, Real and Complex Analysis, McGraw-Hill,
2. H. L. Royden, Real Analysis, Prentice-Hall of India,
3. R. B. Ash; C. A. Dol´eans-Dade, Probabilty and Measure Theory, Academic Press, 2nd Edition.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basic theory of probability starting from axiomatic definition of probability up to limit theorems of probability.

Prerequisite:

Syllabus:

1. 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; Simple symmetric random walk.

Text Books

1. S. Ross, "A First Course in Probability", Pearson Education,
2. D. Stirzaker, "Elementary Probability", Cambridge University Press, Cambridge, 2003.

References

1. K. L. Chung, F. AitSahlia, "Elementary Probability Theory", Undergraduate Texts in Mathematics. Springer-Verlag,
2. P. G. Hoel, S. C. Port, C. J. Stone, "Introduction to Probability Theory", The Houghton Mifflin Series in Statistics. Houghton Mifflin Co.,
3. W. Feller, "An Introduction to Probability Theory and its Applications Vol. 1 and Vol. 2", John Wiley & Sons, 1968, 1971.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the concept of (complex) differentiation and integration of functions defined on the complex plane and their properties.

Prerequisite:

Syllabus:

1. Algebraic and geometric representation of complex numbers; elementary functions including the exponential functions and its relatives (log, cos, sin, cosh, sinh, etc.); concept of holomorphic (analytic) functions, complex derivative and the Cauchy-Riemann equations; harmonic functions. Conformal Mapping, Linear Fractional Transformations, Complex line integrals and Cauchy Integral formula, Representation of holomorpic functions in terms of power series, Morera’s theorem, Cauchy estimates and Liouville’s theorem, zeros of holomorphic functions, Uniform limits of holomorphic functions. Behaviour of holomorphic function near an isolated singularity, Laurent expansions, Counting zeros and poles, Argument principle, Rouche’s theorem, Calculus of residues and evaluation of integrals using contour integration. The Open Mapping theorem, Maximum Modulus Principle, Schwarz Lemma.

Text Books

1. J. B. Conway, "Functions of One Complex Variable", Narosa Publishing House,
2. R. E. Greene, S. G. Krantz, "Function Theory of One Complex Variable", American Mathematical Society, 2011.

References

1. W. Rudin, "Real and Complex Analysis", Tata McGraw-Hill,
2. L. V. Ahlfors, "Complex Analysis", Tata McGraw-Hill,
3. T. W. Gamelin, "Complex Analysis", Undergraduate Texts in Mathematics, Springer,
4. E. M. Stein, R. Shakarchi, "Complex Analysis", Princeton University Press, 2003.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the fundamentals of graph theory and learning the structure of graphs and techniques used to analyze different problems.

Prerequisite:

Syllabus:

1. Graphs, subgraphs, graph isomorphisms, degree sequence, paths, cycles, trees, bipartite graphs, Hamilton cycles, Euler tours, directed graphs, matching, Tutte’s theorem, connectivity, Menger’s theorem, planar graphs, Kuratowski’s theorem, vertex and edge colouring of graphs, network flows, max-flow min-cut theorem, Ramsey theory for graphs, matrices associated with graphs.

Text Books

1. R. Diestel, "Graph Theory", Graduate Texts in Mathematics,
2. Springer, 2010.

References

1. B. Bollobas, "Modern Graph Theory", Graduate Texts in Mathematics,
2. Springer-Verlag,
3. F. Harary, "Graph Theory", Addison-Wesley Publishing Co.,
4. J. A. Bondy, U. S. R. Murty, "Graph Theory", Graduate Texts in Mathematics,
5. Springer, 2008.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the fundamentals of graph theory and learning the structure of graphs and techniques used to analyze different problems

Prerequisite:

Syllabus:

1. Graphs, subgraphs, graph isomorphisms, degree sequence, paths, cycles, trees, bipartite graphs, Hamilton cycles, Euler tours, directed graphs, matching, Tutte’s theorem, connectivity, Menger’s theorem, planar graphs, Kuratowski’s theorem, vertex and edge colouring of graphs, network flows, max-flow min-cut theorem, Ramsey theory for graphs, matrices associated with graphs.

Text Books

1. R. Diestel, "Graph Theory", Graduate Texts in Mathematics,
2. Springer, 2010.

References

1. B. Bollobas, "Modern Graph Theory", Graduate Texts in Mathematics,
2. Springer-Verlag,
3. F. Harary, "Graph Theory", Addison-Wesley Publishing Co.,
4. J. A. Bondy, U. S. R. Murty, "Graph Theory", Graduate Texts in Mathematics,
5. Springer, 2008.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the concept of normed linear space and various properties of operators defined on them.

Prerequisite:

Syllabus:

1. Normed linear spaces and continuous linear transformations, Hahn-Banach theorem (analytic and geometric versions), Baire’s theorem and its consequences - three basic principles of functional analysis (open mapping theorem, closed graph theorem and uniform boundedness principle), Computing the dual of wellknown Banach spaces, Hilbert spaces, Riesz representation theorem, Adjoint operator, Compact operators, Spectral theorem for compact self adjoint operators.

Text Books

1. J. B. Conway, "A Course in Functional Analysis", Graduates Texts in Mathematics 96, Springer,
2. B. Bollob´as, "Linear Analysis", Cambridge University Press, 1999.

References

1. G. F. Simmons, "Introduction to Topology and Modern Analysis", Tata McGraw-Hill, 2013.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the concept of normed linear space and various properties of operators defined on them.

Prerequisite:

Syllabus:

1. Normed linear spaces and continuous linear transformations, Hahn-Banach theorem (analytic and geometric versions), Baire’s theorem and its consequences - three basic principles of functional analysis (open mapping theorem, closed graph theorem and uniform boundedness principle), Computing the dual of wellknown Banach spaces, Hilbert spaces, Riesz representation theorem, Adjoint operator, Compact operators, Spectral theorem for compact self adjoint operators.

Text Books

1. J. B. Conway, "A Course in Functional Analysis", Graduates Texts in Mathematics 96, Springer,
2. B. Bollob´as, "Linear Analysis", Cambridge University Press, 1999.

References

1. G. F. Simmons, "Introduction to Topology and Modern Analysis", Tata McGraw-Hill, 2013.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding of the basic theory of modules, category and functors, algebras.

Prerequisite:

Syllabus:

1. Modules, submodules, module homomorphisms, quotient modules, isomorphism theorems, Direct Sum of modules, finitely generated modules, Free modules, structure theorem of finitely generated modules over PID. Tensor product of modules. Over commutative rings with identity: Categories and Functors, exact functors, Hom and Tensor functors, Localization of modules, Direct and Inverse Limit of modules, Projective, Injective and Flat modules, Ext, Tor. Algebras, Tensor Algebras, Symmetric Algebras, Exterior Algebras, Determinants. Length of Modules, Noetherian and Artinian modules, Hilbert Basis Theorem.

Text Books

1. Dummit, D.S.; Foote, R.M.; "Abstract Algebra", Third Edition, John Wiley & Sons.
2. Rotman, J.; "An Introduction to Homological Algebra", Springer,
3. Sing, Balwant; "Basic Commutative Algebra", World Scientific, 2011.

References

1. Lang, S.; "Algebra", Revised Third Edition, Springer, GTM
2. Weibel, Charles A.; "An Introduction to Homological Algebra", Cambridge University Press,
3. Atiyah, M.F.; McDonald, I.G.; "Introduction to Commutative Algebra", CRC Press, 2018.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning abstract notion of topological spaces, continuous functions between topological spaces, Ursysohn Lemma, Tietze extension theorem and Tychonoff Theorem which they have learned in a particular setting of "Metric Space"
• Lerning basic notions of fundamental groups and covering spaces and some of its applications

Prerequisite:

Syllabus:

1. Topological Spaces, Open and closed sets, Interior, Closure and Boundary of sets, Basis for Topology, Product Topology, Subspace Topology, Metric Topology, Compact Spaces, Locally compact spaces, Continuous functions, Open map, Homeomorphisms, Function Spaces, Separation Axioms: T1, Hausdorff, regular, normal spaces; Uryshon’s lemma, Tietze Extension Theorem, One point compactification, Connected Spaces, Path Connected Spaces, Quotient Topology, Homotopic Maps, Deformation Retract, Contractible Spaces, Fundamental Group, The Brouwer fixed-point theorem.

Text Books

1. J. R.Munkres, "Topology", Prentice-Hall of India,
2. M. A. Armstrong, "Basic Topology", Undergraduate Texts in Mathematics, Springer-Verlag, 1983.

References

1. J. L. Kelley, "General Topology", Graduate Texts in Mathematics, No.
2. Springer-Verlag, New York-Berlin,
3. K. J¨anich, "Topology", Undergraduate Texts in Mathematics. Springer-Verlag,
4. W. G. Chinn, N. E. Steenrod, "First concepts of Topology", The Mathematical Association of America, 1978.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning abstract notion of topological spaces, continuous functions between topological spaces, Ursysohn Lemma, Tietze extension theorem and Tychonoff Theorem which they have learned in a particular setting of "Metric Space".
• Lerning basic notions of fundamental groups and covering spaces and some of its applications

Prerequisite:

Syllabus:

1. Topological Spaces, Open and closed sets, Interior, Closure and Boundary of sets, Basis for Topology, Product Topology, Subspace Topology, Metric Topology, Compact Spaces, Locally compact spaces, Continuous functions, Open map, Homeomorphisms, Function Spaces, Separation Axioms: T1, Hausdorff, regular, normal spaces; Uryshon’s lemma, Tietze Extension Theorem, One point compactification, Connected Spaces, Path Connected Spaces, Quotient Topology, Homotopic Maps, Deformation Retract, Contractible Spaces, Fundamental Group, The Brouwer fixed-point theorem.

Text Books

1. J. R.Munkres, "Topology", Prentice-Hall of India,
2. M. A. Armstrong, "Basic Topology", Undergraduate Texts in Mathematics, Springer-Verlag, 1983.

References

1. J. L. Kelley, "General Topology", Graduate Texts in Mathematics, No.
2. Springer-Verlag, New York-Berlin,
3. K. J¨anich, "Topology", Undergraduate Texts in Mathematics. Springer-Verlag,
4. W. G. Chinn, N. E. Steenrod, "First concepts of Topology", The Mathematical Association of America, 1978.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge on curves and surfaces, manifold and vector field some application on geometry of surfaces.

Prerequisite:

Syllabus:

1. Curves in two and three dimensions, Curvature and torsion for space curves, Existence theorem for space curves, Serret-Frenet formula for space curves, Jacobian theorem, Surfaces in R3 as 2-dimensional manifolds, Tangent spaces and derivatives of maps between manifolds, Geodesics, First fundamental form, Orientation of a surface, Second fundamental form and the Gauss map, Mean curvature, Gaussian Curvature, Differential forms, Integration on surfaces, Stokes formula, Gauss-Bonnet theorem.

Text Books

1. M. P. Do Carmo, "Differential Geometry of Curves and Surfaces", Prentice Hall,
2. Andrew Pressley, "Elementary Differential Geometry", Springer, 2010.

References

1. M. P. Do Carmo, "Differential Forms and Applications", Springer,
2. J. A. Thorpe, "Elementary Topics in Differential Geometry", Undergraduate texts in mathematics, Springer, 2011.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the explicit representations of solutions of four important classes of PDEs, namely, Transport equations, Heat equation, Laplace equation and wave equation for initial value problems.
• Learning the properties of solutions of these equations such as mean value property, maximum principles and regularity.
• Understanding Cauchy-Kowalevski Theorem and uniqueness theorem of Holmgreen for quasilinear equations.

Prerequisite:

Syllabus:

1. Classification of Partial Differential Equations, Cauchy Problem, Cauchy-Kowalevski Theorem, Lagrange-Green identity, The uniquness theorem of Holmgren.
2. Transport equation: Initial value problem, nonhomogeneous problem.
3. Laplace equation: Fundamental solution, Mean Value formula, properties of Harmonic functions, Green’s function, Energy methods, Harnack’s inequality.
4. Heat Equation: Fundamental solution, Mean value formula, properties of solutions.
5. Wave equation: Solution by spherical means, Nonhomogeneous problem, properties of solutions. Textbooks:
6. L. C. Evans, "Partial Differential Equations", Graduate Studies in Mathematics 19, American Mathematical Society,
7. F. John, "Partial Differential Equations", Springer International Edition, 2009.

References

1. G. B. Folland, "Introduction to Partial Differential Equations", Princeton University Press,
2. S. Kesavan, "Topics in Functional Analysis and Applications", John Wiley & Sons, 1989.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basic properties of fields including the fundamental theorem of Galois theory.

Prerequisite:

Syllabus:

1. Field extensions, algebraic extensions, Ruler and compass constructions, splitting fields, algebraic closures, separable and inseparable extensions, cyclotomic polynomials and extensions, automorphism groups and fixed fields, Galois extensions, Fundamental theorem of Galois theory, Fundamental theorem of algebra, Finite fields, Galois group of polynomials, Computations of Galois groups over rationals, Solvable groups, nilpotent groups, Solvability by radicals, Transcendental extensions.

Text Books

1. D. S. Dummit, R. M. Foote, "Abstract Algebra", Wiley-India edition, 2013.

References

1. I. N. Herstein, "Topics in Algebra", Wiley-India edition,
2. M. Artin, "Algebra", Prentice-Hall of India,
3. J. Rotman, "Galois Theory", Universitext, Springer-Verlag,
4. S. Lang, "Algebra", Revised Third Edition. Spinger.

Course: iPhD-Core

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basics of fundamental group (π1) and singular homology.
• Learning different techniques to compute the fundamental group such as homotopy invariance and Van-Kampen Theorem.
• Learning different techniques to compute singular homology of a space, including homotopy invariance, Mayer-Vietoris, excision, long exact sequence.

Prerequisite:

Syllabus:

1. Homotopy Theory: Simply Connected Spaces, Covering Spaces, Universal Covering Spaces, Deck Transformations, Path lifting lemma, Homotopy lifting lemma, Group Actions, Properly discontinuous action, free groups, free product with amalgamation, Seifert-Van Kampen Theorem, Borsuk-Ulam Theorem for sphere, Jordan Separation Theorem. Homology Theory: Simplexes, Simplicial Complexes, Triangulation of spaces, Simplicial Chain Complexes, Simplicial Homology, Singular Chain Complexes, Cycles and Boundary, Singular Homology, Relative Homology, Short Exact Sequences, Long Exact Sequences, Mayer-Vietoris sequence, Excision Theorem, Invariance of Domain.

Text Books

1. J. R. Munkres, "Topology", Prentice-Hall of India,
2. A. Hatcher, "Algebraic Topology", Cambridge University Press, 2009.

References

1. G. E. Bredon, "Topology and Geometry", Graduates Texts in Mathematics 139, Springer, 2009.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the representation of finite groups via character theory.

Prerequisite:

Syllabus:

1. Group representations, Maschke’s theorem and completely reducibility, Characters, Inner product of Characters, Orthogonality relations, Burnside’s theorem, induced characters, Frobenius reciprocity, induced representations, Mackey’s Irreducibility Criterion, Character table of some well-known groups, Representation theory of the symmetric group: partitions and tableaux, constructing the irreducible representations.

Text Books

1. G. James, M. Liebeck, "Representations and Characters of Groups", Cambridge University Press, 2010.

References

1. J. L. Alperin, R. B. Bell, "Groups and Representations", Graduate Texts in Mathematics 162, Springer,
2. B. Steinberg, "Representation Theory of Finite Groups", Universitext, Springer,
3. J-P. Serre, "Linear Representations of Finite Groups", Graduate Texts in Mathematics 42, Springer-Verlag,
4. B. Simon, "Representations of Finite and Compact Groups", Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Leraning some important theorems in complex analysis such as Riemann mapping theorem, Weirstrass factorization theorem, Runge’s theorem, Hardamard factorization theorem, Little Picard’s theorem and Great Picard’s theorem.
• Learning some basic techniques of harmonic functions and characterization of Dirichlet Region.

Prerequisite:

Syllabus:

1. Review of basic Complex Analysis: Cauchy-Riemann equations, Cauchy’s theorem and estimates, power series expansions, maximum modulus principle, Classification of singularities and calculus of residues. Space of continuous functions, Arzela’s theorem, Spaces of analytic functions, Spaces of meromorphic functions, Riemann mapping theorem, Weierstrass Factorization theorem, Runge's theorem, Simple connectedness, Mittag-Leffler's theorem, Analytic continuation, Schwarz reflection principle, Mondromy theorem, Jensen’s formula, Genus and order of an entire function, Hadamard factorization theorem, Little Picard theorem, Great Picard theorem, Harmonic functions.

References

1. L. V. Ahlfors, "Complex Analysis", Tata McGraw-Hill,
2. J. B. Conway, "Functions of One Complex Variable II", Graduate Texts in Mathematics 159, Springer-Verlag,
3. W. Rudin, "Real and Complex Analysis", Tata McGraw-Hill,
4. R. Remmert, "Theory of Complex Functions", Graduate Texts in Mathematics 122, Springer, 2008.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the concept of topological vector space, as a generalisation of normed linear spaces, and various properties of operators defined on them.

Prerequisite:

Syllabus:

1. Definition and examples of topological vector spaces (TVS) and locally convex spaces (LCS); Linear operators; Hahn-Banach Theorems for TVS/ LCS (analytic and geometric forms); Uniform boundedness principle; Open mapping theorem; Closed graph thoerem; Weak and weak* vector topologies; Bipolar theorem; dual of LCS spaces; Krein-Milman theorem for TVS; Krien-Smulyan theorem for Banach spaces; Inductive and projective limit of LCS.

References

1. W. Rudin, "Functional Analysis", Tata McGraw-Hill,
2. A. P. Robertson, W. Robertson, "Topological Vector Spaces", Cambridge Tracts in Mathematics 53, Cambridge University Press,
3. J. B. Conway, "A Course in Functional Analysis", Graduates Texts in Mathematics 96, Springer, 2006.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the theory of discrete time and continuous time Markov chains.

Prerequisite:

Syllabus:

1. 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.

References

1. P. G. Hoel, S. C. Port, C. J. Stone, "Introduction to Stochastic Processes", Houghton Mifflin Co.,
2. R. Durrett, "Essentials of Stochastic Processes", Springer Texts in Statistics, Springer,
3. G. R. Grimmett, D. R. Stirzaker, "Probability and Random Processes", Oxford University Press,
4. S. M. Ross, "Stochastic Processes", Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, 1996

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the fundamentals of classical algebraic geometry.
• Learning about the theory of Riemann surfaces, divisors, line bundles, Chern Classes and the Riemann Roch Theorem.

Prerequisite:

Syllabus:

1. Prime ideals and primary decompositions, Ideals in polynomial rings, Hilbert Basis theorem, Noether normalisation lemma, Hilbert’s Nullstellensatz, Affine and Projective varieties, Zariski Topology, Rational functions and morphisms, Elementary dimension theory, Smoothness, Curves, Divisors on curves, Bezout’s theorem, Riemann-Roch for curves, Line bundles on Projective spaces.

References

1. K. Hulek, "Elementary Algebraic Geometry", Student Mathematical Library 20, American Mathematical Society,
2. I. R. Shafarevich, "Basic Algebraic Geometry 1: Varieties in Projective Space", Springer,
3. J. Harris, "Algebraic geometry", Graduate Texts in Mathematics 133, Springer-Verlag,
4. M. Reid, "Undergraduate Algebraic Geometry", London Mathematical Society Student Texts 12, Cambridge University Press,
5. K. E. Smith et. al., "An Invitation to Algebraic Geometry", Universitext, Springer-Verlag,
6. R. Hartshorne, "Algebraic Geometry", Graduate Texts in Mathematics 52, Springer-Verlag, 1977.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the different algebraic techniques used in the study of the graphs

Prerequisite:

Syllabus:

1. Adjacency matrix of a graph and its eigenvalues, Spectral radius of graphs, Regular graphs and Line graphs, Strongly regular graphs, Cycles and Cuts, Laplacian matrix of a graph, Algebraic connectivity, Laplacian spectral radius of graphs, Distance matrix of a graph, General properties of graph automorphisms, Transitive and Arc-tranisitve graphs, Symmetric graphs.

References

1. N. Biggs, "Algebraic Graph Theory", Cambridge University Press,
2. C. Godsil, G. Royle, "Algebraic Graph Theory", Graduate Texts in Mathematics 207, Springer-Verlag,
3. R. B. Bapat, "Graphs and Matrices", Universitext, Springer, Hindustan Book Agency, New Delhi, 2010.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basic properties of number fields, computation of class numbers and zeta functions.

Prerequisite:

Syllabus:

1. Number Fields and Number rings, prime decomposition in number rings, Dedekind domains, Ideal class group, Galois theory applied to prime decomposition, Gauss reciprocity law, Cyclotomic fields and their ring of integers, finiteness of ideal class group, Dirichlet unit theorem, valuations and completions of number fields, Dedekind zeta function and distribution of ideal in a number ring.

References

1. D. A. Marcus, "Number Fields", Universitext, Springer-Verlag,
2. G. J. Janusz, "Algebraic Number Fields", Graduate Studies in Mathematics 7, American Mathematical Society,
3. S. Alaca, K. S. Williams, "Introductory Algebraic Number Theory", Cambridge University Press,
4. S. Lang, "Algebraic Number Theory", Graduate Texts in Mathematics 110, Springer-Verlag,
5. A. Frohlich, M. J. Taylor, "Algebraic Number Theory", Cambridge Studies in Advanced Mathematics 27, Cambridge University Press,
6. J. Neukirch, "Algebraic Number Theory", Springer-Verlag, 1999.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning data structure, design and analysis algorithms.
• Understanding some important algorithms like sortings, graph theoretics, polynomial related and optimization related.

Prerequisite:

Syllabus:

1. Algorithm analysis, asymptotic notation, probabilistic analysis; Data Structure: stack, queues, linked list, hash table, binary search tree, red-black tree; Sorting: heap sort, quick sort, sorting in linear time; Algorithm design: divide and conquer, greedy algorithms, dynamic programming; Algebraic algorithms: Winograd’s and Strassen’s matrix multiplication algorithm, evaluation of polynomials, DFT, FFT, efficient FFT implementation; Graph algorithms: breadth-first and depth-first search, minimum spanning trees, single-source shortest paths, all-pair shortest paths, maximum flow; NP-completeness and approximation algorithms.

References

1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, "The Design and Analysis of Computer Algorithms", Addison-Wesley Publishing Co.,
2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to Algorithms", MIT Press, Cambridge,
3. E. Horowitz, S. Sahni, "Fundamental of Computer Algorithms", Galgotia Publication,
4. D. E. Knuth, "The Art of Computer Programming Vol. 1, Vol. 2, Vol 3", Addison-Wesley Publishing Co., 1997, 1998, 1998.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of cryptography and cryptanalysis.
• Understanding the theory and design of cryptographic schemes like stream ciphers, block ciphers and public key ciphers like RSA, El-Gamal, elliptic curve cryptosystem.
• Learning about data authentication, integrity and secret sharing.

Prerequisite:

Syllabus:

1. Overview of 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.

References

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

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the structures of finite fields, factorization of polynomials, some applications towards cryptography, coding theory and combinatorics.

Prerequisite:

Syllabus:

1. Structure of finite fields: characterization, roots of irreducible polynomials, traces, norms and bases, roots of unity, cyclotomic polynomial, representation of elements of finite fields, Wedderburn’s theorem; Polynomials over finite field: order of polynomials, primitive polynomials, construction of irreducible polynomials, binomials and trinomials, factorization of polynomials over small and large finite fields, calculation of roots of polynomials; Linear recurring sequences: LFSR, characteristic polynomial, minimal polynomial, characterization of linear recurring sequences, Berlekamp-Massey algorithm; Applications of finite fields: Applications in cryptography, coding theory, finite geometry, combinatorics.

References

1. R. Lidl, H. Neiderreiter, "Finite Fields", Cambridge university press,
2. G. L. Mullen, C. Mummert, "Finite Fields and Applications", American Mathematical Society,
3. A. J. Menezes et. al., "Applications of Finite Fields", Kluwer Academic Publishers,
4. Z-X. Wan, "Finite Fields and Galois Rings", World Scientific Publishing Co., 2012.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning how to measure information and encoding of information.
• Understanding the theory and techniques of error correcting codes like Reed-Muller codes, BCH codes, Reed-Solomon codes, Algebraic codes.

Prerequisite:

Syllabus:

1. Information Theory: Entropy, Huffman coding, Shannon-Fano coding, entropy of Markov process, channel and mutual information, channel capacity;
2. Error correcting codes: Maximum likelihood decoding, nearest neighbour decoding, linear codes, generator matrix and parity-check matrix, Hamming bound, Gilbert-Varshamov bound, binary Hamming codes, Plotkin bound, nonlinear codes, Reed-Muller codes, Cyclic codes, BCH codes, Reed-Solomon codes, Algebraic codes.

References

1. R. W. Hamming, "Coding and Information Theory", Prentice-Hall,
2. N. J. A. Sloane, F. J. MacWilliams, "Theory of Error Correcting Codes", North-Holland Mathematical Library 16, North-Holland,
3. S. Ling, C. Xing, "Coding Theory: A First Course", Cambridge University Press,
4. W. C. Haffman, V. Pless, "Fundamentals of Error-Coding Codes", Cambridge University Press,
5. S. Lin, "An Introduction to Error-Correcting Codes", Prentice-Hall, 1970.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the propositional logic and first order theory.
• Understanding the completeness and compactness theorems with Godel’s incompleteness theorem.

Prerequisite:

Syllabus:

1. Propositional Logic, Tautologies and Theorems of propositional Logic, Tautology Theorem. First Order Logic: First order languages and their structures, Proofs in a first order theory, Model of a first order theory, validity theorems, Metatheorems of a first order theory, e. g., theorems on constants, equivalence theorem, deduction and variant theorems etc. Completeness theorem, Compactness theorem, Extensions by definition of first order theories, Interpretations theorem, Recursive functions, Arithmatization of first order theories, Godel’s first Incompleteness theorem, Rudiments of model theory including Lowenheim-Skolem theorem and categoricity.

References

1. J. R. Shoenfield, "Mathematical logic", Addison-Wesley Publishing Co.,
2. E. Mendelson, "Introduction to Mathematical Logic", Chapman & Hall, 1997.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning calculus in Banach Spaces, degree theory and it’s application for fixed point theorems of Brouwer and Schauder.
• Learning homotopy, homotopy extension and invariance theorems and its applications.

Prerequisite:

Syllabus:

1. Calculus in Banach spaces, inverse and multiplicit function theorems, fixed point theorems of Brouwer, Schauder and Tychonoff, fixed point theorems for nonexpansive and set-valued maps, predegree results, compact vector fields, homotopy, homotopy extension, invariance theorems and applications.

References

1. S. Kesavan, "Nonlinear Functional Analysis", Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the concepts of C*-algebra, von-Neuman algebra and toeplitz operators and the notion of index for Fredholm operators.

Prerequisite:

Syllabus:

1. Compact operators on Hilbert Spaces. (a) Fredholm Theory (b) Index, C*-algebras - noncommutative states and representations, Gelfand-Neumark representation theorem, Von-Neumann Algebras; Projections, Double Commutant theorem, functionalCalculus, Toeplitz operators.

References

1. W. Arveson, "An invitation to C*-algebras", Graduate Texts in Mathematics, No.
2. Springer-Verlag,
3. N. Dunford and J. T. Schwartz, "Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space", Interscience Publishers John Wiley i& Sons
4. R. V. Kadison and J. R. Ringrose, "Fundamentals of the theory of operator algebras. Vol. I. Elementary theory", Pure and Applied Mathematics, 100, Academic Press, Inc.,
5. V. S. Sunder, "An invitation to von Neumann algebras", Universitext, Springer-Verlag, 1987.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning Automata and Language theory by studying automata and context free language.
• Learning Computability theory by studying Turing machine and halting problem.
• Learning Complexity theory by studying P and NP class problems.

Prerequisite:

Syllabus:

1. 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.

References

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

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge on Haar measure, convolution structure on Lie group with emphasize to harmonic analysis on the groups Circle and real line.

Prerequisite:

Syllabus:

1. Topological Groups: Basic properties of topological groups, subgroups, quotient groups. Examples of various matrix groups. Connected groups.
2. Haar measure: Discussion of Haar measure without proof on ℝ, 𝕋, ℤ and simple matrix groups, Convolution, the Banach algebra L1(G) and convolution with special emphasis on L1(ℝ), L1(𝕋), L1(ℤ).
3. Basic Representation Theory: Unitary representation of groups, Examples and General properties, The representations of Group and Group algebras, C*-algebra of a group, GNS construction, Positive definite functions, Schur’s Lemma.
4. Abelian Groups: Fourier transform and its properties, Approximate identities in L1(G) , Classical Kernels on ℝ, The Fourier inversion Theorem, Plancherel theorem on ℝ, Plancherel measure on ℝ, 𝕋, ℤ.
5. Dual Group of an Abelian Group: The Dual group of a locally compact abelian group, Computation of dual groups for ℝ, 𝕋, ℤ Pontryagin’s Duality theorem.

References

1. G. B. Folland, "A Course in Abstract Harmonic Analysis", CRC Press,
2. H. Helson, "Harmonic Analysis", Texts and Readings in Mathematics, Hindustan Book Agency,
3. Y. Katznelson, "An Introduction to Harmonic Analysis", Cambridge University Press,
4. L. H. Loomis, "An Introduction to Abstract Harmonic Analysis", Dover Publication,
5. E. Hewitt, K. A. Ross, "Abstract Harmonic Analysis Vol. I", Springer-Verlag,
6. W. Rudin, "Real and Complex Analysis", Tata McGraw-Hill, 2013.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning 𝑝-adic numbers, quadratic forms, Dirichlet series and modular forms.

Prerequisite:

Syllabus:

1. Review of Finite fields, Gauss Sums and Jacobi Sums, Cubic and biquadratic reciprocity, Polynomial equations over finite fields, Theorems of Chevally and Warning, Quadratic forms over prime fields. Ring of 𝑝-adic integers, Field of 𝑝-adic numbers, completion, 𝑝-adic equations, Hensel’s lemma, Hilbert symbol, Quadratic forms with 𝑝-adic coefficients. Dirichlet series: Abscissa of convergence and absolute convergence, Riemann Zeta function and Dirichlet L-functions. Dirichlet’s theorem on primes in arithmetic progression. Functional equation and Euler product for L-functions. Modular Forms and the Modular Group, Eisenstein series, Zeros and poles of modular functions, Dimensions of the spaces of modular forms, The 𝑗-invariant L-function associated to modular forms, Ramanujan τ function.

References

1. J.-P. Serre, "A Course in Arithmetic", Graduate Texts in Mathematics 7, Springer-Verlag,
2. K. Ireland, M. Rosen, "A Classical Introduction to Modern Number Theory", Graduate Texts in Mathematics 84, Springer-Verlag,
3. H. Hasse, "Number Theory", Classics in Mathematics, Springer-Verlag,
4. W. Narkiewicz, "Elementary and Analytic Theory of Algebraic Numbers", Springer Monographs in Mathematics, Springer-Verlag,
5. F. Q. Gouvêa, " -adic Numbers", Universitext, Springer-Verlag, 1997.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about measure theoretic probability starting from probability spaces to theory of martingales.

Prerequisite:

Syllabus:

1. 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.

References

1. A. Gut, "Probability: A Graduate Course", Springer Texts in Statistics, Springer,
2. K. L. Chung, "A Course in Probability Theory", Academic Press,
3. S. I. Resnick, "A Probability Path", Birkhauser,
4. P. Billingsley, "Probability and Measure", Wiley Series in Probability and Statistics, John Wiley & Sons,
5. J. Jacod, P. Protter, "Probability Essentials", Universitext, Springer-Verlag,
6. S. R. S. Varadhan, "Probability Theory", Courant Lecture Notes, Vol. 7, AMS, 2001.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the use of different algebraic technique to study the combinatorial problems

Prerequisite:

Syllabus:

1. Catalan Matrices and Orthogonal Polynomials, Catalan Numbers and Lattice Paths, Combinatorial Interpretation of Catalan Numbers, Symmetric Polynomials and Functions, Schur Functions, Jacobi-Trudi identity, RSK Algorithm, Standard Tableaux, Young diagrams and -binomial coefficients, Plane Partitions, Group actions on boolean algebras, Enumeration under group action, Walks in graphs, Cubes and the Radon transform, Sperner property, Matrix-Tree Theorem.

References

1. R. P. Stanley, "Algebraic Combinatorics", Undergraduate Texts in Mathematics, Springer,
2. M. Aigner, "A Course in Enumeration", Graduate Texts in Mathematics 238, Springer,
3. R. P. Stanley, "Enumerative Combinatorics Vol. 2", Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 1999.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the theoretical study of cryptography which puts foundation for the study and design of real-life cryptography.

Prerequisite:

Syllabus:

1. Introduction to cryptography and computational model, computational difficulty, pseudorandom generators, zero-knowledge proofs, encryption schemes, digital signature and message authentication schemes, cryptographic protocol.

References

1. O. Goldreich, "Foundations of Cryptography - Vol. I and Vol. II", Cambridge University Press, 2001,
2. S. Goldwasser, Mihir Bellare, "Lecture Notes on Cryptography", 2008, available online from http://cseweb.ucsd.edu/ mihir/papers/gb.html

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding different kinds of incidence structures such as projective spaces, affine spaces, generalized quadrangles, polar spaces and quadratic sets.

Prerequisite:

Syllabus:

1. Definitions and Exampleas, projective planes, affine planes, projective spaces, affine spaces, collineations of projective and affine spaces, fundamental theorem of projective and affine spaces, polar spaces, generalized quadrangles, quadrics and quadratic sets.

References

1. J. Ueberberg, "Foundations of Incidence Geometry", Springer Monographs in Mathematics, Springer,
2. L. M. Batten, "Combinatorics of Finite Geometries", Cambridge University Press,
3. Bart De Bruyn,"An Introduction to Incidence Geometry", Frontiers in Mathematics, Birkhauser/Springer, Cham
4. Gyorgy Kiss and Tamas Szonyi, "Finite Geometries", CRC Press, Boca Raton, FL
5. G. E. Moorhouse, "Incidence Geometry", 2007, available online from http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basics of Lie Algebra

Prerequisite:

Syllabus:

1. Definitions and Examples, Derivations, Ideals, Homomorphisms, Nilpotent Lie Algebras and Engel’s theorem, Solvable Lie Algebras and Lie’s theorem, Jordan decomposition and Cartan’s criterion, Semisimple Lie algebras, Casimir operator and Weyl’s theorem, Representations of , Root space decomposition, Abstract root systems, Weyl group and Weyl chambers, Classification of irreducible root systems, Abstract theory of weights, Isomorphism and conjugacy theorems, Universal enveloping algebras and PBW theorem, Representation theory of semi-simple Lie algebras, Verma modules and Weyl character formula.

References

1. J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Graduate Texts in Mathematics 9, Springer-Verlag,
2. K. Erdmann, M. J. Wildon, "Introduction to Lie Algebras", Springer Undergraduate Mathematics Series, Springer-Verlag,
3. J.-P. Serre, "Complex Semisimple Lie Algebras", Springer Monographs in Mathematics, Springer-Verlag,
4. N. Jacobson, "Lie Algebras", Dover Publications, 1979.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of distribution Theory, Sobolev Spaces and their properties.

Prerequisite:

Syllabus:

1. Distribution Theory, Sobolev Spaces, Embedding theorems, Trace theorem. Dirichlet, Neumann and Oblique derivative problem, Weak formulation, Lax–Milgram, Maximum Principles– Weak and Strong Maximum Principles, Hopf Maximum Principle, Alexandroff-Bakelmann-Pucci Estimate.

References

1. L. C. Evans, "Partial Differential Equations", Graduate Studies in Mathematics 19, American Mathematical Society,
2. H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations", Universitext, Springer,
3. R. A. Adams, J. J. F. Fournier, "Sobolev Spces", Pure and Applied Mathematics 140, Elsevier/Academic Press,
4. S. Kesavan, "Topics in Functional Analysis and Applications", John Wiley & Sons,
5. M. Renardy, R. C. Rogers, "An Introduction to Partial Differential Equations", Springer, 2008.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning random graphs and their applications.

Prerequisite:

Syllabus:

1. Models of random graphs and of random graph processes; illustrative examples; random regular graphs, configuration model; appearance of the giant component small subgraphs; long paths and Hamiltonicity; coloring problems; eigenvalues of random graphs and their algorithmic applications; pseudo-random graphs.

References

1. N. Alon, J. H. Spencer, "The Probabilistic Method", John Wiley & Sons, 2008
2. B. Bollobás, "Random Graphs", Cambridge Studies in Advanced Mathematics 73, Cambridge University Press,
3. S. Janson, T. Luczak, A. Rucinski, "Random Graphs", Wiley-Interscience,
4. R. Durrett, "Random Graph Dynamics", Cambridge University Press,
5. J. H. Spencer, "The Strange Logic of Random Graphs", Springer-Verlag, 2001.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning how to use probabilistic techniques to different areas of mathematics and computer science.

Prerequisite:

Syllabus:

1. Inequalities of Markov and Chebyshev (median algorithm), first and second moment method (balanced allocation), inequalities of Chernoff (permutation routing) and Azuma (chromatic number), rapidly mixing Markov chains (random walk in hypercubes, card shuffling), probabilistic generating functions (random walk in -dimensional lattice)

References

1. R. Motwani, P. Raghavan, "Randomized Algorithms", Cambridge University Press,
2. M. Mitzenmacher, E. Upfal, "Probability and Computing: Randomized algorithms and probabilistic analysis", Cambridge University Press, 2005.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge of smooth manifolds, tangent and cotangent spaces, vector bundles, (co)tangent bundles, vector fields, differential forms, exterior differentiation, De-Rham cohomology, integration on manifolds, homotopy invariance of De-Rham cohomology and the statement of Poincare Duality.

Prerequisite:

Syllabus:

1. Differentiable manifolds and maps: Definition and examples, Inverse and implicit function theorem, Submanifolds, immersions and submersions.
2. The tangent and cotangent bundle: Vector bundles, (co)tangent bundle as a vector bundle, Vector fields, flows, Lie derivative.
3. Differential forms and Integration: Exterior differential, closed and exact forms, Poincaré lemma, Integration on manifolds, Stokes theorem, De Rham cohomology.

References

1. Michael Spivak, "A comprehensive introduction to differential geometry", Vol. 1, 3rd edition,
2. Frank Warner, "Foundations of differentiable manifolds and Lie groups", Springer-Verlag, 2nd edition,
3. John Lee, "Introduction to smooth manifolds", Springer Verlag, 2nd edition,
4. Louis Auslander and Robert E. MacKenzie, "Introduction to differentiable manifolds", Dover, 2nd edition, 2009.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the various properties of commutative rings, various class of commutative rings, and dimension theory.

Prerequisite:

Syllabus:

1. Commutative rings, ideals, operations on ideals, prime and maximal ideals, nilradicals, Jacobson radicals, extension and contraction of ideals, Modules, free modules, projective modules, exact sequences, tensor product of modules, Restriction and extension of scalars, localization and local rings, extended and contracted ideals in rings of fractions, Noetherian modules, Artinian modules, Primary decompositions and associate primes, Integral extensions, Valuation rings, Discrete valuation rings, Dedekind domains, Fractional ideals, Completion, Dimension theory.

Text Books

1. M. F. Atiyah, I. G. Macdonald, "Introduction to Commutative Algebra", Addison-Wesley Publishing Co., 1969.

References

1. R. Y. Sharp, "Steps in Commutative Algebra", London Mathematical Society Student Texts,
2. Cambridge University Press,
3. D. S. Dummit, R. M. Foote, "Abstract Algebra", Wiley-India edition, 2013.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• It is a unique style of course where the mathematics students having interest in computation can learn to compute different algebraic problems in computer. Here students will learn the computation of the problems related (i) linear algebra, (ii) non-linear system of equations like Grobner bases, (iii) polynomial, (iv) algebraic number theory and (v) elliptic curve.

Prerequisite:

Syllabus:

1. Linear algebra and lattices: Asymptotically fast matrix multiplication algorithms, linear algebra algorithms, normal forms over fields, Lattice reduction; Solving system of non-linear equations: Gröbner basis, Buchberger’s algorithms, Complexity of Gröbner basis computation; Algorithms on polynomials: GCD, Barlekamp-Massey algorithm, factorization of polynomials over finite field, factorization of polynomials over and ; Algorithms for algebraic number theory: Representation and operations on algebraic numbers, trace, norm, characteristic polynomial, discriminant, integral bases, polynomial reduction, computing maximal order, algorithms for quadratic fields; Elliptic curves: Implementation of elliptic curve, algorithms for elliptic curves.

References

1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, "The Design and Analysis of Computer Algorithms", Addison-Wesley Publishing Co.,
2. H. Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics 138, Springer-Verlag,
3. D. Cox, J. Little, D. O’shea, "Ideals, Varieties and Algorithms: An introduction to computational algebraic geometry and commutative algebra", Undergraduate Texts in Mathematics, Springer-verlag, 2007.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the elementary properties of Dirichlet series and distribution of primes.

Prerequisite:

Syllabus:

1. Arithmetic functions, Averages of arithmetical functions, Distribution of primes, finite abelian groups and characters, Gauss sums, Dirichlet series and Euler products, Reimann Zeta function, Dirichlet L-functions, Analytic proof of the prime number theorem, Dirichlet Theorem on primes in arithmetic progression.

References

1. T. M. Apostol, "Introduction to Analytic Number Theory", Springer International Student Edition,
2. K. Chandrasekharan, "Introduction to Analytic Number Theory", Springer-Verlag,
3. H. Iwaniec, E. Kowalski, "Analytic Number Theory", American Mathematical Society Colloquium Publications 53, American Mathematical Society, 2004.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the basic facts about classical groups defined over fields such as General Linear groups, Special Linear groups, Symplectic groups, Orthogonal groups and Unitary groups.

Prerequisite:

Syllabus:

1. General and special linear groups, bilinear forms, Symplectic groups, symmetric forms, quadratic forms, Orthogonal geometry, orthogonal groups, Clifford algebras, Hermitian forms, Unitary spaces, Unitary groups.

References

1. L. C. Grove, "Classical Groups and Geometric Algebra", Graduate Studies in Mathematics 39, American Mathematical Society,
2. E. Artin, "Geometric Algebra", John Wiley & sons, 1988.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the basics of Ergodic Theory.

Prerequisite:

Syllabus:

1. Measure preserving systems; examples: Hamiltonian dynamics and Liouville’s theorem, Bernoulli shifts, Markov shifts, Rotations of the circle, Rotations of the torus, Automorphisms of the Torus, Gauss transformations, Skew-product, Poincare Recurrence lemma: Induced transformation: Kakutani towers: Rokhlin’s lemma. Recurrence in Topological Dynamics, Birkhoff’s Recurrence theorem, Ergodicity, Weak-mixing and strong-mixing and their characterizations, Ergodic Theorems of Birkhoff and Von Neumann. Consequences of the Ergodic theorem. Invariant measures on compact systems, Unique ergodicity and equidistribution. Weyl’s theorem, The Isomorphism problem; conjugacy, spectral equivalence, Transformations with discrete spectrum, Halmos–von Neumann theorem, Entropy. The Kolmogorov-Sinai theorem. Calculation of Entropy. The Shannon Mc-Millan–Breiman Theorem, Flows. Birkhoff’s ergodic Theorem and Wiener’s ergodic theorem for flows. Flows built under a function.

References

1. Peter Walters, "An introduction to ergodic theory", Graduate Texts in Mathematics,
2. Springer-Verlag,
3. Patrick Billingsley, "Ergodic theory and information", Robert E. Krieger Publishing Co.,
4. M. G. Nadkarni, "Basic ergodic theory", Texts and Readings in Mathematics,
5. Hindustan Book Agency,
6. H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory", Princeton University Press,
7. K. Petersen, "Ergodic theory", Cambridge Studies in Advanced Mathematics,
8. Cambridge University Press, 1989.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowldege on Fourier Series, Fourier transforms and celebrated differentiation theorem and important operators like Hilbert transform and Maximal function.

Prerequisite:

Syllabus:

1. Fourier series and its convergences, Dirichlet kernel, Frejer kernel, Parseval formula and its applications. Fourier transforms,the Schwartz space, Distribution and tempered distribution, Fourier Inversion and Plancherel theorem. Fourier analysis on -spaces. Maximal functions and boundedness of Hilbert transform. Paley-Wiener Theorem for distribution. Poisson summation formula, Heisenberg uncertainty Principle, Wiener’s Tauberian theorem.

References

1. Y. Katznelson, "An Introduction to Harmonic Analysis", Cambridge University Press,
2. E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton Mathematical Series 32, Princeton University Press,
3. G. B. Folland, "Fourier Analysis and its Applications", Pure and Applied Undergraduate Texts 4, America Mathematical Society,
4. A. Terras, "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane", Second Edition, Springer, 2013.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

Prerequisite:

Syllabus:

1. General Properties: Definition of Lie groups, subgroups, cosets, group actions on manifolds, homogeneous spaces, classical groups. Exponential and logarithmic maps, Adjoint representation, Lie bracket, Lie algebras, subalgebras, ideals, stabilizers, center Baker-Campbell-Hausdorff formula, Lie’s Theorems. Structure Theory of Lie Algebras: Solvable and nilpotent Lie algebras (with Lie/Engel theorems), semisimple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria, Jordan decomposition. Complex semisimple Lie algebras, Toral subalgebras, Cartan subalgebras, Root decomposition and root systems. Weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, BGG resolution, Weyl character formula.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. J. Faraut, "Analysis on Lie Groups", Cambridge Studies in Advanced Mathematics 110, Cambridge University Press,
3. B. C. Hall, "Lie Groups, Lie algebras and Representations", Graduate Texts in Mathematics 222, Springer-Verlag,
4. W. Fulton, J. Harris, "Representation Theory: A first course", Springer-Verlag,
5. J. E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Graduate Texts in Mathematics 9, Springer-Verlag,
6. A. Kirillov, "Introduction to Lie Groups and Lie Algebras", Cambridge Studies in Advanced Mathematics 113, Cambridge University Press,
7. V. S. Varadharajan, "Lie Groups, Lie Algebras and their Representations", Springer-Verlag, 1984.

Course: iPhD-Elective

Duration: 56 Outcome of the Course Learning the concepts and various structure theorems of C*-algebra and von- Hours

Credit: 4

Outcome:

Prerequisite:

Syllabus:

1. Banach algebras/C*–algebras: Definition and examples; Spectrum of a Banach algebra; Gelfand transform; Gelfand-Naimark theorem for commutative Banach algebras/ C*-algebras; Functional calculus for C*-algebras; Positive cone in a C*-algebra; Existance of an approximate identity in a C*-algebra; Ideals and Quotients of a C*-algebra; Positive linear functionals on a C*-algebra; GNS construction. Locally convex topologies on the algebras of bounded operators on a Hilbert space, von-Neumann’s bi-commutant theorem; Kaplansky’s density theorem. Ruan’s characterization of Operator Spaces (if time permites).

References

1. R. V. Kadison, J. R. Ringrose, "Fundamentals of the Theory of Operator Algebras Vol. I", Graduate Studies in Mathematics 15, American Mathematical Society,
2. G. K. Pedersen, "C*–algebras and their Automorphism Groups", London Mathematical Society Monographs 14, Academic Press,
3. V. S. Sunder, "An Invitation to von Neumann Algebras", Universitext, Springer-Verlag,
4. M. Takesaki, "Theory of Operator Algebras Vol. I", Springer-Verlag, 2002.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the first principles of representations and understanding the important examples of 3 different types of groups, viz., compact, nilpotent and solvable groups.

Prerequisite:

Syllabus:

1. Introduction to topological group, Haar measure on locally compact group, Representation theory of compact groups, Peter Weyl theorem, Linear Lie groups, Exponential map, Lie algebra, Invariant Differentail operators, Representation of the group and its Lie algebra. Fourier analysis on and . Representation theory of Heisenberg group . Representation of Euclidean motion group.

References

1. J. E. Humphreys, "Introduction to Lie algebras and representation theory", Springer-Verlag,
2. S. C. Bagchi, S. Madan, A. Sitaram, U. B. Tiwari, "A first course on representation theory and linear Lie groups", University Press,
3. Mitsou Sugiura, "Unitary Representations and Harmonic Analysis", John Wiley & Sons,
4. Sundaram Thangavelu, "Harmonic Analysis on the Heisenberg Group", Birkhauser,
5. Sundaram Thangavelu, "An Introduction to the Uncertainty Principle", Birkhauser, 2003.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Knowledge on representaion on compact lie groups with examples SU(2), SO(n).

Prerequisite:

Syllabus:

1. Review of General Theory: Locally compact groups, Computation of Haar measure on ℝ, 𝕋, SU(2), SO(3) and some simple matrix groups, Convolution, the Banach algebra L1(G).
2. Representation Theory: General properties of representations of a locally compact group, Complete reducibility, Basic operations on representations, Irreducible representations.
3. Representations of Compact groups: Unitarilzibality of representations, Matrix coefficients, Schur's orthogonality relations, Finite dimensionality of irreducible representations of compact groups. Various forms of Peter-Weyl theorem, Fourier analysis on Compact groups, Character of a representation. Schur’s orthogonality relations among characters. Weyl’s Chracter formula, Computing the Unitary dual of SU(2), SO(3); Fourier analysis on SO(n).

References

1. T. Brocker, T. Dieck, "Representations of Compact Lie Groups", Springer-Verlag,
2. J. L. Clerc, "Les Représentatios des Groupes Compacts, Analyse Harmonique" (J. L. Clerc et. al., ed.), C.I.M.P.A.,
3. G. B. Folland, "A Course in Abstract Harmonic Analysis", CRC Press,
4. M. Sugiura, "Unitary Representations and Harmonic Analysis", John Wiley & Sons,
5. E. B. Vinberg, "Linear Representations of Groups", Birkhäuser/Springer,
6. A. Wawrzyńczyk, "Group Representations and Special Functions", PWN–Polish Scientific Publishers, 1984.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning modular forms over SL2 and their congruence subgroups, and their Hecke theory.

Prerequisite:

Syllabus:

1. SL2(ℝ) and its congruence subgroups, Modular forms for SL2(ℝ), Modular forms for congruence subgroups, Modular forms and differential operators, Hecke theory, L-series, Theta functions and transformation formula.

References

1. J.-P. Serre, "A Course in Arithmetic", Graduate Texts in Mathematics 7, Springer-Verlag,
2. N. Koblitz, "Introduction to Elliptic Curves and Modular Forms", Graduate Texts in Mathematics 97, Springer-Verlag,
3. J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier, "The 1-2-3 of Modular Forms", Universitext, Springer-Verlag,
4. F. Diamond, J. Shurman, "A First Course in Modular Forms", Graduate Texts in Mathematics 228, Springer-Verlag,
5. S. Lang, "Introduction to Modular Forms", Springer-Verlag,
6. G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Forms", Princeton University Press, 1994.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning elliptic curves and the structure of their rational points.

Prerequisite:

Syllabus:

1. Congruent numbers, Elliptic curves, Elliptic curves in Weierstrass form, Addition law, Mordell–Weil Theorem, Points of finite order, Points over finite fields, Hasse-Weil L-function and its functional equation, Complex multiplication.

References

1. J. H. Silverman, J. Tate, "Rational Points on Elliptic Curves", Undergraduate Texts in Mathematics, Springer-Verlag,
2. N. Koblitz, "Introduction to Elliptic Curves and Modular Forms", Graduate Texts in Mathematics 97, Springer-Verlag,
3. J. H. Silverman, "The Arithmetic of Elliptic Curves", Graduate Texts in Mathematics 106, Springer,
4. A. W. Knapp, "Elliptic Curves", Mathematical Notes 40, Princeton University Press,
5. J. H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Graduate Texts in Mathematics 151, Springer-Verlag, 1994.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about the theory of Brownian motion and it applications to stochastic differential equations.

Prerequisite:

Syllabus:

1. Brownian Motion, Martingale, Stochastic integrals, extension of stochastic integrals, stochastic integrals for martingales, Itô’s formula, Application of Itô’s formula, stochastic differential equations.

References

1. H. H. Kuo, "Introduction to Stochastic Integration", Springer,
2. J. M Steele, "Stochastic Calculus and Financial Applications", Springer-Verlag,
3. F. C. Klebaner, "Introduction to Stochastic Calculus with Applications", Imperial College, 2005.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the fundamentals of Lie groups and Lie Algebras.
• Learning about (bi)invariant vector fields, integration on Lie Groups, Cartan’s Theorem.

Prerequisite:

Syllabus:

1. Review of Several variable Calculus: Directional Derivatives, Inverse Function Theorem, Implicit function Theorem, Level sets in ℝn, Taylor’s theorem, Smooth function with compact support.
2. Manifolds: Differentiable manifold, Partition of Unity, Tangent vectors, Derivative, Lie groups, Immersions and submersions, Submanifolds.
3. Vector Fields: Left invariant vector fields of Lie groups, Lie algebra of a Lie group, Computing the Lie algebra of various classical Lie groups.
4. Flows: Flows of a vector field, Taylor’s formula, Complete vector fields.
5. Exponential Map: Exponential map of a Lie group, One parameter subgroups, Frobenius theorem (without proof).
6. Lie Groups and Lie Algebras: Properties of Exponential function, product formula, Cartan’s Theorem, Adjoint representation, Uniqueness of differential structure on Lie groups.
7. Homogeneous Spaces: Various examples and Properties.
8. Coverings: Covering spaces, Simply connected Lie groups, Universal covering group of a connected Lie group. Finite dimensional representations of Lie groups and Lie algebras.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces", Graduate Studies in Mathematics 34, American Mathematical Society,
3. S. Kumaresan, "A Course in Differential Geometry and Lie Groups", Texts and Readings in Mathematics 22, Hindustan Book agency,
4. F. W. Warner, "Foundations of Differentiable Manifolds and Lie Groups", Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning the representation theory of compact Lie groups and the group SL(2,C).
• Learning classifications of all simple Lie algebras through root system.

Prerequisite:

Syllabus:

1. General theory of representations, operations on representations, irreducible representations, Schur’s lemma, Unitary representations and complete reducibility. Compact Lie groups, Haar measure on compact Lie groups, Schur’s Theorem, characters, Peter-Weyl theorem, universal enveloping algebra, Poincare-Birkoff-Witt theorem, Representations of Lie(SL(2,C)). Abstract root systems, Weyl group, rank 2 root systems, Positive roots, simple roots, weight lattice, root lattice, Weyl chambers, simple reflections, Dynkin diagrams, classification of root systems, Classification of semisimple Lie algebras. Representations of Semisimple Lie algebras, weight decomposition, characters, highest weight representations, Verma modules, Classification of irreducible finite-dimensional representations, Weyl Character formula, The representation theory of SU(3), Frobenius Reciprocity theorem, Spherical Harmonics.

References

1. D. Bump, "Lie Groups", Graduate Texts in Mathematics 225, Springer,
2. J. Faraut, "Analysis on Lie Groups", Cambridge Studies in Advanced Mathematics 110, Cambridge University Press,
3. B. C. Hall, "Lie Groups, Lie algebras and Representations", Graduate Texts in Mathematics 222, Springer-Verlag,
4. W. Fulton, J. Harris, "Representation Theory: A first course", Springer-Verlag,
5. A. Kirillov, "Introduction to Lie Groups and Lie Algebras", Cambridge Studies in Advanced Mathematics 113, Cambridge University Press,
6. A. W. Knapp, "Lie Groups: Beyond an introduction", Birkäuser,
7. B. Simon, "Representations of Finite and Compact Groups", Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Learning about the mathematical modeling of simple stock markets and techniques to analyze them.

Prerequisite:

Syllabus:

1. Financial market models in finite discrete time, Absence of arbitrage and martingale measures, Valuation and hedging in complete markets, Basic facts about Brownian motion, Stochastic integration, Stochastic calculus: Itô’s formula, Girsanov transformation, Itô’s representation theorem, Black-Scholes formula

References

1. J. Jacod, P. Protter, "Probability Essentials", Universitext, Springer-Verlag,
2. D. Lamberton, B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance", Chapman-Hall,
3. H. Föllmer, A. Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter, 2011.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the technique used for constructing combinatorial designs and its relation with linear codes.

Prerequisite:

Syllabus:

1. Incidence structures, affine planes, translation plane, projective planes, conics and ovals, blocking sets. Introduction to Balanced Incomplete Block Designs (BIBD), Symmetric BIBDs, Difference sets, Hadamard matrices and designs, Resolvable BIBDs, Latin squares. 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.

References

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

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding the vector order structure and its relation with Functional Analysis.

Prerequisite:

Syllabus:

1. Cones and orderings; order convexity; order units; approximate order units; bases. Positive linear mappings and functionals; extension and separation theorems; decomposition of linear functionals into positive linear functionals. Vector lattices; basic theory. Norms and orderings; duality of ordered spaces; (approximate) order unit spaces; base normed spaces. Normed and Banach lattices; AM-spaces, AL-spaces; Kakutani theorems for AM-spaces and AL-spaces. Matrix ordered spaces: matricially normed spaces; matricial ordered normed spaces; matrix order unit spaces; Arveson-Hahn-Banach extension theorem.

Text Books

1. G.J.O. Jameson, "Lecture Notes in Mathematics" 141 Springer-Verlag,
2. N.C. Wong and K.F. Ng, "(2) Partially ordered topological vector spaces", Oxford University Press,
3. C.D. Aliprantis and O. Burkinshaw, "Positive operators", Academic Press,
4. H.H. Schaefer, "Banach lattices and positive operators", Berlin: Springer, 1974.

References

1. W.A. J. Luxemburg and A.C. Zaanen, "Riesz Spaces", Elsevier,
2. A.C. Zaanen, "Introduction to operator theory in Riesz spces (Vol 1 Vol 2)", Springer, 1997

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding analytic and harmonic functions on the unit disc.
• Understanding properties H^p spaces, for 1≤p<∞.
• Understanding invarint subspaces for the shift operator on H^2 space.

Prerequisite:

Syllabus:

1. Fourier Series: Cesaro Means, Characterization of Types of Fourier Series;
2. Analytic and Harmonic Functions in the Unit Disc: The Cauchy and Poisson Kernels, Boundary Values, Fatou’s Theorem, H^p Spaces;
3. The Space : The Helson-Lowdenslager Approach, Szego’s Theorem, Completion of the Discussion of H1;
4. Factorization for H^p functions: Inner and Outer Functions, Blaschke Products and Singular Functions, The Factorization Theorem, Absolute Convergence of Taylor Series, Functions of Bounded Characteristic;
5. Analytic Functions with Continuous Boundary Values: Conjugate Harmonic Functions, Theorems of Fatou and Rudin;
6. The Shift Operator: The Shift Operator on H^2, Invariant Subspaces for H^2 of the Half-plane, Isometries, The Shift Operator on L^2.

References

1. Kenneth Hoffman, Banach spaces of analytic functions, Reprint of the 1962 original, Dover Publications, Inc., New York,
2. Walter Rudin, Real and complex analysis, Third edition. McGraw-Hill Book Co., New York,
3. Duren, Peter L., Theory of spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 1970.

Course: iPhD-Elective

Duration: 56 Hours

Credit: 4

Outcome:

• Understanding contractive operators by exhibiting as a compression of unitary operators.
• Understanding Hardy classes and space.
• Understanding dilations of commuting and non-commuting contractions.

Prerequisite:

Syllabus:

1. Contractions and Their Dilations: Unilateral shifts, Wold decomposition, Bilateral shifts, Contractions, Canonical decomposition, Isometric and unitary dilations, Matrix construction of the unitary dilation, a discussion on rational dilation; Geometrical and Spectral Properties of Dilations: Structure of the minimal unitary dilations, Isometric dilations, Dilation of commutants; Functional Calculus: Hardy classes, Inner and outer functions, The classes and , The role of outer functions, Contractions of class ; Operator-Valued Analytic Functions: The spaces and , Inner and outer functions, Lemmas on Fourier representation, Factorizations, Analytic kernels; Functional Models: Characteristic functions, Functional models for a given contraction, Functional models for analytic functions; A discussion on Commuting and non-commuting contractions and their dilations.

References

1. Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici and László Kérchy, Harmonic analysis of operators on Hilbert space; Second edition. Revised and enlarged edition. Universitext. Springer, New York,
2. Vern Ival Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics,
3. Cambridge University Press, Cambridge,
4. Jim Agler, John Harland and Benjamin J. Raphael, Classical function theory, operator dilation theory, and machine computation on multiply-connected domains. (English summary) Mem. Amer. Math. Soc. 191 (2008), no.
5. Nikolai K. Nikolskii, Operators, functions, and systems: an easy reading. Vol. 1 and Vol
6. Hardy, Hankel, and Toeplitz, American Mathematical Society, Providence, RI, 2002.