Course: M402

Approval:

Credit: 4

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.

Reference Book

1. J. L. Alperin, R. B. Bell, “Groups and Representations”, Graduate Texts in Mathematics 162, Springer, 1995.
2. B. Steinberg, “Representation Theory of Finite Groups”, Universitext, Springer, 2012.
3. J-P. Serre, “Linear Representations of Finite Groups”, Graduate Texts in Mathematics 42, Springer-Verlag, 1977.
4. B. Simon, “Representations of Finite and Compact Groups”, Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

Text Book

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

Course: M403

Approval:

Credit: 4

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.

Reference Book

1. R. Y. Sharp, “Steps in Commutative Algebra”, London Mathematical Society Student Texts, 51. Cambridge University Press, 2000.
2. D. S. Dummit, R. M. Foote, “Abstract Algebra”, Wiley-India edition, 2013.

Text Book

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

Course: M404

Approval:

Credit: 4

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.

Reference Book

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

Text Book

1.  J. R. Munkres, “Topology”, Prentice-Hall of India, 2013.  
2. A. Hatcher, “Algebraic Topology”, Cambridge University Press, 2009.

Course: M482

Approval:

Credit: 4

Review of matrix algebra (optional), data matrix, summary statistics, graphical representations (3 hrs)
Distribution of random vectors, moments and characteristic functions, transformations,
some multivariate distributions: multivariate normal, multinomial, dirichlet distribution, limit theorems (5 hrs)
Multivariate normal distribution: properties, geometry, characteristics function, moments, distributions of linear combinations, conditional distribution and multiple correlation (5 hrs)
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 (8
hrs)
Inference about mean vector: testing for normal mean, Hotelling T2
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 (10 hrs)
Techniques of dimension reduction, principle component analysis: definition of principle components and their estimation, introductory factor analysis, multidimensional
scaling (10 hrs)
Classification problem: linear and quadratic discriminant analysis, logistic regression,
support vector machine (8 hrs)
Cluster analysis: non-hierarchical and hierarchical methods of clustering (5 hrs)

Reference Book

1. K. V. Mardia, J. T. Kent, and J. M. Bibby (1980), “Multivariate Analysis”, Academic Press.
2. T. W. Anderson (2003), “An Introduction to Multivariate Statistical Analysis”, Wiley.
3. C. Chatfield and A. J. Collins (1980), “Introduction to Multivariate Analysis”, Chapman & Hall.
4. R. A. Johnson and D. W. Wichern, (2007), “Applied Multivariate Statistical Analysis”, 6th edition, Pearson.
5. Brian Everitt and Torsten Hothorn (2011), “An Introduction to Applied Multivariate Analysis with R”, Springer.
6. M. L. Eaton (1983), “Multivariate Statistics”, John Wiley.

Course: M483

Approval:

Credit: 4

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

Reference Book

1. Michael Spivak, “A comprehensive introduction to differential geometry”, Vol. 1, 3rd edition, 1999.
2. Frank Warner, “Foundations of differentiable manifolds and Lie groups”, Springer Verlag, 2nd edition, 1983.
3. John Lee, “Introduction to smooth manifolds”, Springer Verlag, 2nd edition, 2013.
4. Louis Auslander and Robert E. MacKenzie, “Introduction to differentiable manifolds”, Dover, 2nd edition, 2009.

Course: M652

Approval:

Credit: 4

Cauchy-Riemann equations, Cauchy's theorem and estimates, Zeros, Poles and Singularities, The open mapping theorem, The argument principle, Maximum modulus principle, Schwarz lemma, Residues and the residue calculus.Normal families, Arzela's theorem, Product developments, functions with prescribed zeroes and poles, Hadamard's theorem, Conformal mappings, Riemann mapping theorem, the linear fractional transformations.

Reference Book

1. L. V. Ahlfors: Complex analysis (McGraw-Hill), 1978.
2. J. B. Conway: Functions of one complex variable II (Springer), 1995.
3. W. Rudin: Real and Complex Analysis (McGraw-Hill), 1987.
4. R. Remmert: Theory of Complex Functions, Springer 1998.
5. R. V. Churchill and J. W. Brown: Complex Variables and Applications (McGraw-Hill).

Course: M653

Approval:

Credit: 4

Ordinary Differential Equations: Initial and boundary value problems, Basic existence, Uniqueness theorems for a system of ODE, Gronwall’s lemma, Continuous dependence on initial data, Linear systems with variable coefficients, Variation of parameter formula, Floquet theory, Systems of linear equations with constant coefficients, Stability of equilibrium positions.Partial Differential Equations: Single and systems of PDE, First order PDE, Semi-linear and nonlinear equations (Monge’s method), Four important linear PDE, Transport equations, Laplace equations, Fundamental solution, Mean value formulas, Green’s functions, Energy methods, Heat equation, fundamental solution, Mean value formula, Energy methods, Wave equations, Solutions by spherical mean, Energy method, Maximum principle for elliptic and parabolic equations with applications.

Reference Book

1. V. I. Arnold, Ordinary Differential Equations , Prentice Hall of India.
2. Brauer and Nohel, Qualitative Theory of Differential Equations, Dover Publications.
3. Coddington and Levinson, Ordinary Differential Equations, Tata Mcgraw-Hill.
4. Fritz John, Partial Differential Equation, Narosa Publications.
5. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer International Edition.
6. L. C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, Vol 19.

Course: M654

Approval:

Credit: 4

Combinatorics:Counting principles, Generating functions, Recurrence relation, Polya’s enumeration theory, partially ordered sets.Graph Theory:Graphs, Trees, Blocks, Connectivity, Eulerian and Hamiltonian graphs, Planer graphs, Graph colouring.Design Theory: Block Designs, Balanced incomplete block design, Difference sets and Automorphism, Latin squares, Hadamard matrices, Projective planes, Generalized quadrangles.Algorithm:Algorithm, Asymptotic analysis, Complexity hierarchy, NP-complete problems.

Reference Book

1. F. Roberts and B. Tesman: Applied Combinatorics. Pearson Education, 2005.
2. M. Aigner, A course in Enumeration, Springer.
3. R. P. Stanley, Enumerative Combinatorics, Cambridge University Press.
4. F. Harary, Graph Theory, Narosa Publishing House.
5. G.A. Bondy and U.S.R. Murty: Graph Theory. Springer, 2008.
6. W. D. Wallis, Introduction to Combinatorial Designs, Chapman & Hall/CRC
7. D. R. Stinson and D. Stinson, Combinatorial Designs: Construction and Analysis, Springer.
8. Thomas Cormen, Charles Leiserson, Ronald Rivest: Introduction to Algorithms. PHI, 1998.

Course: M655

Approval:

Credit: 4

Basic definitions, Eulerian and Hamiltonian graphs, Planarity, Colourability, Four colour problem, Matching and Hall’s marriage theorem, Max-flow Min-cut theorem, Ramsey theory, Line graphs, Enumeration, Digraphs. Matroids, Groups and Graphs, Matrices and graphs, Eigenvalues of graphs, The Laplacian of a graph, Strongly regular graphs.

Reference Book

1. D. B. West, Introduction to Graph Theory, Prentice Hall of India.
2. F. Harary, Graph Theory, Narosa Publishing House.
3. B. Bollobas, Extremal Graph Theory , Dover Publications.
4. R. Diestel, Graph Theory, Springer International Edition.
5. G. A. Bondy and U. S. R. Murty, Graph Theory, Springer
6. C. Godsil and G. Royle, Algebraic Graph Theory, Springer International Edition.

Course: M656

Approval:

Credit: 4

The Fundamental Theorem of Arithmetic, Distribution of prime numbers, Congruences, Chinese remainder theorem, Congruences with prime-power modulus, Fermat's little theorem, Wilson's theorem, Euler function and its applications, Group of units, Primitive roots, Quadratic residues and Quadratic reciprocity law, Arithmetic functions, Mobius Inversion formula, Dirichlet product, Sum of squares, Introduction to Zeta function and Dirichlet Series.

Reference Book

1. G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers (Oxford).
2. J. A. Jones and J. M. Jones: Elementary Number Theory (Springer).
3. I. Niven, H. S. Zuckerman & H. L. Montgomery: The Theory of Numbers (Wiley)
4. T. M. Apostol: Introduction to Analytic Number Theory (Springer)

Course: M657

Approval:

Credit: 4

Review of Basic undergraduate probability: Random variables, Standard discrete and continuous distributions, Expectation, Variance, Conditional Probability.Discrete time Markov chains: countable state space, classification of statesCharacteristic functions, modes of convergences, Borel-Cantelli Lemma, Central Limit Theorem, Law of Large numbersConvergence Theorems in Markov Chains

Reference Book

1. W. Feller: Introduction to Probability Theory and ins Applications Vol.I & Vol. II (Wiley)
2. S. M. Ross: Introduction to Probability Models (AP)
3. Hoel, Port & Stone: Introduction to Stochastic Processes (HMC)
4. S. M. Ross: Stochastic Processes (Wiley)

Course: M658

Approval:

Credit: 4

Martingale Theory: Radon-Nikoydm Theorem, Doob-Meyer decomposition.Weak convergence of probability measures,Brownian motion, Markov processes and Stationary processes.

Reference Book

1. O. Kallenberg: Foundation of Modern Probability (Springer)
2. P. Billingsley: Convergence of probability measures.(John Wiley & Sons, Inc.)
3. D. Revuz & M. Yor: Continuous martingales and Brownian motion (Springer-Verlag)
4. S. M. Ross: Stochastic Processes (Wiley)
5. J. L. Doob: Stochastic Processes (Wiley)

Course: M451

Approval:

Credit: 4

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.

Reference Book

1. L. V. Ahlfors, “Complex Analysis”, Tata McGraw-Hill, 2013.
2. J. B. Conway, “Functions of One Complex Variable II”, Graduate Texts in Mathematics 159, Springer-Verlag, 1996.
3. W. Rudin, “Real and Complex Analysis”, Tata McGraw-Hill, 2013.
4. R. Remmert, “Theory of Complex Functions”, Graduate Texts in Mathematics 122, Springer, 2008.

Course: M452

Approval:

Credit: 4

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.

Reference Book

1. W. Rudin, “Functional Analysis”, Tata McGraw-Hill, 2007.A. P. Robertson, W. Robertson, “Topological Vector Spaces”, Cambridge Tracts in Mathematics 53, Cambridge University Press, 1980.J. B. Conway, “A Course in Functional Analysis”, Graduates Texts in Mathematics 96, Springer, 2006.

Course: M453

Approval:

Credit: 4

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.

Reference Book

1. K. Hoffman, R. Kunze, “Linear Algebra”, Prentice-Hall of India, 2012.
2. S. Roman, “Advanced Linear Algebra”, Graduate Texts in Mathematics 135, Springer, 2008.
3. R. A. Horn, C. R. Johnson, “Matrix Analysis”, Cambridge University Press, 2010.

Course: M454

Approval:

Credit: 4

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

Reference Book

1. L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics 19, American Mathematical Society, 2010.
2. F. John, “Partial Differential Equations”, Springer International Edition, 2009.
3. G. B. Folland, “Introduction to Partial Differential Equations”, Princeton University Press, 1995.
4. S. Kesavan, “Topics in Functional Analysis and Applications”, John Wiley & Sons, 1989.

Course: M456

Approval:

Credit: 4

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.

Reference Book

1. J. Harris, “Algebraic geometry”, Graduate Texts in Mathematics 133, Springer-Verlag, 1995.
2. M. Reid, “Undergraduate Algebraic Geometry”, London Mathematical Society Student Texts 12, Cambridge University Press, 1988.
3. K. E. Smith et. al., “An Invitation to Algebraic Geometry”, Universitext, Springer-Verlag, 2000.
4. K. Hulek, “Elementary Algebraic Geometry”, Student Mathematical Library 20, American Mathematical Society, 2003.
5. I. R. Shafarevich, “Basic Algebraic Geometry 1: Varieties in Projective Space”, Springer, 2013.
6. R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics 52, Springer-Verlag, 1977.

Course: M457

Approval:

Credit: 4

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.

Reference Book

1. N. Biggs, “Algebraic Graph Theory”, Cambridge University Press, 1993.
2. C. Godsil, G. Royle, “Algebraic Graph Theory”, Graduate Texts in Mathematics 207, Springer-Verlag, 2001.
3. R. B. Bapat, “Graphs and Matrices”, Universitext, Springer, Hindustan Book Agency, New Delhi, 2010.

Course: M458

Approval:

Credit: 4

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.

Reference Book

1. D. A. Marcus, “Number Fields”, Universitext, Springer-Verlag, 1977.
2. G. J. Janusz, “Algebraic Number Fields”, Graduate Studies in Mathematics 7, American Mathematical Society, 1996.
3. S. Alaca, K. S. Williams, “Introductory Algebraic Number Theory”, Cambridge University Press, 2004.
4. S. Lang, “Algebraic Number Theory”, Graduate Texts in Mathematics 110, Springer-Verlag, 1994.
5. A. Frohlich, M. J. Taylor, “Algebraic Number Theory”, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, 1993.
6. J. Neukirch, “Algebraic Number Theory”, Springer-Verlag, 1999.

Course: M460

Approval:

Credit: 4

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.

Reference Book

1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, “The Design and Analysis of Computer Algorithms”, Addison-Wesley Publishing Co., 1975.
2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, “Introduction to Algorithms”, MIT Press, Cambridge, 2009.
3. E. Horowitz, S. Sahni, “Fundamental of Computer Algorithms”, Galgotia Publication, 1987.
4. D. E. Knuth, “The Art of Computer Programming Vol. 1, Vol. 2, Vol 3”, Addison Wesley Publishing Co., 1997, 1998, 1998.

Course: M463

Approval:

Credit: 4

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.

Reference Book

1. R. Lidl, H. Neiderreiter, “Finite Fields”, Cambridge university press, 2000.
2. G. L. Mullen, C. Mummert, “Finite Fields and Applications”, American Mathematical Society, 2007.
3. A. J. Menezes et. al., “Applications of Finite Fields”, Kluwer Academic Publishers, 1993.
4. Z-X. Wan, “Finite Fields and Galois Rings”, World Scientific Publishing Co., 2012.

Course: M464

Approval:

Credit: 4

Information Theory: Entropy, Huffman coding, Shannon-Fano coding, entropy of Markov process, channel and mutual information, channel capacity;
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.

Reference Book

1. R. W. Hamming, “Coding and Information Theory”, Prentice-Hall, 1986.
2. N. J. A. Sloane, F. J. MacWilliams, “Theory of Error Correcting Codes”, North-Holland Mathematical Library 16, North-Holland, 2007.
3. S. Ling, C. Xing, “Coding Theory: A First Course”, Cambridge University Press, 2004.
4. V. Pless, “Introduction to the Theory of Error-Correcting Codes”, Wiley-Interscience Publication, John Wiley & Sons, 1998.
5. S. Lin, “An Introduction to Error-Correcting Codes”, Prentice-Hall, 1970.

Course: M465

Approval:

Credit: 4

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, Godels first Incompleteness theorem, Rudiments of model
theory including Lowenheim-Skolem theorem and categoricity.

Reference Book

1. J. R. Shoenfield, “Mathematical logic”, Addison-Wesley Publishing Co., 1967.
2. E. Mendelson, “Introduction to Mathematical Logic”, Chapman & Hall, 1997.

Course: M466

Approval:

Credit: 4

Reference Book

1. G. De Barra, “Measure theory and integration”.J. Nevue, “Mathematical foundations of the calculus of probability”, Holden-Day, Inc., 1965.I. K. Rana, “An introduction to measure and integration”, Narosa Publishing House.P. Billingsley, “Probability and measure”, John Wiley & Sons, Inc., 1995. W. Rudin, “Real and complex analysis”, McGraw-Hill Book Co., 1987.K. R. Parthasarathy, “Introduction to probability and measure”, The Macmillan Co. of India, Ltd., 1977.

Course: M467

Approval:

Credit: 4

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.

Reference Book

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

Course: M468

Approval:

Credit: 4

Reference Book

1. W. Arveson, “An invitation to C*-algebras”, Graduate Texts in Mathematics, No. 39. Springer-Verlag, 1976.
2. N. Dunford and J. T. Schwartz, “Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space”, Interscience Publishers John Wiley i& Sons 1963.
3. 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., 1983.
4. V. S. Sunder, “An invitation to von Neumann algebras”, Universitext, Springer-Verlag, 1987.

Course: M470

Approval:

Credit: 4

Reference Book

1. G. B. Folland, “A Course in Abstract Harmonic Analysis”, CRC Press, 2000.
2. H. Helson, “Harmonic Analysis”, Texts and Readings in Mathematics, Hindustan Book Agency, 2010.
3. Y. Katznelson, “An Introduction to Harmonic Analysis”, Cambridge University Press, 2004.
4. L. H. Loomis, “An Introduction to Abstract Harmonic Analysis”, Dover Publication, 2011.
5. E. Hewitt, K. A. Ross, “Abstract Harmonic Analysis Vol. I”, Springer-Verlag, 1979.
6. W. Rudin, “Real and Complex Analysis”, Tata McGraw-Hill, 2013.

Course: M471

Approval:

Credit: 4

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.

Reference Book

1. J.-P. Serre, “A Course in Arithmetic”, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.
2. K. Ireland, M. Rosen, “A Classical Introduction to Modern Number Theory”, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
3. H. Hasse, “Number Theory”, Classics in Mathematics, Springer-Verlag, 2002.
4. W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Numbers”, Springer Monographs in Mathematics, Springer-Verlag, 2004.
5. F. Q. Gouvˆea, “p-adic Numbers”, Universitext, Springer-Verlag, 1997.

Course: M474

Approval:

Credit: 4

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

Reference Book

1. O. Goldreich, “Foundations of Cryptography - Vol. I and Vol. II”, Cambridge University Press, 2001, 2004.
2. S. Goldwasser, Mihir Bellare, “Lecture Notes on Cryptography”, 2008, available online from http://cseweb.ucsd.edu/ mihir/papers/gb.html

Course: M475

Approval:

Credit: 4

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.

Reference Book

1. J. Ueberberg, “Foundations of Incidence Geometry”, Springer Monographs in Mathematics, Springer, 2011.
2. L. M. Batten, “Combinatorics of Finite Geometries”, Cambridge University Press, 1997.
3. E. E. Shult, “Points and Lines”, Universitext, Springer, 2011.
4. L. M. Batten, A. Beutelspacher, “The Theory of Finite Linear Spaces: Combinatorics of points and lines”, Cambridge University Press, 1993
5. G. E. Moorhouse, “Incidence Geometry”, 2007, available online from http://www.uwyo.edu/moorhouse/handouts/incidence geometry.pdf

Course: M476

Approval:

Credit: 4

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.

Reference Book

1. J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory”, Graduate Texts in Mathematics 9, Springer-Verlag, 1978.
2. K. Erdmann, M. J. Wildon, “Introduction to Lie Algebras”, Springer Undergraduate Mathematics Series, Springer-Verlag, 2006.
3. J.-P. Serre, “Complex Semisimple Lie Algebras”, Springer Monographs in Mathematics, Springer-Verlag, 2001.
4. N. Jacobson, “Lie Algebras”, Dover Publications, 1979.

Course: M477

Approval:

Credit: 4

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) zero
sum games, Introduction to non-linear programming and techniques.

Reference Book

1. J. K. Strayer, “Linear Programming and its Applications”, Undergraduate Texts in Mathematics, Springer-Verlag, 1989.
2. P. R. Thie, G. E. Keough, “An Introduction to Linear Programming and Game Theory”, John Wiley & Sons, 2008.
3. L. Brickman, “Mathematical Introduction to Linear Programming and Game Theory”, Undergraduate Texts in Mathematics, Springer-Verlag, 1989.
4. D. G. Luenberger, Y. Ye, “Linear and Nonlinear Programming”, International Series in Operations Research & Management Science 116, Springer, 2008.

Course: M478

Approval:

Credit: 4

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.

Reference Book

1. L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics 19, American Mathematical Society, 2010.
2. H. Brezis, “Functional Analysis, Sobolev Spaces and Partial Differential Equations”, Universitext, Springer, 2011.
3. R. A. Adams, J. J. F. Fournier, “Sobolev Spces”, Pure and Applied Mathematics 140, Elsevier/Academic Press, 2003.
4. S. Kesavan, “Topics in Functional Analysis and Applications”, John Wiley & Sons, 1989.
5. M. Renardy, R. C. Rogers, “An Introduction to Partial Differential Equations”, Springer, 2008.

Course: M479

Approval:

Credit: 4

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.

Reference Book

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, 2001.
3. S. Janson, T. Luczak, A. Rucinski, “Random Graphs”, Wiley-Interscience, 2000. 
4. R. Durrett, “Random Graph Dynamics”, Cambridge University Press, 2010.
5. J. H. Spencer, “The Strange Logic of Random Graphs”, Springer-Verlag, 2001.

Course: M480

Approval:

Credit: 4

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)

Reference Book

1. R. Motwani, P. Raghavan, “Randomized Algorithms”, Cambridge University Press, 2004.
2. M. Mitzenmacher, E. Upfal, “Probability and Computing: Randomized algorithms and probabilistic analysis”, Cambridge University Press, 2005

Course: M482

Approval:

Credit: 4

Review of matrix algebra (optional), data matrix, summary statistics, graphical representations (3 hrs)
Distribution of random vectors, moments and characteristic functions, transformations,
some multivariate distributions: multivariate normal, multinomial, dirichlet distribution, limit theorems (5 hrs)
Multivariate normal distribution: properties, geometry, characteristics function, moments, distributions of linear combinations, conditional distribution and multiple correlation (5 hrs)
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 (8
hrs)
Inference about mean vector: testing for normal mean, Hotelling T2
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 (10 hrs)
Techniques of dimension reduction, principle component analysis: definition of principle components and their estimation, introductory factor analysis, multidimensional
scaling (10 hrs)
Classification problem: linear and quadratic discriminant analysis, logistic regression,
support vector machine (8 hrs)
Cluster analysis: non-hierarchical and hierarchical methods of clustering (5 hrs)

Reference Book

1. K. V. Mardia, J. T. Kent, and J. M. Bibby (1980), “Multivariate Analysis”, Academic Press.
2. T. W. Anderson (2003), “An Introduction to Multivariate Statistical Analysis”, Wiley.
3. C. Chatfield and A. J. Collins (1980), “Introduction to Multivariate Analysis”, Chapman & Hall.
4. R. A. Johnson and D. W. Wichern, (2007), “Applied Multivariate Statistical Analysis”, 6th edition, Pearson.
5. Brian Everitt and Torsten Hothorn (2011), “An Introduction to Applied Multivariate Analysis with R”, Springer.
6. M. L. Eaton (1983), “Multivariate Statistics”, John Wiley.

Course: M483

Approval:

Credit: 4

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

Reference Book

1. Michael Spivak, “A comprehensive introduction to differential geometry”, Vol. 1, 3rd edition, 1999.
2. Frank Warner, “Foundations of differentiable manifolds and Lie groups”, Springer Verlag, 2nd edition, 1983.
3. John Lee, “Introduction to smooth manifolds”, Springer Verlag, 2nd edition, 2013.
4. Louis Auslander and Robert E. MacKenzie, “Introduction to differentiable manifolds”, Dover, 2nd edition, 2009.

Course: M484

Approval:

Credit: 4

Introduction to simple linear regression, least square estimation and hypothesis testing of model parameters, prediction, interval estimation in simple linear regression, Coefficient of determination, estimation by maximum likelihood, multiple linear regression, matrix representation of the regression model, estimation and testing of model parameters and prediction, model adequacy checking-residual analysis, PRESS statistics, outlier detection, lack of fit test, serial correlation and Durbin-Watson test, transformation and weighting to correct model inadequacies-variance-stabilizing transformation, generalized and weighted least squares, diagnostics for influential observations, Cook’s D test, multicollinearity-sources and effects, diagnosis and treatment for multicollinearity, ridge regression and LASSO, bootstrap estimation, dummy variable model, variable selection and model building–stepwise methods, polynomial regression and interaction regression models, nonlinear regression, generalized linear models-logistic regression and Poisson regression.

Reference Book

1. Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining, “Introduction to Linear Regression Analysis”, 5th Edition, Wiley, 2012. 

2. N. R. Draper and H. Smith (1998), Applied Regression Analysis, 3rd Edition, New York: Wiley. 

3. Michael H. Kutner, Chris J. Nachtsheim, and John Neter, “Applied Linear Statistical Models”, McGraw-Hill/Irwin; 5th edition, 2004. 

4. Seber, G. A. F. and Lee, A. J., “Linear Regression Analysis”, John Wiley and Sons, 2nd Edition, 2003.
5. N. H. Bingham, John M. Fry, “Regression: Linear Models in Statistics”, Springer Undergraduate Mathematics Series, 2010. 


Course: M485

Approval:

Credit: 4

Examples and objectives of time series, stationary time series and autocorrelation function, estimation and elimination of trend and seasonal components, testing for noise sequence, moving average process, autoregressive processes and ARMA processes, estimation of autocorrelation function, methods of forecasting-Durbin-Levinson algorithm and Innovations algorithm, the Wold decomposition, ARMA models-the auto-covariance and partial auto-covariance function, forecasting ARMA processes, spectral analysis-spectral densities, periodogram, modeling with ARMA processes, Yule-Walker estimation, maximum likelihood estimation, diagnostic checking, non-stationary time series-ARIMA models, identification techniques, forecasting ARIMA models, seasonal ARIMA models, multivariate time series, ARCH and GARCH models.

Reference Book

1. Peter J. Brockwell and Richard A. Davis, “Introduction toTime Series and Forecasting”, Springer Texts in Statistics, 2010.
2. Chris. Chatfield, “The analysis of time series: An introduction”, 6th edition, Chapman & Hall/CRC, 2004. 

3. J. D. Cryer and K.-S. Chan, “Time series analysis with applications in R”, 2nd edition, Springer, 2008.
4. R. H. Shumway and D. S. Stoffer, “Time series analysis and its applications with R examples”, 3rd edition, Springer, 2011. 


Course: M552

Approval:

Credit: 4

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.

Reference Book

1. T. M. Apostol, “Introduction to Analytic Number Theory”, Springer International Student Edition, 2000.
2. K. Chandrasekharan, “Introduction to Analytic Number Theory”, Springer-Verlag, 1968. 
3. H. Iwaniec, E. Kowalski, “Analytic Number Theory”, American Mathematical Society Colloquium Publications 53, American Mathematical Society, 2004.

Course: M553

Approval:

Credit: 4

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.

Reference Book

1. L. C. Grove, “Classical Groups and Geometric Algebra”, Graduate Studies in Mathematics 39, American Mathematical Society, 2002.
2. E. Artin, “Geometric Algebra”, John Wiley & sons, 1988.

Course: M554

Approval:

Credit: 4

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 forflows. Flows built under a function.

Reference Book

1. Peter Walters, “An introduction to ergodic theory”, Graduate Texts in Mathematics, 79. Springer-Verlag, 1982.Patrick Billingsley, “Ergodic theory and information”, Robert E. Krieger Publishing Co., 1978.M. G. Nadkarni, “Basic ergodic theory”, Texts and Readings in Mathematics, 6.Hindustan Book Agency, 1995.H. Furstenberg, “Recurrence in ergodic theory and combinatorial number theory”, Princeton University Press, 1981.K. Petersen, “Ergodic theory”, Cambridge Studies in Advanced Mathematics, 2.Cambridge University Press, 1989.

Course: M555

Approval:

Credit: 4

Reference Book

1. Y. Katznelson, “An Introduction to Harmonic Analysis”, Cambridge University Press, 2004.
2. E. M. Stein, G. Weiss, “Introduction to Fourier Analysis on Euclidean Spaces”, Princeton Mathematical Series 32, Princeton University Press, 1971.
3. G. B. Folland, “Fourier Analysis and its Applications”, Pure and Applied Undergraduate Texts 4, America Mathematical Society, 2010.

Course: M556

Approval:

Credit: 4

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.

Reference Book

1. D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
2. J. Faraut, “Analysis on Lie Groups”, Cambridge Studies in Advanced Mathematics 110, Cambridge University Press, 2008.
3. B. C. Hall, “Lie Groups, Lie algebras and Representations”, Graduate Texts in Mathematics 222, Springer-Verlag, 2003.
4. W. Fulton, J. Harris, “Representation Theory: A first course”, Springer-Verlag, 1991.
5. J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory”, Graduate Texts in Mathematics 9, Springer-Verlag, 1978.
6. A. Kirillov, “Introduction to Lie Groups and Lie Algebras”, Cambridge Studies in Advanced Mathematics 113, Cambridge University Press, 2008.

Course: M557

Approval:

Credit: 4

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).

Reference Book

1. R. V. Kadison, J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras Vol. I”, Graduate Studies in Mathematics 15, American Mathematical Society, 1997.
2. G. K. Pedersen, “C*–algebras and their Automorphism Groups”, London Mathematical Society Monographs 14, Academic Press, 1979.
3. V. S. Sunder, “An Invitation to von Neumann Algebras”, Universitext, Springer-Verlag, 1987.
4. M. Takesaki, “Theory of Operator Algebras Vol. I”, Springer-Verlag, 2002.

Course: M558

Approval:

Credit: 4

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.

Reference Book

1. J. E. Humphreys, “Introduction to Lie algebras and representation theory”, Springer-Verlag, 1978.
2. S. C. Bagchi, S. Madan, A. Sitaram, U. B. Tiwari, “A first course on representation theory and linear Lie groups”, University Press, 2000.
3. Mitsou Sugiura, “Unitary Representations and Harmonic Analysis”, John Wiley & Sons, 1975.
4. Sundaram Thangavelu, “Harmonic Analysis on the Heisenberg Group”, Birkhauser, 1998.
5. Sundaram Thangavelu, “An Introduction to the Uncertainty Principle”, Birkhauser, 2003. 

Course: M559

Approval:

Credit: 4

Reference Book

1. T. Brocker, T. Dieck, “Representations of Compact Lie Groups”, Springer-Verlag, 1985.
2. J. L. Clerc, “Les Repr ́esentatios des Groupes Compacts, Analyse Harmonique” (J.L. Clerc et. al., ed.), C.I.M.P.A., 1982.
3. G. B. Folland, “A Course in Abstract Harmonic Analysis”, CRC Press, 2000. 
4. M. Sugiura, “Unitary Representations and Harmonic Analysis”, John Wiley &Sons, 1975.
5. E. B. Vinberg, “Linear Representations of Groups”, Birkh ̈ auser/Springer, 2010. 
6. A. Wawrzy ́ nczyk, “Group Representations and Special Functions”, PWN–Polish Scientific Publishers, 1984.

Course: M560

Approval:

Credit: 4

Reference Book

1. J.-P. Serre, “A Course in Arithmetic”, Graduate Texts in Mathematics 7, Springer- Verlag, 1973.
2. N. Koblitz, “Introduction to Elliptic Curves and Modular Forms”, Graduate Texts in Mathematics 97, Springer-Verlag, 1993.
3. J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier, “The 1-2-3 of Modular Forms”, Universitext, Springer-Verlag, 2008.
4. F. Diamond, J. Shurman, “A First Course in Modular Forms”, Graduate Texts in Mathematics 228, Springer-Verlag, 2005.
5. S. Lang, “Introduction to Modular Forms”, Springer-Verlag, 1995.
6. G. Shimura, “Introduction to the Arithmetic Theory of Automorphic Forms”, Princeton University Press, 1994.

Course: M561

Approval:

Credit: 4

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.

Reference Book

1. J. H. Silverman, J. Tate, “Rational Points on Elliptic Curves”, Undergraduate Texts in Mathematics, Springer-Verlag, 1992.
2. N. Koblitz, “Introduction to Elliptic Curves and Modular Forms”, Graduate Texts in Mathematics 97, Springer-Verlag, 1993.
3. J. H. Silverman, “The Arithmetic of Elliptic Curves”, Graduate Texts in Mathematics 106, Springer, 2009.
4. A. W. Knapp, “Elliptic Curves”, Mathematical Notes 40, Princeton University Press, 1992.
5. J. H. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Graduate Texts in Mathematics 151, Springer-Verlag, 1994

Course: M562

Approval:

Credit: 4

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

Reference Book

1. H. H. Kuo, “Introduction to Stochastic Integration”, Springer, 2006.
2. J. M Steele, “Stochastic Calculus and Financial Applications”, Springer-Verlag, 2001.
3. F. C. Klebaner, “Introduction to Stochastic Calculus with Applications”, Imperial College, 2005.

Course: M563

Approval:

Credit: 4

Review of Several variable Calculus: Directional Derivatives, Inverse Function Theorem, Implicit function Theorem, Level sets in R n , 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, Universalcovering group of a connected Lie group. Finite dimensional representations
of Lie groups and Lie algebras.

Reference Book

1. D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
2. S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, Graduate Studies in Mathematics 34, American Mathematical Society, 2001.
3. S. Kumaresan, “A Course in Differential Geometry and Lie Groups”, Texts and Readings in Mathematics 22, Hindustan Book agency, 2002.
4. F. W. Warner, “Foundations of Differentiable Manifolds and Lie Groups”, Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

Course: M564

Approval:

Credit: 4

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, Therepresentation theory of SU (3), Frobenius Reciprocity theorem, Spherical
Harmonics.

Reference Book

1. D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
2. J. Faraut, “Analysis on Lie Groups”, Cambridge Studies in Advanced Mathematics 110, Cambridge University Press, 2008.
3. B. C. Hall, “Lie Groups, Lie algebras and Representations”, Graduate Texts in Mathematics 222, Springer-Verlag, 2003.
4. W. Fulton, J. Harris, “Representation Theory: A first course”, Springer-Verlag, 1991.
5. A. Kirillov, “Introduction to Lie Groups and Lie Algebras”, Cambridge Studies in Advanced Mathematics 113, Cambridge University Press, 2008.
6. A. W. Knapp, “Lie Groups: Beyond an introduction”, Birk ̈ auser, 2002.
7. B. Simon, “Representations of Finite and Compact Groups”, Graduate Studies in Mathematics 10, American Mathematical Society, 2009.

Course: M565

Approval:

Credit: 4

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, Itˆo’s representation theorem, BlackScholes formula

Reference Book

1. J. Jacod, P. Protter, “Probability Essentials”, Universitext, Springer-Verlag, 2003.
2. D. Lamberton, B. Lapeyre, “Introduction to Stochastic Calculus Applied to Finance”, Chapman-Hall, 2008.
3. H. F ̈ollmer, A. Schied, “Stochastic Finance: An Introduction in Discrete Time”, de Gruyter, 2011.

Course: M568

Approval:

Credit: 4

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.

Reference Book

1. 1.W.A. J. Luxemburg and A.C. Zaanen, \Riesz Spaces“, Elsevier, 1971.
2. 2. A.C. Zaanen, \Introduction to operator theory in Riesz spces (Vol 1 & Vol 2)“, Springer, 1997

Text Book

1. 1.G.J.O. Jameson, \Lecture Notes in Mathematics“ 141 Springer-Verlag,1970.
2. 2. N.C. Wong and K.F. Ng, \(2) Partially ordered topological vector spaces“, Oxford University Press, 1973.
3. 3. C.D. Aliprantis and O. Burkinshaw, \Positive operators“, Academic Press, 1985.
4. 4. H.H. Schaefer, \Banach lattices and positive operators“, Berlin: Springer,1974.

Course: MA601

Approval:

Credit: 8

Group Theory: Dihedral groups, Permutation groups, Group actions, Sylow’s theorems,
Simplicity of the alternating groups, Direct and semidirect products, Solvable groups,
Nilpotent groups, Jordan Holder Theorem, free groups.Ring Theory: Properties of Ideals, Chinese remainder theorem, Field of fractions, Euclidean
domains, Principal ideal domains, Unique factorization domains, Polynomial Rings,
Irreducibility criteria, Matrix rings.Module Theory: Examples, quotient modules, isomorphism theorems, Generation of
modules, free modules, tensor products of modules, Exact sequences - Projective, Injective
and Flat modules.

Reference Book

1.  D. S. Dummit and R. M. Foote, Abstract Algebra. John Wiley & Sons, 2004.
2. T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer, 1980.
3. M. Artin, Algebra, Prentice Hall, 1991.
4. N. Bourbaki, Algebra, Springer, 1989.
5. C Musili, Introduction to Rings and Modules, Narosa Publishing House.
6. N. S. Gopalakrishnan, University Algebra, New Age International

Course: MA602

Approval:

Credit: 8

Linear Algebra: Matrix of a Linear transformation, dual vector spaces, determinants, Tensor
algebras, Symmetric algebras, Exterior algebras,Modules over PIDs: Basic theory, Structure theorem for finitely generated abelian groups,
Rational and Jordan canonical forms.Field Theory: Algebraic extensions, Splitting fields, Algebraic closures, Separable and Inseparable
extensions, Cyclotomic polynomials and extensions, Galois extensions, Fundamental
Theorem of Galois theory, Finite fields, Composite extensions, Simple extensions, Cyclotomic
extensions and Abelian extensions over rational field, Galois groups of polynomials,
Fundamental theorem of algebra, Solvable and Radical extensions, Computation of Galois
groups over rational field.

Reference Book

1. D. S. Dummit and R. M. Foote, Abstract Algebra. John Wiley & Sons, 2004.
2. T. W. Hungerford, Algebra, Graduate Texts in Mathematics, 73, Springer, 1980.
3. M. Artin, Algebra, Prentice Hall, 1991.
4. T. T. Moh: Algebra, World Scientic, 1992
5. N. Bourbaki, Algebra, Springer, 1989.

Course: MA603

Approval:

Credit: 8

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

Reference Book

1. D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
2. G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
3. P. R. Halmos, Measure Theory, Springer-Verlag, 1974.
4. H. L. Royden, Real Analysis, Macmillan 1988.
5. W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.
6. Elliott H. Lieb and Michael Loss , Analysis, American Mathematical Society, 2001

Course: MA604

Approval:

Credit: 8

Banach spaces: Review of Banach spaces, Hahn-Banach Theorem and its applications, Baire Category theorem and its applications like Closed graph theorem, Open mapping theorem.Topological Vector spaces: Weak and weak* topologies, locally convex topological vector spaces.
Hilbert spaces: Review of Hilbert spaces and operator Theory, Compact operators, Schauder's theorem on the spectral theory of compact operators.Banach algebras: Elementary properties,
Resolvent and spectrum, Spectral radius formula,
Ideals and homomorphisms, Gelfand transforms, Gelfand theorem for commutative Banach algebras.

Reference Book

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

Course: MA605

Approval:

Credit: 8

Topological spaces, Continuous maps between topological spaces, product topology,Quotient spaces, Connectedness, Compactness, Path connected spaces, separation axioms,Tychono spaces, Urysohn's lemma and metrization theoremDierentiable functions on Rn, Jacobian criteria, Taylor's theorem, Inverse function theorem,Implicit function theorem, Maxima-minima, Lagrange multiplier

Reference Book

1. Armstrong, Basic Topology, Springer, 1983
2. Munkres, Topology, Pearson Education, 2005.
3.  J. Dugundji, Topology
4. J. J. Duistermaat, J. A. C. Kolk: Multidimensional Real Analysis I: Dierentiation
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 Dierential Geometry and Lie Groups,TRIM series
9.  T. M. Apostol: Calculus: Multi-Variable Calculus and Linear Algebra With Applications to DierentialEquations And Probability- Vol 2.

Course: MA606

Approval:

Credit: 8

Homotopy Theory: Fundamental groups and its functorial properties, examples,Van- Kampen Theorem, Computation of fundamental group of S1.Covering spaces: Covering spaces, Computation of fundamental groups using cover- ings. Theclassication of covering spaces. Deck transformations.Simply connected spaces: Simply connected spaces-Universal covering spaces of locally simplyconnected and pathwise connected spaces. - Universal covering group of connected subgroupsof General Linear groups.Homology groups: Ane spaces, simplexes and chains - Homology groups - Properties ofHomology groups. - Chain Complexes, Relation Between one dimensional Homotopy andHomology groups. Computation of Homology groups Sn, Brouwer's xed point theorem.

Reference Book

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

Course: MA607

Approval:

Credit: 8

Review of basic Complex Analysis: Cauchy-Riemann equations, Cauchy's theoremand estimates, power series expansions, maximum modulus principle, Classication ofsingularities and calculus of residues; Normal families, Arzela-Ascoli theorem, Riemann mappingtheorem; Weierstrass factorization theorem, Runges theorem, Mittag-Leers theorem;Hadamard factorization theorem, Analytic Continuation, Gamma and Zeta functions

Reference Book

1. L. V. Ahlfors, Complex Analysis, Tata McGraw-Hill, 2013.
2. J. B. Conway, \Functions of one complex variable“, Second edition. Graduate Texts in Mathematics, 11. Springer-Verlag, New York-Berlin, 1978.
3. R. Narasimhan and Y. Nievergelt, \Complex analysis in one variable“, Second edition. Birkhuser Boston,Inc., Boston, MA, 2001.
4. W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 2013.
5. Wolfgang Fischer, Ingo Lieb, A Course in Complex Analysis: From Basic Results to Advanced Topics,Springer, 2012
6. Eberhard Freitag, Rolf Busam, Complex Analysis, Springer, 2005
7. Stein and Shakarchi, Complex Analysis, Princeton University Press, 2003
8. Gamelin, Complex Analysis, Springer, 2000