Course: M402

Approval: UG-Core, PG-Elective

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: UG-Core, PG-Elective

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: UG-Core, PG-Elective

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: UG-Elective, PG-Elective

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: UG-Elective, PG-Elective

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: PG-Elective

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: PG-Elective

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: PG-Elective

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: PG-Elective

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: PG-Elective

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: PG-Elective

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: PG-Elective

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: M101

Approval: UG-Core

Credit: 3

Method of Mathematical Proofs: Induction, Construction, Contradiction, Contrapositive.
Set: Union and Intersection of sets, Distributive laws, De Morgan's Law, Finite and infinite sets.
Relation: Equivalence relation and equivalence classes.
Function: Injections, Surjections, Bijections, Composition of functions, Inverse function, Graph of a function.
Countable and uncountable sets, Natural numbers via Peano arithmetic, Integers, Rational numbers, Real Numbers and Complex Numbers. Matrices, Determinant, Solving system of linear equations, Gauss elimination method, Linear mappings on R2 and R3, Linear transformations and Matrices.
Symmetry of Plane Figures: Translations, Rotations, Reflections, Glide-reflections, Rigid motions.

Reference Book

1. G. Polya, “How to Solve It”, Princeton University Press, 2004.
2. K. B. Sinha et. al., “Understanding Mathematics”, Universities Press (India), 2003.
3. M. Artin, “Algebra”, Prentice-Hall of India, 2007 (Chapters 1, 4, 5).
4. J. R. Munkres, “Topology”, Prentice-Hall of India, 2013 (Chapter 1).

Course: M102

Approval: UG-Core

Credit: 3

Concept of ordered field, Bounds of a set, ordered completeness axiom and
characterization of R as a complete ordered field. Archimedean property
of real numbers. Modulus of real numbers, intervals, neighbourhood of a
point.
Sequences of Real Numbers: Definition and examples, Bounded sequences, Convergence of sequences, Uniqueness of limit, Algebra of limits,
Monotone sequences and their convergence, Sandwich rule.
Series: Definition and convergence, Telescopic series, Series with non-negative terms.
Tests for convergence [without proof]: Cauchy condensation test, Comparison test, Ratio test, Root test, Absolute and conditional convergence, Alternating series and Leibnitz test.
Limit of a function at a point, Sequential
criterion for the limit of a function at a point. Algebra of limits, Sandwich theorem, Continuity at a point and on intervals, Algebra of continuous
functions. Discontinuous functions, Types of discontinuity.
Differentiability: Definition and examples, Geometric and physical interpretations, Algebra
of differentiation, Chain rule, Darboux Theorem, Rolle’s Theorem, Mean
Value Theorems of Lagrange and Cauchy. Application of derivatives: Increasing and decreasing functions, Maxima and minima of functions. Higher
order derivatives, Leibnitz rule, L’Hopital rule.

Reference Book

1. K. A. Ross, “Elementary Analysis”, Undergraduate Texts in Mathematics, Springer, 2013.
2. S. K. Berberian, “A First Course in Real Analysis”, Undergraduate Texts in Mathematics, Springer-Verlag, 1994.

Text Book

1. R.G Bartle, D.R. Sherbert, “Introduction to Real Analysis” , John Wiley & Sons, 1992.

Course: M141

Approval: UG-Core

Credit: 2

Introduction to computers, Linux and Shell Programming, Latex, Gnuplot.

Reference Book

1.  Peter Norton, “Introduction to Computers”, McGraw-Hill Education, 2004.
2. D. Morley, C. S. Parker “Fundamentals of Computers”, Ceneage Learning India Pvt. Ltd., 2011.
3. S. G. Kochan, P. Wood, “Unix Shell Programming”, Macmillan Computer Publications, 2003.
4. L. Lamport, “Latex: A Documentation Preparation System User’s Guide and Reference Manual”, Pearson/Addision Wesley, 1994.
5. Gnuplot documentation from official website of gnuplot.

Course: M142

Approval: UG-Core

Credit: 2

Programming language: C/C++; Algorithm and data structure: Stack, Queue, Linked list, Searching, Sorting.

Reference Book

1. B. W. Kernighan, D. M. Ritchie, “The C Programming Language”, Prentice-Hall, 2009.
2. E. Balaguruswamy, “Programming in ANSI C”, Tata McGraw-Hill, 2012.
3. B. Stroustrup, “The C++ Programming Language”, Pearson, 2014.
4. E. Balaguruswamy, “Object Oriented Programming With C++”, McGraw-Hill, 2013.
5. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, “Introduction to Algorithms”, MIT Press, Cambridge, 2009.

Course: M201

Approval: UG-Core

Credit: 4

Countability of a set, Countability of rational numbers, Uncountability of
real numbers. Limit point of a set, Bolzano-Weirstrass theorem, Open sets,
Closed sets, Dense sets. Subsequence, Limit superior and limit inferior of
a sequence, Cauchy criterion for convergence of a sequence, Monotone subsequence. Tests of convergence of series, Abel’s and Dirichlet’s tests for series, Riemann rearrangement theorem. Continuous functions on closed and
bounded intervals, Intermediate value theorem, Monotone functions, Continuous monotone functions and their invertibility, Discontinuity of monotone functions. Uniform continuity, Equivalence of continuity and uniform
continuity on closed and bounded intervals, Lipschitz condition, Other sufficient condition for uniform continuity. Riemann Integration: Darboux’s
integral, Riemann sums and their properties, Algebra of Riemann integrable
functions, Class of Riemann integrable functions, Mean value theorem, Fundamental theorems of calculus, Change of variable formula (statement only),
Riemann-Stieltjes integration (definition). Taylor’s theorem and Taylor’s series, Elementary functions. Improper integral, Beta and Gamma functions.

Reference Book

1. T. M. Apostol, “Calculus Vol. I”, Wiley-India edition, 2009.
2. S. K. Berberian, “A First Course in Real Analysis”, Undergraduate Texts in Mathematics, Springer-Verlag, 1994.

Text Book

1. R. G. Bartle, D. R. Sherbert, “Introduction to Real Analysis”, John Wiley & Sons, 1992.
2. K. A. Ross, “Elementary Analysis”, Undergraduate Texts in Mathematics, Springer, 2013.

Course: M202

Approval: UG-Core

Credit: 4

Groups, subgroups, normal subgroups, quotient groups, homomorphisms,
isomorphism theorems, automorphisms, permutation groups, group actions,
Sylow’s theorem, direct products, finite abelian groups, semi-direct products, free groups.

Reference Book

1. I. N. Herstein, “Topics in Algebra”, Wiley-India edition, 2013.
2. M. Artin, “Algebra”, Prentice-Hall of India, 2007.

Text Book

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

Course: M203

Approval: UG-Core

Credit: 4

Pigeonhole principle, Counting principles, Binomial coefficients, Principles
of inclusion and exclusion, recurrence relations, generating functions, Catalan numbers, Stirling numbers, Partition numbers, Schr ̈oder numbers, Block
designs, Latin squares, Partially ordered sets, Lattices, Boolean algebra.

Reference Book

1. J. H. van Lint, R. M. Wilson, “A Course in Combinatorics”, Cambridge University Press, 2001. 
2. I. Anderson, “A First Course in Discrete Mathematics”, Springer Undergraduate Mathematics Series, 2001.
3. R. P. Stanley, “Enumerative Combinatorics Vol. 1”, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2012.

Text Book

1. R. A. Brualdi, “Introductory Combinatorics”, Pearson Prentice Hall, 2010.
2. J. P. Tremblay, R. Manohar, “Discrete Mathematical Structures with Application to Computer Science”, Tata McGraw-Hill Edition, 2008.

Course: M204

Approval: UG-Core

Credit: 4

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, pathconnected sets. Sequences of functions, Pointwise convergence
and uniform convergence, Arzela-Ascoli Theorem, Weierstrass Approximation Theorem, power series, radius of convergence, uniform convergence
and Riemann integration, uniform convergence and differentiation, Stone
Weierstrass theorem for compact metric spaces.

Reference Book

1. R. R. Goldberg, “Methods of Real Analysis”, John Wiley & Sons, 1976.
2. G. B. Folland, “Real Analysis”, Wiley-Interscience Publication, John Wiley & Sons, 1999.

Text Book

1. G. F. Simmons, “Introduction to Topology and Modern Analysis”, Tata McGraw-Hill, 2013.
2. S. Kumaresan, “Topology of Metric Spaces”, Narosa Publishing House, 2005.

Course: M205

Approval: UG-Core

Credit: 4

System of Linear Equations, Matrices and elementary row operations, Rowreduced echelon form of matrices, Vector spaces, subspaces, quotient spaces,
bases and dimension, direct sums, Linear transformations and their matrix
representations, Dual vector spaces, transpose of a linear transformation,
Polynomial rings (over a field), Determinants and their properties, Eigenvalues and eigenvectors, Characteristic polynomial and minimal polynomial,
Triangulation and Diagonalization, Simultaneous Triangulation and diagonalization, Direct-sum decompositions, Primary decomposition theorem.

Reference Book

1. S. H. Friedberg, A. J. Insel, L. E. Spence, “Linear Algebra”, Prentice Hall, 1997. 
2. A. Ramachandra Rao, P. Bhimasankaram, “Linear Algebra”, Texts and Readings in Mathematics, 19. Hindustan Book Agency, New Delhi, 2000.
3. M. Artin, “Algebra”, Prentice-Hall of India, 2007.

Text Book

1. K. Hoffman, R. Kunze, “Linear Algebra”, Prentice-Hall of India, 2012.

Course: M206

Approval: UG-Core

Credit: 4

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.

Reference Book

1. K. L. Chung, F. AitSahlia, “Elementary Probability Theory”, Undergraduate Texts in Mathematics. Springer-Verlag, 2003. 
2. P. G. Hoel, S. C. Port, C. J. Stone, “Introduction to Probability Theory”, The Houghton Mifflin Series in Statistics. Houghton Mifflin Co., 1971.
3. W. Feller, “An Introduction to Probability Theory and its Applications Vol. 1 and Vol. 2”, John Wiley & Sons, 1968, 1971.

Text Book

1. S. Ross, “A First Course in Probability”, Pearson Education, 2012. 
2. D. Stirzaker, “Elementary Probability”, Cambridge University Press, Cambridge, 2003.

Course: M207

Approval: UG-Core

Credit: 4

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.

Reference Book

1. T. M. Apostol, “Introduction to Analytic Number Theory”, Springer International Student Edition, 2000. 
2. G. A. Jones, J. M. Jones, “Elementary Number Theory”, Springer Undergraduate Mathematics Series. Springer-Verlag, 1998.

Text Book

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

Course: M208

Approval: UG-Core

Credit: 4

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, maxflow min-cut theorem, Ramsey theory for graphs, matrices associated with
graphs.

Reference Book

1. B. Bollob ́as, “Modern Graph Theory”, Graduate Texts in Mathematics, 184. Springer- Verlag, 1998.
2. F. Harary, “Graph Theory”, Addison-Wesley Publishing Co., 1969.
3. J. A. Bondy, U. S. R. Murty, “Graph Theory”, Graduate Texts in Mathematics, 244. Springer, 2008.

Text Book

1. R. Diestel, “Graph Theory”, Graduate Texts in Mathematics, 173. Springer, 2010.

Course: M301

Approval: UG-Core

Credit: 4

Outer measure, measurable sets, Lebesgue measure, measurable functions,
Lebesgue integral, Basic properties of Lebesgue integral, convergence in
measure, differentiation and Lebesgue measure. L p Spaces, Holder and
Minkowski inequalities, Riesz-Fisher theorem, Radon-Nykodin theorem, Riesz
representation theorem. Fourier series, L 2 -convergence properties of Fourier
series, Fourier transform and its properties.

Reference Book

1. G. de Barra, “Measure Theory and Integration”, New Age International, New Delhi, 2003.
2. W. Rudin, “Principles of Mathematical Analysis”, Tata McGraw-Hill, 2013.

Text Book

1. H. L. Royden, “Real Analysis”, Prentice-Hall of India, 2012.
2. G. B. Folland, “Real Analysis”, Wiley-Interscience Publication, John Wiley & Sons, 1999.

Course: M302

Approval: UG-Core

Credit: 4

Rings, ideals, quotient rings, ring homomorphisms, isomorphism theorems,
prime ideals, maximal ideals, Chinese remainder theorem, Field of fractions, Euclidean Domains, Principal Ideal Domains, Unique Factorization
Domains, Polynomial rings, Gauss lemma, irreducibility criteria.Modules, submodules, quotients modules, module homomorphisms, isomorphism theorems, generators, direct product and direct sum of modules,
free modules, finitely generated modules over a PID, Structure theorem
for finitely generated abelian groups, Rational form and Jordan form of a
matrix, Tensor product of modules.

Reference Book

1. I. N. Herstein, “Topics in Algebra”, Wiley-India edition, 2013.
2. M. Artin, “Algebra”, Prentice-Hall of India, 2007.

Text Book

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

Course: M303

Approval: UG-Core

Credit: 4

Classifications of Differential Equations: origin and applications, family
of curves, isoclines. First order equations: separation of variable, exactequation, integrating factor, Bernoulli equation, separable equation, homogeneous equations, orthogonal trajectories, Picard’s existence and uniqueness theorems. Second order equations: variation of parameter, annihilator
methods. 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. Linear
system: general properties, fundamental matrix solution, constant coefficient system, asymptotic behavior, exact and adjoint equation, oscillatory
equations, Green’s function. Sturm-Liouville theory. Partial Differential
Equations: Classifications of PDE, method of separation of variables, characterstic method.

Reference Book

1. G. F. Simmons, S. G. Krantz, “Differential Equations”, Tata Mcgraw-Hill Edition, 2007.
2. B. Rai, D. P. Choudhury, “A Course in Ordinary Differential Equation”, Narosa Publishing House, New Delhi, 2002.
3. R. P. Agarwal, D. O Regan, “Ordinary and Partial Differential Equations”, Univer sitext. Springer, 2009.

Text Book

1. S. L. Ross, “Differential Equations”, Wiley-India Edition, 2009.
2. E. A. Coddington, “An Introduction to Ordinary Differential Equations”, Prentice Hall of India, 2012

Course: M304

Approval: UG-Core

Credit: 4

Reference Book

1. J. L. Kelley, “General Topology”, Graduate Texts in Mathematics, No. 27. Springer- Verlag, New York-Berlin, 1975.
2. K. J ̈ anich, “Topology”, Undergraduate Texts in Mathematics. Springer-Verlag, 1984

Text Book

1. J. R.Munkres, “Topology”, Prentice-Hall of India, 2013.
2. M. A. Armstrong, “Basic Topology”, Undergraduate Texts in Mathematics, Springer-Verlag, 1983.

Course: M305

Approval: UG-Core

Credit: 4

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 S 2 in sampling from a
normal distribution, Exact sampling distributions: χ 2 , 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 χ 2 , t, F .

Reference Book

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

Text Book

1. H. J. Larson, “Introduction to Probability Theory and Statistical Inference”, John Wiley & Sons, 1982.
2. V. K. Rohatgi, “Introduction to Probability Theory and Mathematical Statistics”, John Wiley & Sons, 1976.

Course: M306

Approval: UG-Core

Credit: 4

Differentiability of functions from an open subset of R n to R m and properties, chain rule, partial and directional derivatives, Continuously differentiable functions, Inverse function theorem, Implicit function theorem, Interchange of order of differentiation, Taylor’s series, Extrema of a function,
Extremum problems with constraints, Lagrange multiplier method with applications, Integration of functions of several variables, Change of variable
ormula (without proof) with examples of applications of the formula, spherical coordinates, Stokes theorem (without proof), Deriving Green’s theorem,
Gauss theorem and Classical Stokes theorem.

Reference Book

1. W. Kaplan, “Advanced Calculus”, Addison-Wesley Publishing Company, 1984.
2. T. M. Apostol, “Mathematical Analysis”, Narosa Publishing House, 2013.

Text Book

1. W. Fleming, “Functions of Several Variables”, Undergraduate Texts in Mathematics. Springer-Verlag, 1977.
2. T. M. Apostol, “Calculus Vol. II”, Wiley-India edition, 2009.

Course: M307

Approval: UG-Core

Credit: 4

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.

Reference Book

1. I. N. Herstein, “Topics in Algebra”, Wiley-India edition, 2013.M. Artin, “Algebra”, Prentice-Hall of India, 2007.J. Rotman, “Galois Theory”, Universitext, Springer-Verlag, 1998.

Text Book

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

Course: M308

Approval: UG-Core

Credit: 4

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.

Reference Book

1. W. Rudin, “Real and Complex Analysis”, Tata McGraw-Hill, 2013.
2. L. V. Ahlfors, “Complex Analysis”, Tata McGraw-Hill, 2013.
3. T. W. Gamelin, “Complex Analysis”, Undergraduate Texts in Mathematics, Springer, 2006.
4. E. M. Stein, R. Shakarchi, “Complex Analysis”, Princeton University Press, 2003.

Text Book

1. J. B. Conway, “Functions of One Complex Variable”, Narosa Publishing House, 2002.
2. R. E. Greene, S. G. Krantz, “Function Theory of One Complex Variable”, American Mathematical Society, 2011.

Course: M310

Approval: UG-Core

Credit: 4

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 R 3 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.

Reference Book

1. M. P. Do Carmo, “Differential Forms and Applications”, Springer, 1994.
2. J. A. Thorpe, “Elementary Topics in Differential Geometry”, Undergraduate texts in mathematics, Springer, 2011.

Text Book

1. M. P. Do Carmo, “Differential Geometry of Curves and Surfaces”, Prentice Hall, 1976.
2. Andrew Pressley, “Elementary Differential Geometry”, Springer, 2010.

Course: M311

Approval: UG-Core

Credit: 4

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

Reference Book

1. 1. S. D. Conte, C. De Boor, “Elementary Numerical Analysis, Tata McGraw-Hill, 2006.
2. 2. W. H. Press et. al., “Numerical Recipes - The Art of Scientific Computing”, Cambridge University Press, 2007.

Text Book

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

Course: M401

Approval: UG-Core

Credit: 4

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 self
adjoint compact operators.

Reference Book

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

Text Book

1. J. B. Conway, “A Course in Functional Analysis”, Graduates Texts in Mathematics 96, Springer, 2006.
2. B. Bollob ́as, “Linear Analysis”, Cambridge University Press, 1999.

Course: M402

Approval: UG-Core, PG-Elective

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: UG-Core, PG-Elective

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: UG-Core, PG-Elective

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: M451

Approval: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective, PG-Elective

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: UG-Elective, PG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: UG-Elective

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: MA701

Approval: iPhD-Core

Credit: 8

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

System of linear equations, matrices, Gauss elimination, Basis, dimension of a vec- tor 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, Gram- Schmidt 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.

Reference Book

1. Artin, M.; Algebra, Prentice Hall, 1991.
2. Lax, P., Linear Algebra and its applications, John Wiley & Sons, Second edi- tion, 2007.
3. Rose, H.E.; Linear Algebra, Birkhauser, 2002.

Text Book

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

Course: MA702

Approval: iPhD-Core

Credit: 8

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.

Sequences of functions, Pointwise convergence and uniform convergence, Arzela- Ascoli Theorem, Weierstrass Approximation Theorem, power series, radius of con- vergence, uniform convergence and Riemann integration, uniform convergence and differentiation.

Reference Book

1. S. Kumaresan, Topology of Metric Spaces, Narosa Publishing House, 2005.
2. G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2013.
3. Tom M. Apostol, Mathematical analysis, Addison-Wesley, 1974.
4. R. P. Boas, A Primer of Real Functions, MAA/AMS, Carus Monographs, Volume 13, Fourth Edition, 1960.

Text Book

1. N. L. Carothers, Real Analysis, Cambridge University Press, 2012.
2. W. Rudin, Principles of mathematical analysis, McGraw-Hill Book Co., 1976.

Course: MA703

Approval: iPhD-Core

Credit: 8

Outcome: Learning the elementary properties of rings of integers including divisibility, congru- ences, continued fractions and Gauss reciprocity laws.

Divisibility, Primes, Fundamental theorem of arithmetic, Congruences, Chinese re- mainder theorem, Linear congruences, Congruences with prime-power modulus, Fer- mat

Reference Book

1. T. M. Apostol, Introduction to Analytic Number Theory, Springer International Student Edition, 2000.
2. G. A. Jones, J. M. Jones, Elementary Number Theory, Springer Undergraduate Mathe- matics Series. Springer-Verlag, 1998.

Text Book

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

Course: MA704

Approval: iPhD-Core

Credit: 8

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 properties of solutions as maximum principle, asymptotic be- haviour and phase portrait analysis of 2nd order equations. • Learning characteristics method for solving 1st order partial Differential Equa- tions. Classifications of Differential Equations: origin and applications, family of curves, isoclines. First order equations: separation of variable, exact equation, integrating factor, Bernoulli equation, separable equation, homogeneous equations, orthogonal trajectories, Picard’s existence and uniqueness theorems. Second order equations: variation of parameter, annihilator methods. Series solution: power series solutions about regular and singular points. Method of Frobenius, Bessel’s equation and Legen- dre equations. Wronskian determinant, Phase portrait analysis for 2nd order system, comparison and maximum principles for 2nd order equations. Linear system: gen- eral properties, fundamental matrix solution, constant coefficient system, asymptotic behavior, exact and adjoint equation, oscillatory equations, Green’s function. Sturm- Liouville theory. Partial Differential Equations: Classifications of PDE, method of separation of variables, characteristic method.

Reference Book

1. G. F. Simmons, S. G. Krantz, Differential Equations, Tata Mcgraw-Hill Edition, 2007.
2. B. Rai, D. P. Choudhury, A Course in Ordinary Differential Equation, Narosa Publishing House, New Delhi, 2002.
3. R. P. Agarwal, D. OR´egan, Ordinary and Partial Differential Equations, Universitext. Springer, 2009.
4. C. M. Bender, S. A. Orszag, Advanced mathematical methods for Scientists and Engineers, Springer Verlag, 1999.

Text Book

1. S. L. Ross, Differential Equations, Wiley-India Edition, 2009.
2. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall of India, 2012.

Course: MA705

Approval: iPhD-Core

Credit: 8

Outcome: Learning the notion of limits, continuity, differentiation and integration in the higher dimensional euclidean spaces.

Differentiability of functions from an open subset of Rn to Rm and properties, chain rule, partial and directional derivatives, Continuously differentiable functions, Inverse function theorem, Implicit function theorem, Interchange of order of differentiation, Taylor

Reference Book

1. W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Company, 1984.
2. T. M. Apostol, Mathematical Analysis, Narosa Publishing House, 2013.

Text Book

1. W. Fleming, Functions of Several Variables, Undergraduate Texts in Mathematics. Springer- Verlag, 1977.
2. T. M. Apostol, Calculus Vol. II, Wiley-India edition, 2009.

Course: MA706

Approval: iPhD-Core

Credit: 8

Outcome: Understanding the properties of group actions and their various applications. Under- standing the various ring structures, especially polynomial rings over fields.

Group Theory: Dihedral and Permutation groups, normal subgroups, group homo- morphisms (Review only). Group isomorphism theorems, Group actions, Sylow

Reference Book

1. Hungerford, T. W.; Algebra, Graduate Texts in Mathematics, 73, Springer.
2. Burton, David; A First Course in Rings and Ideals. Addison-Wesley.
3. Artin, M.; Algebra, Prentice Hall, 1991.

Text Book

1. Dummit, D. S.; Foote, R. M.; Abstract Algebra, Third Edition, John Wiley & Sons.

Course: MA707

Approval: iPhD-Core

Credit: 8

Outcome: Upon successful completion of the course, students will be familiar with various ad- vanced concepts and techniques from measure theory.

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.

Reference Book

1. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1986.
2. H. L. Royden, Real Analysis, Prentice-Hall of India, 2012.
3. R. B. Ash; C. A. Dol´eans-Dade, Probability and Measure Theory, Academic Press, 2nd Edition.

Text Book

1. G. B. Folland, Real Analysis, Wiley-Interscience Publication, John Wiley & Sons, 1999.
2. S. Kesavan, Measure and Integration, Texts and Readings in Mathematics 77, Hindustan Book Agency, 2019.

Course: MA708

Approval: iPhD-Core

Credit: 8

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

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

Reference Book

1. K. L. Chung, F. AitSahlia, Elementary Probability Theory, Undergraduate Texts in Mathematics. Springer-Verlag, 2003.
2. P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Probability Theory, The Houghton Mifflin Series in Statistics. Houghton Mifflin Co., 1971.
3. W. Feller, An Introduction to Probability Theory and its Applications Vol. 1 and Vol. 2, John Wiley & Sons, 1968, 1971.

Text Book

1. S. Ross, A First Course in Probability, Pearson Education, 2012.
2. D. Stirzaker, Elementary Probability, Cambridge University Press, Cambridge, 2003.

Course: MA709

Approval: iPhD-Core

Credit: 8

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

Algebraic and geometric representation of complex numbers; elementary functions in- cluding the exponential functions and its relatives (log, cos, sin, cosh, sinh, etc.); con- cept 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

Reference Book

1. W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 2013.
2. L. V. Ahlfors, Complex Analysis, Tata McGraw-Hill, 2013.
3. T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer, 2006.
4. E. M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press, 2003.

Text Book

1. J. B. Conway, Functions of One Complex Variable, Narosa Publishing House, 2002.
2. R. E. Greene, S. G. Krantz, Function Theory of One Complex Variable, American Mathe- matical Society, 2011.

Course: MA710

Approval: iPhD-Core

Credit: 8

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

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 coloring of graphs, network flows, max-flow min-cut theorem, Ramsey theory for graphs, matrices associated with graphs.

Reference Book

1. B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, 184. Springer-Verlag, 1998.
2. F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.
3. J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244. Springer, 2008.

Text Book

1. R. Diestel, Graph Theory, Graduate Texts in Mathematics, 173. Springer, 2010.

Course: MA801

Approval: iPhD-Core

Credit: 8

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

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 well-known Banach spaces, Hilbert spaces, Riesz representation theorem, Adjoint operator, Compact operators, Spectral theorem for compact self-adjoint operators.

Reference Book

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

Text Book

1. J. B. Conway, A Course in Functional Analysis, Graduates Texts in Mathematics 96, Springer, 2006.
2. B. Bollob´as, Linear Analysis, Cambridge University Press, 1999.

Course: MA802

Approval: iPhD-Core

Credit: 8

Outcome: Understanding of the basic theory of modules, category and functions, algebras.

Modules, submodules, module homomorphisms, quotient modules, isomorphism the- orems, 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 mod- ules, Projective, Injective and Flat modules, Ext, Tor. Algebras, Tensor Algebras, Symmetric Algebras, Exterior Algebras, Determinants. Length of Modules, Noethe- rian and Artinian modules, Hilbert Basis Theorem.

Reference Book

1. Lang, S.; Algebra, Revised Third Edition, Springer, GTM 211.
2. Weibel, Charles A.; An Introduction to Homological Algebra, Cambridge University Press, 1995.
3. Atiyah, M.F.; McDonald, I.G.; Introduction to Commutative Algebra, CRC Press, 2018.

Text Book

1. Dummit, D.S.; Foote, R.M.; Abstract Algebra, Third Edition, John Wiley & Sons.
2. Rotman, J.; An Introduction to Homological Algebra, Springer, 2009.
3. Sing, Balwant; Basic Commutative Algebra, World Scientific, 2011.

Course: MA803

Approval: iPhD-Core

Credit: 8

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’. Learning basic notions of fundamental groups and covering spaces and some of its applications.

Topological Spaces, Open and closed sets, Interior, Closure and Boundary of sets, Basis for Topology, Product Topology, Subspace Topology, Metric Topology, Com- pact Spaces, Locally compact spaces, Continuous functions, Open map, Homeomor- phisms, 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.

Reference Book

1. J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975.
2. K. J¨anich, Topology, Undergraduate Texts in Mathematics. Springer-Verlag, 1984.
3. W. G. Chinn, N. E. Steenrod, First concepts of Topology, The Mathematical Association of America, 1978.

Text Book

1. J. R.Munkres, Topology, Prentice-Hall of India, 2013.
2. M. A. Armstrong, Basic Topology, Undergraduate Texts in Mathematics, Springer-Verlag, 1983.

Course: MA804

Approval: iPhD-Core

Credit: 8

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

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.

Reference Book

1. M. P. Do Carmo, Differential Forms and Applications, Springer, 1994.
2. J. A. Thorpe, Elementary Topics in Differential Geometry, Undergraduate texts in math- ematics, Springer, 2011.

Text Book

1. M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.
2. Andrew Pressley, Elementary Differential Geometry, Springer, 2010.

Course: MA805

Approval: iPhD-Core

Credit: 8

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 Holm- green for quasilinear equations.

Classification of Partial Differential Equations, Cauchy Problem, Cauchy-Kowalevski Theorem, Lagrange-Green identity, The uniqueness theorem of Holmgren, Transport equation: Initial value problem, nonhomogeneous problem. Laplace 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. G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995.
2. S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, 1989.

Text 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.

Course: MA806

Approval: iPhD-Core

Credit: 8

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

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 rationales, Solvable groups, nilpotent groups, Solvability by radicals, Transcendental extensions.

Reference Book

1. I. N. Herstein, Topics in Algebra, Wiley-India edition, 2013.
2. M. Artin, Algebra, Prentice-Hall of India, 2007.
3. J. Rotman, Galois Theory, Universitext, Springer-Verlag, 1998.
4. S. Lang, Algebra, Revised Third Edition. Spinger.

Text Book

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

Course: MA901

Approval: iPhD-Core

Credit: 8

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.

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, In- variance 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: MA851

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the representation of finite groups via character theory.

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 Mathemat- ics 10, American Mathematical Society, 2009.

Text Book

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

Course: MA852

Approval: iPhD-Elective

Credit: 8

Outcome: Learning 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.

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: MA853

Approval: iPhD-Elective

Credit: 8

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

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 theorem; 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.
2. A. P. Robertson, W. Robertson, Topological Vector Spaces, Cambridge Tracts in Mathematics 53, Cambridge University Press, 1980.
3. J. B. Conway, A Course in Functional Analysis, Graduates Texts in Mathematics 96, Springer, 2006.

Course: MA854

Approval: iPhD-Elective

Credit: 8

Outcome: Understanding the theory of discrete time and continuous time Markov chains.

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.

Reference Book

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

Course: MA855

Approval: iPhD-Elective

Credit: 8

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.

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

Course: MA856

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the different algebraic techniques used in the study of the graphs

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-transitive 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: MA857

Approval: iPhD-Elective

Credit: 8

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

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: MA858

Approval: iPhD-Elective

Credit: 8

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

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: MA859

Approval: iPhD-Elective

Credit: 8

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.

Overview of Cryptography and cryptanalysis, some simple cryptosystems (e.g., shift, substitution, affine, knapsack) and their cryptanalysis, classification of cryptosystem, 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.

Reference Book

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

Course: MA860

Approval: iPhD-Elective

Credit: 8

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

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: MA861

Approval: iPhD-Elective

Credit: 8

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.

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. W. C. Haffman, V. Pless, Fundamentals of Error-Coding Codes, Cambridge University Press, 2003.
5. S. Lin, An Introduction to Error-Correcting Codes, Prentice-Hall, 1970.

Course: MA862

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the propositional logic and first order theory.
Understanding the completeness and compactness theorems with Godel's incompleteness theorem.

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, Meta theorems 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.

Reference Book

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

Course: MA863

Approval: iPhD-Elective

Credit: 8

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.

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: MA864

Approval: iPhD-Elective

Credit: 8

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

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, L? functional Calculus, Toeplitz operators.

Course: MA865

Approval: iPhD-Elective

Credit: 8

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.

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.

Reference Book

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

Course: MA866

Approval: iPhD-Elective

Credit: 8

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

Topological Groups: Basic properties of topological groups, subgroups, quotient groups. Examples of various matrix groups. Connected groups. 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). 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. 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. 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.

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: MA867

Approval: iPhD-Elective

Credit: 8

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

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 ? 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. Gouv ea, p-adic Numbers, Universitext, Springer-Verlag, 1997.

Course: MA868

Approval: iPhD-Elective

Credit: 8

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

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.

Reference Book

1. A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, Springer, 2013.
2. K. L. Chung, A Course in Probability Theory, Academic Press, 2001.
3. S. I. Resnick, A Probability Path, Birkhauser, 1999.
4. P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics, John Wiley & Sons, 2012.
5. J. Jacod, P. Protter, Probability Essentials, Universitext, Springer-Verlag, 2003.
6. S. R. S. Varadhan, Probability Theory, Courant Lecture Notes, Vol. 7, AMS, 2001.

Course: MA869

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the use of different algebraic technique to study the combinatorial problems

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.

Reference Book

1. R. P. Stanley, Algebraic Combinatorics, Undergraduate Texts in Mathematics, Springer, 2013.
2. M. Aigner, A Course in Enumeration, Graduate Texts in Mathematics 238, Springer, 2007.
3. R. P. Stanley, Enumerative Combinatorics Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 1999.

Course: MA870

Approval: iPhD-Elective

Credit: 8

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

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: MA871

Approval: iPhD-Elective

Credit: 8

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

Definitions and Examples, 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. Bart De Bruyn,An Introduction to Incidence Geometry, Frontiers in Mathematics, Birkhauser/Springer, Cham 2016.
4. Gyorgy Kiss and Tamas Szonyi, Finite Geometries, CRC Press, Boca Raton, FL 2020.
5. G. E. Moorhouse, Incidence Geometry, 2007, available online from http://www.uwyo.edu/moorhouse/handouts/incidence geometry.pdf

Course: MA872

Approval: iPhD-Elective

Credit: 8

Outcome: Understanding the basics of Lie Algebra

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, Semi simple 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 Semi simple Lie Algebras, Springer Monographs in Mathematics, Springer-Verlag, 2001.
4. N. Jacobson, Lie Algebras, Dover Publications, 1979.

Course: MA873

Approval: iPhD-Elective

Credit: 8

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

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: MA874

Approval: iPhD-Elective

Credit: 8

Outcome: Learning random graphs and their applications.

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: MA875

Approval: iPhD-Elective

Credit: 8

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

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: MA876

Approval: iPhD-Elective

Credit: 8

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.

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, Poincar e 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: MA877

Approval: iPhD-Elective

Credit: 8

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

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, Addison-Wesley Publishing Co., 1969.

Course: MA878

Approval: iPhD-Elective

Credit: 8

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.

Linear algebra and lattices: Asymptotically fast matrix multiplication algorithms, linear algebra algorithms, normal forms over fields, Lattice reduction; Solving system of non-linear equations: Gro bner basis, Buchberger's algorithms, Complexity of Gro bner 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.

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. H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, 1993.
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: MA879

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the elementary properties of Dirichlet series and distribution of primes.

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: MA880

Approval: iPhD-Elective

Credit: 8

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.

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: MA881

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the basics of Ergodic Theory.

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, Halmosvon 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.

Reference Book

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

Course: MA882

Approval: iPhD-Elective

Credit: 8

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

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.

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.
4. A. Terras, Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincar e Upper Half-Plane, Second Edition, Springer, 2013.

Course: MA883

Approval: iPhD-Elective

Credit: 8

Outcome: Learning the rudiments of Lie groups and irreducible representations of compact Lie groups parametrized by Weyl Character formula.

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), semi simple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria, Jordan decomposition. Complex semi simple 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.

Course: MA884

Approval: iPhD-Elective

Credit: 8

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

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; Existence 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: MA885

Approval: iPhD-Elective

Credit: 8

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

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 Differential 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: MA886

Approval: iPhD-Elective

Credit: 8

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

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). Representation Theory: General properties of representations of a locally compact group, Complete reducibility, Basic operations on representations, Irreducible representations. 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 Character formula, Computing the Unitary dual of SU (2), SO(3); Fourier analysis on SO(n).

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, Analyze 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. Wawrzyn czyk, Group Representations and Special Functions, PWN Polish Scientific Publishers, 1984.

Course: MA887

Approval: iPhD-Elective

Credit: 8

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

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.

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: MA888

Approval: iPhD-Elective

Credit: 8

Outcome: Learning elliptic curves and the structure of their rational points.

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: MA889

Approval: iPhD-Elective

Credit: 8

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

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.

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: MA890

Approval: iPhD-Elective

Credit: 8

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

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 subgroup, 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.

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: MA891

Approval: iPhD-Elective

Credit: 8

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.

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 semi simple Lie algebras. Representations of Semi simple 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.

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: MA892

Approval: iPhD-Elective

Credit: 8

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

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, Black-Scholes 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: MA893

Approval: iPhD-Elective

Credit: 8

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

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.

Reference Book

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

Course: MA894

Approval: iPhD-Elective

Credit: 8

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

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.

Reference Book

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

Text Book

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

Course: MA895

Approval: iPhD-Elective

Credit: 8

Outcome: Understanding analytic and harmonic functions on the unit disc.
Understanding properties Hp spaces, for 1 ? p < ?.
Understanding invariant subspaces for the shift operator on H2 space.

Fourier Series: Cesaro Means, Characterization of Types of Fourier Series; Analytic and Harmonic Functions in the Unit Disc: The Cauchy and Poisson Kernels, Boundary Values, Fatou's Theorem, Hp Spaces; The Space H1: The Helson-Lowdenslager Approach, Szego's Theorem, Completion of the Discussion of H1; 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; Analytic Functions with Continuous Boundary Values: Conjugate Harmonic Functions, Theorems of Fatou and Rudin; The Shift Operator: The Shift Operator on H2, Invariant Subspaces for H2 of the Half-plane, Isometries, The Shift Operator on L2.

Reference Book

1. Kenneth Hoffman, Banach spaces of analytic functions, Reprint of the 1962 original, Dover Publications, Inc., New York, 1988.
2. Walter Rudin, Real and complex analysis, Third edition. McGraw-Hill Book Co., New York, 1987.
3. Duren, Peter L., Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 1970.

Course: MA896

Approval: iPhD-Elective

Credit: 8

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

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 H? and H?, The role of outer functions, Contractions of class C0; Operator-Valued Analytic Functions: The spaces L2(U ) and H2(U ), 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.

Reference Book

1. B ela Sz.-Nagy, Ciprian Foias, Hari Bercovici and La szlo K erchy, Harmonic analysis of operators on Hilbert space; Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010.
2. Vern Ival Paulsen, completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.
3. 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. 892.
4. Nikolai K. Nikolskii, Operators, functions, and systems: an easy reading. Vol. 1 and Vol 2. Hardy, Hankel, and Toeplitz, American Mathematical Society, Providence, RI, 2002.