MATHEMATICS

Integrated Ph.D. Programme

(Mathematical Sciences)

 

Course No.     Credits    Course Title                       

Core:                 (core courses compulsory)

 

MA 212            3:0      Algebra I

MA 219            3:0      Linear Algebra

MA 221           3:0     Analysis I

MA 222           3:0     Analysis II

MA 223             3:0    Functional Analysis

MA 224           3:0    Complex Analysis

MA 229            3:0    Calculus on Manifolds

MA 231           3:0    Topology I

MA 241            3:0   Ord. Differential Eqns.                                     

MA 242            3:0    Partial Diff. Eqns.

MA 261            3:0    Probability Models

MA 331            3:0    Topology & Geometry 

Soft Core:

 

MA 213            3:0      Algebra II

MA 361            3:0      Probability Theory       

Project:

 

MA  201         0 : 7    Project 

Elective Courses:

MA  251        3:0    Numerical  Methods

MA 312         3:0    Commutative Algebra

MA 320          3:0    Representation Theory of  

                                Compact Lie Groups

MA 322        3:0     Fourier / Harmonic

                                Analysis                              

MA 326        3:0     Fourier Analysis

MA 329        3:0    Topics in Several 

                               Complex  Variables      

MA 330        3:0     Topology  II

MA 332        3:0     Algebraic Topology

MA 333        3:0     Topics in Riemannian  

                                Geometry

MA 368        3:0      Topics in Probability

                                 & Stochastic Processes

MA  369       3:0      Random Matrix Theory      

 

 

MA  212  (AUG)  3:0

Algebra    I

 

Groups: Review of groups, subgroups, homomorphisms, normal subgroups, quotient groups, isomorphism theorems. Group actions and its applications,  Sylow theorems.  Structure of finitely generated  abelian  groups, free  groups. Rings: Review  of  rings, homomorphisms, ideals and isomorphism theorems. Prime  ideals, maximal ideals.  Chinese  remainder  theorem.  Euclidean domains, principal  ideal  domains, unique  factorization domains. Factorization  in  polynomial  rings. Modules: Modules, homomorphisms and  exact sequences.  Free modules. Hom and  tensor  products.  Structure  theorem  for  modules  over  PIDs.

 

C S Praneshachar/B J Venkatachala

 

Artin, M., Algebra, Prentice Hall of India, 1994.

Dummit, D.S., and Foote, R. M., Abstract Algebra, John Wiley and Sons, 2001.

Hungerford, T.W., Algebra, Springer, India, 2004.

Herstein, I.N., Topics in Algebra, John Wiley & Sons, 1995.

 

 

MA  213 (JAN)  3:0

Algebra II

 

Representation theory: Representations of finite groups, irreducible representations, complete reducibility, Schur’s lemma, characters, orthogonality,  class functions,  regular representations and induced representations, the group  algebra. Linear groups: Representations of the group  SU2.

 

T. Bhattacharyya

 

Artin, M., Algebra, Prentice Hall of India, 1994.

Fulton, W., and Harris, J., Representation Theory, Springer Verlag,  1991.

Serre, J. P., Linear Representations of Finite Groups, Springer Verlag, 1977.

 

 

MA 219  (AUG)  3:0

Linear   Algebra

 

Vector  spaces:  Basis and dimension, direct  sums. Determinants: Theory of determinants,  Cramer’s  rule. Linear  transformations: Rank-nullity theorem, algebra  of  linear transformations,  dual spaces. Linear operators, eigenvalues and eigenvectors, characteristic  polynomial,  the Cayley-Hamilton theorem, minimal   polynomial, algebraic and geometric  multiplicities,  diagonalization, the Jordan canonical form. Symmetry: Group of motions of the plane, Discrete groups  of  motion, finite subgroups of  S0(3). Bilinear  forms: Symmetric,  skew  symmetric   and Hermitian  forms. Sylvester’s law of inertia. Spectral theorem  for Hermitian  and   normal   operators   on  finite  dimensional   vector  spaces. Linear groups: Classical  linear  groups, SU2   and SL2(R).

 

Basudeb Datta

 

Artin, M., Algebra, Prentice Hall of India, 1994.

Herstein, I.N., Topics in Algebra, Vikas Publications, 1972.

Strang, G., Linear Algebra and its Applications, Third Edn, Saunders, 1988.

Halmos, P., Finite dimensioinal Vector Spaces, Springer Verlag (UTM), 1987.

 

MA 221 (AUG) 3:0

Analysis   I

 

Review of real and complex number systems, topology  of metric spaces. Continuity  and 

differentiability. The intermediate value theorem. Mean  value  theorems and Taylor’s theorem, Intermediate  value  theorem. The Riemann-Stieltjes integral. Introduction to functions of several variables, differentiablility, directional and total derivatives. Sequences and series of functions, uniform convergence, the Weierstrass approximation theorem.

 

 Gautam Bharali

 

Royden, H. L., Real Analysis, Macmillan, 1988.

Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1986.

 

MA 222 (JAN) 3:0

Analysis    II

 

Construction of the Lebesgue measure, measurable  functions, limit  theorems.  Lebesgue integration.  Different notions of convergence and convergence theorems. Product measures  and  Fubini’s theorem. Signed measure and the Radon-Nikodym theorem, change of variables, complex  measures.

 

 A K Nandakumaran

 

De  Barra, G.,  Introduction  to  Measure  Theory,  Von  Nostrand  Reinhold,  1974.

Hewitt, E.,  and   Stromberg,  K.,  Real  and  Abstract  Analysis,  Springer,  1969.

Royden, H. L.,  Real  Analysis,  Macmillan, 1988.

Rudin, W., Real and  Complex  Analysis, McGraw-Hill,  1986.

 

 

 

MA 223 (AUG) 3:0

Functional  Analysis

 

Basic topological concepts, metric spaces, normed linear spaces, Banach spaces, bounded linear functionals and dual spaces, Hahn-Banach  theorem. Bounded linear operators, open-mapping heorem, closed graph theorem. The Banach-Steinhaus theorem. Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on  a  Hilbert  space. The spectral  theorem  for compact,   self-adjoint  operators.

 

 Manjunath Krishnapur

 

Goffman, C., and  Pedrick, G., First Course in Functional Analysis, Prentice Hall India, 1995.

Conway, J.B., A Course in Functional  Analysis, Springer,  1990.

Taylor, A.E., Introduction to Functional Analysis, Wiley International Edition, 1958.

 

 

MA 224 (JAN) 3:0

Complex  Analysis

 

Complex numbers, complex-analytic functions, Cauchy’s integral formula, power series, Liouville’s  theorem. The maximum-modulus theorem. Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal  mappings. The Schwarz lemma, automorphisms of the disc. Normal families and Montel’s theorem. The Riemann  mapping  theorem.

 

 S  Thangavelu

 

Ahlfors, L.V., Complex Analysis, McGraw-Hill, 1979.

Conway, J.B., Functions of One Complex Variable, Springer Verlag, 1978.

Gamelin, T.W., Complex Analysis, UTM, Springer, 2001.

 

 

MA 229 (JAN) 3:0

Calculus on Manifolds

 

Functions of several real variables: Differentiability, the total derivative and directional derivatives. The inverse and the implicit function theorems, Sard’s theorem. Integration on Euclidean spaces: Volume integrals and iterated integrals, Fubini’s theorem, the change-of-variables formula, partitions of unity. Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes’ theorem and applications.

 

 Gadadhar Misra

 

Spivak, M., Calculus on Manifolds, W.A. Benjamin, Inc. 1965.

Apostol, T.A., Mathematical Analysis, Narosa, Indian Edn.

Munkres, J., Analysis on Manifolds, Westview Press, 1997

Rudin, W., Principles of Mathematical Analysis, McGraw Hill, 1976.

 

MA  231  (AUG)  3 : 0

Topology   I

 

Open and closed sets, continuous functions, the metric topology, the product topology, the ordered topology, the quotient topology. Connectedness and path connectedness, local path connectedness. Compactness, countability axioms, separation axioms. Complete metric spaces, the Baire category theorem. Urysohn’s embedding theorem.  Topological groups, orbit spaces

 

 E K Narayanan/H Seshadri

 

Armstrong, M.A., Basic Topology, Springer, India, 2004.

Janich, K., Topology, Springer Verlag, UTM, 1984.

Munkres, K.R., Topology, Pearson Education, 2005.

Simmons, G.F., Topology and Modern Analysis, McGraw-Hill, 1963.

 

MA 241  (JAN)  3:0

Ordinary  Differential Equations

 

Basic concepts: Phase space, existence and uniquness theorems, dependence on initial conditions, flows. Linear  systems: The fundamental  matrix, stability of equilibrium  points. Sturm-Liouville  theory. Nonlinear  systems and their  stability: The Poincare-Bendixon  theorem, perturbed  linear  systems,  Lyapunov  methods.

 

 G  Rangarajan

 

Coddington, E.A., and Levinson, N., Theory of Ordinary Differential Equations, Tata McGraw Hill, 1972.

Birkhoff, G., and Rota, G.-C., Ordinary Differential Equations, Wiley, 1989.

Hartman, P., Ordinary Differential Equations, Birkhaeuser, 1982.

Perko, L., Differential Equations and Dynamical Systems, Springer Verlag, 1991.

 

 

MA  242  (AUG)  3: 0

Partial  Differential  Equations

 

First-order partial differential equations and Hamilton-Jacobi equations. Cauchy problem and  classification of second-order  equations,  Holmgren’s  uniqueness  theorem. Laplace’s equation,  diffusion  equation, wave  equation. Some methods of solutions, method of separation of variables.

 

 Kaushal Verma

 

Garabedian, P. R., Partial Differential Equations, John Wiley and Sons, 1964.

Prasad. P., and Ravindran, R., Partial Differential Equations, Wiley Eastern, 1985.

Renardy, M., and Rogers, R.C., An Introduction to Partial Differential Equations, Springer Verlag, 1992.

Fritz John, Partial differential Equations, Springer, Intl Students Edn), 1971.

 

MA 251 (AUG) 3:0

Numerical Methods

 

Numerical solution of algebraic and transcendental equations, iterative algorithms, convergence, Newton-Raphson procedure, solutions of polynomial and simultaneous linear equations, Gauss method, relaxation procedure, error estimates. Numerical integration, Euler-Maclaurin formula, Newton-Cotes formulae, error estimates, Gaussian quadratures, extensions to multiple integrals. Methods of Euler, Adams, Runge-Kutta and   predictor-corrector procedures, stability of solution.  Solution of boundary value problems: shooting method with least-square convergence criterion, quasi-linearization method, parametric differentiation technique, invariant finite-difference techniques, stability and convergence of the solution, method of characteristics.

 

Mahesh Kumari

 

Gupta, A., and Bose, S.C., Introduction to Numerical Analysis, Academic Publishers, 1989.

Conte, S.D., and de Boor C., Elementary Numerical Analysis, McGraw-Hill, 1980.

Hildebrand, F.B., Introduction to Numerical Analysis, Tata McGraw-Hill, 1988.

 

MA 261 (AUG) 3:0

Probability Models

 

Sample spaces, events, probability, discrete and continuous random variables. Conditioning and independence, Bayes’  formula. Moments and the moment generating function, characteristic function. Laws of large numbers, central limit theorem, Markov chains, Poisson processes.

 

 M K Ghosh

Ross, S.M., Introduction to Probability Models, Academic Press 1993.

Taylor, H.M., and Karlin, S., An Introduction to Stochastic Modelling, Academic Press, 1994.

 

 

iyah, M. F. and  Macdonald,  I. G.                                   MA 312  (JAN)  3:0

Commutative Algebra

 

Noetherian  rings  and  Modules, localisations,  exact  sequences,  Hom,  tensor  products,  Hilbert’s  Null-stellensatz,  integral  dependence,  going-up  and  going  down  theorems,  Noether’s  normalization  lemma,  discrete  valuation  rings  and  Dedekind  domains.

 

D  P  Patil

 

Atiyah, M.F., and Macdonald, I.G., Introduction to Commutative Algebra, Addison Wesley, 1969.

Raghavan., S.B. Singh and Sridharan, R., Homological  Methods in Commutative Algebra, TIFR Mathematical Pamphlet. No.5, Oxford University Press, 1977.  

Serre, J.P., Local Algebra (translated from French), Springer Monographs in Mathematics, Springer Verlag, 2000.

Zariski, O., and Samuel P., Commutative Algebra, Vols. I and  II,  Van Nostrand, 1958 and 1960.

 

 

MA  320  (AUG)  3:0

Representation  Theory of Compact Lie Groups

 

Lie  groups, Lie  algebras, matrix groups, representations, Schur’s  orthogonality  relations, Peter-Weyl  theorem, structure  of  compact  semisimple Lie groups, maximal  tori,  roots and rootspaces, classification  of fundamental systems Weyl group, Highest weight theorem, Weyl  Integration  formula, Weyl’s  character formula.

 

E  K Narayanan

 

Varadarajan, V.S., Lie algebras and their representations, Springer 1964.

Hall, B.C., Lie groups, Lie algebras and representations, Springer 2003.

Barry Simon, Representations of finite and compact groups, AMS, 1996.

Knapp, A.W., Representation theory of semi simple lie groups. An overview based on examples, Princeton Univ. Press, 2002.

 

 MA 322 (AUG) 3:0

Fourier/ Harmonic Analysis

 

Prerequisites:  MA  326

Harmonic Analysis in phase space: Heisenberg group, Fourier-Wigner transform, Weyl transform, Time-Frequency analysis, Short-time Fourier transform, Gabor frames, Zak transform, wavelets, modulation spaces, uncertainty principles.

 

S Thangavelu

 

Folland, G.B., Harmonic Analysis in phase space, Ann. Math. Stud., Princeton University Press, Princeton, 1989.

Groechenig, K., Foundations of  time frequency analysis, Birkhauser, Boston, 2000.

 

MA 326 (JAN) 3:0

Fourier Analysis

           

Introduction to Fourier Series. Plancherel theorem, basis approximation theorems, Dini’s Condition etc.  Introduction to Fourier transform; Plancherel theorem Wiener-Tauberian theorems, interpolation of operators, maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions. Introduction to singular integral operators.

 

S Thangavelu

 

Dym, H. and Mckean, H.P., Fourier series and integrals, 1972.

Stein, E.M., Singular integrals and differentiability properties of functions, 1970.

Stein, E.M., and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, 1975.

Sadosky, C., Interpolation of operators and Singular integrals, 1979.

 

MA 329 (AUG) 3:0
Topics in Several Complex Variables
 

Prerequisites:  MA  224

 

Interpolation in several complex variables. Classical theorems of Nevanlinna-Pick type, Rudin-Carleson type theorems in one and higher dimensions; d-bar methods. 

 
Kaushal Verma
 
Hormander, L., Introduction to Complex Analysis in Several Variables, Third Edn, North-Holland, 1990.

Krantz, S.G., Function Theory of Several Complex Variables, AMS Chelsea Publishing, 2001.

Rudin, W., Function Theory in the Unit Ball of Cn, Springer Verlag, 1980.

 

 
MA 329 (JAN) 3:0
Topics in  Complex  Analysis
 

Prerequisites:  MA  224

 
Operator space structures on function algebras.  The MIN and the MAX, dual spaces and tensor product norms,  the Schwarz Lemma.  Extremal Metrics.  Applications to operator theory.
 
Gadadhar Misra
 

Pisier, G., Introduction to Operator Theory, London Mathematical Society Lecture Note Series 294, Cambridge Univ., Press, Cambridge, 2003.

Paulsen, V., Completely bounded maps and operator algrbras, Cambridge Studies in Advanced  Mathematics, 78, Cambridge Univ., Press, Cambridge, 2002.

 

 

MA 330 (AUG) 3:0

Topology   II

 

Point Set Topology: Continuous functions, metric topology, connectedness, path connectedness, compactness, countability axioms, separation axioms, complete metric spaces, function spaces, quotient topology, topological groups, orbit spaces. The fundamental group: Homotopy of  maps, multiplication of paths, the fundamental group, induced homomorphisms, the  fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.

 

E K Narayanan/H Seshadri

 

Armstrong, M.A., Basic Topology, Springer, India, 2004.

Hatcher, A., Algebraic Topology, Cambridge Univ. Press, 2002.

Janich, K., Topology, Springer Verlag, UTM, 1984.

 

MA 331 (AUG) 3:0

Topology and Geometry

 

Manifolds: Differentiable manifolds, differentiable maps and tangent spaces, regular values and Sard's theorem, vector fields, submersions and immersions, Lie  groups, the Lie algebra of a Lie  group. Fundamental Groups: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.

 

H Seshadri

 

Brickell, F., and Clark, R.S., Differentiable Manifolds, Van Nostrand Reinhold Co., London, 1970.

Guillemin, V. and Pollack, A., Differential Topology, Prentice Hall, 1974.

Kosniowski, C., A First Course in Algebraic Topology, Cambridge Univ. Press, 1980.

Milnor, John W., Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, Princeton Univ. Press, 1997.

Munkres, J.R., Elements of Algebraic Topology, Addison Wesley, 1984.

 

 

MA  332  (JAN)  3:0

Algebraic  Topology

 

Homology: Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients. Cohomology: Cohomology groups, relative cohomology, cup products, Kunneth formula, cap Product, orientation on manifolds, Poincare  duality.

 

 Siddhartha Gadgil

 

Hatcher, A., Algebraic Topology, Cambridge Univ. Press, 2002. 

Greenberg, M.J., Lectures on Algebraic Topology, W A Bejamin  Inc., London, 1973.

Munkres, J.R., Elements of Algebraic Topology, Addison Wesley, 1984.   

Spanier, E.H., Algebraic Topology, Tata McGraw Hill, 1966.

 

MA 333 (JAN) 3:0

Riemannian Geometry

 

Prerequisites :  MA 331

 

Riemannian metric, Levi-Civita connection, geodesics, the exponential map, Hopf-Rinow theorem, curvature tensor, first and second variational formula, Jacobi fields, Myers Bonnet theorem, Bishop-Gromov volume comparison theorem, Cartan-Hadamard theorem, Synge’s theorem, de Rham cohomology and the Bochner techniques. 

 

Hemangi Shah/Siddhartha Gadgil

 

Gallot, S., Hulin, D., and Lafontaine J., Riemannian Geometry, Third Edn, Springer, Universitext, 2004. 
Petersen, P., Riemannian Geometry, Springer, GTM, 1998.
 

MA 361 (JAN) 3:0

Probability Theory

 

Probability  measures  and  random variables,  pi  and  lambda  systems,  expectation,  moment generating   function,  characteristic  function,  laws  of  large  numbers,  limit  theorems,  conditional contribution  and  expectation,  martingales,  infinitely  divisible  laws  and  stable  laws.

 

M K Ghosh/A Chakraborty

 

Breiman, L., Probability, Addison Wesley, 1968.

Laha, R.G., and Rohatgi, V.K., Probability Theory, John Wiley, 1979.

Borkar, V.S., Probability Theory: An Advanced Course, Springer Verlag, 1995.

 

MA 368 (AUG)  3:0

Topics in Probability and Stochastic Processes

 

Discrete Parameter Martingales and Applications, Ergodic Theory, Random Walks, Branching Processes.

 

Srikanth K Iyer

 

Chung, K.L., A course in Probability Theory, Third Edn, Academic Press, San Diego, 2001.

Feller, W., Introduction to Probability Theory and Applications, Vol. I & II,  22nd Edn, John Wiley and Sons, 1971.

Shiryaev, A.N., Probability, Springer, NY, 1966.

Spitzer, F., Principles of Random Walk, Second Edn, Springer, NY, 1976.

Athreya, K.B., and Ney, P.E., Branching Processes, Springer, NY, 1972.

Williams, D., Probability with Martingales, Cambridge Univ., Press, 1991.

 

MA 369 (JAN) 3:0

Random Matrix Theory

 

Basic models of random matrices, limiting spectral distributions by various methods, free probability.  Fluctuation behaviour of eigenvalues in the bulk and the edge.

 

Manjunath Krishmapur

Anderson, G.A., Guionnet, A., and Zeitouni, O., Cambridge Studies in Advanced Mathematics, No. 118, Cambridge Univ. Press.

Zhidong Bai and Jack Silverstein, Spectral Analysis of  Large Dimensional Random Matrices, Springer Series in Statistics.

Peter Forrester, Log-gases and Random Matrices (fourthcoming book available at

http://www.ms.unimelb.edu.au/-matpjf/matpjf,htm).

 

MA 392  (AUG)  3:0

Probabilistic Graphical Models

 

Graphical models provide a way of modeling high dimensional random structures and have found wide applications.  The popular  Hidden  Markov  Models, Markov  Random  fields.  LDA, fall within this framework.  A  graphical model  is  a graph  whose nodes are  random variables.  The graphical model  formalism  uses  the  structure of the graph  to code independence relations.  The goal  of  this course  is  to provide a systematic introduction to the underlying probability  and  statistical  issues. Inconsistent format.

 

Some of  the topics that will be covered are :

 

Basic probability and statistics:  Independence, Conditional independence, Multivariate Normal  distribution.  Estimation of  parameters, Maximum  Likelihood,  Bayesian  methods.  Exponential families. Directed  graphical  models  (Bayesian networks):  D-separation  and  conditional 

Independence.  Markov equivalence.  I-equivalence.  Undirected Graphical Models (Markov Networks).  Markov Networks and Independence.  Gibbs distribution and Markov networks. Gaussian networks: Gaussian Bayesian Networks.  Gaussian markov random fields. Hidden Markov models,  Kalman filters, Markov random fields, Generative modeling of  data, LDA. Exact inference in Bayesian networks: Junction tree algorithm, Belief  Propagation, Forward – Backward algorithm in HMM. Approximation inference: Variational techniques, MCMC  techniques, Gibbs sampling. Parameter learning:  Learning in fully observed models, multinomial  and multivariate  learning, EM  algorithm. Structure learning: Search over DAGs, Search over DAG patterns, Model averaging, AIC, BIC.

 

R V RAMAMOORTHI

 

Daphne Koller and Nir Friedman, Probabilistic Graphical Models,

Richard Neapolitan, Learning  Bayesian

Parts of these two texts will form the core of the course.  As additional topics are discussed          

relevant  references  will be  provided.