The PhD Programme

Overview

Ph.D. PROGRAMME IN MATHEMATICS offers an exciting and unique opportunity to students for pursuing research in the following areas:

  • Pure Mathematics
  • Applied Mathematics & Scientific Computing
  • Probability & Statistics
  • Theoretical Computer Science

Under this programme students undergo a substantial amount of relevant course work consisting of advance topics in Aglebra, Analysis, Topology, Numerical Analysis, Solid & Fluid Mechanics and Statistics, followed by research work under the supervision of an Advisor, who is decided by the Department, taking into account the aptitude, the needs and the preferences of the student.

Salient Features

  • Nurtures creative and analytical abilities of scholars.
  • Provides flexible approach for self-study and seminar courses in different areas.
  • Encourages interdisciplinary research for bridging the gap between theoretical and applied areas.
  • Cultivates communication skills through seminars and tutorials.
  • Offers opportunity to engage in developmental research work.

About the Department

The Department of Mathematics is well recognised for teaching and research. It has a large faculty with research interests covering a wide range of fields:

  • Algebra, Combinatorics, Topology and Geometry
  • Functional Analysis, Harmonic Analysis, Partial Differential Equations and Systems & Control.
  • Numerical Analysis & Scientific Computing.
  • Statistical Inference, Applied Probability, Statistical Quality Control and Biostatistics.
  • Algorithms & Complexity and Combinatorial Optimization,

Collabortive research with other Science and Engineering Departments of the Institute is encouraged. Faculty members undertake projects sponsored by organizations such as National Board for Higher Mathematics, Indian National Science Academy, Board of Research in Nuclear Science, Council of Scientific & Industrial Research, Department of Science and Technology, Department of Bio-Technology and Indian Council of Medical Research etc. A strong group of Industrial Mathematics has evolved in the Department for providing Industry-Academic linkage.

Job Placement

The graduates of this Department have been placed in various academic institutions - IITs, IISc and several universities in India and abroad. The Training and Placement Office of the Institute arranges Campus Interviews for the students with prospective employer in Industry and R & D Organisations with strong inputs from the Department. In recent years our students have been hired by prestigious organisations like TCS, ISRO, Infotech, TRDDC, CDAC, ORG etc.

Admission Requirement and Financial Aid

The Institute offers Teaching Assistantships requiring eight hours of work per week. Students can also be supported by scholarships / fellowships of other organizations such as National Board for Higher Mathematics, Council of Scientific & Industrial Research, University Grants Commission, Department of Science & Technology. Admissions take pace twice a year in June and in December on the basis of a test and an interview. The candidates should have obtained first class at the Masters degree in Mathematics/Statistics/Computer Science and must have a valid GATE score or an award of NBHM/CSIR/UGC Research Fellowship.

QUALIFYING EXAMINATION REQUIREMENT

For students who have joined the Ph.D. program before July 2017, the following rules apply.

    1. Each student in the Ph.D. programme in the Department of Mathematics is required to take a Qualifying Examination to be administered by the DPGC, Department of Mathematics.

    2. The qualifying examination shall consist of written examinations on any two of the following six topics: (i) Algebra, (ii) Analysis, (iii) Geometry and Topology, (iv) Differential Equations, (v) Probability and (vi) Statistics.

    3. The qualifying examination shall be conducted in the week before the beginning of the Autumn semester as well as the Spring semester. The results of the qualifying examination shall be made known before the date of registration.

    4. Each student is permitted at most two attempts for each of the two chosen topics. It is expected that every student aspiring to do Ph.D. will successfully complete the qualifying examination requirement by the end of the first academic year. In any case, the maximum time allotted for a student to pass the qualifying examination shall be three semesters. If a student fails to pass the examination in one or both of the chosen topics, then he/she may be permitted a change of topic. However, the ceiling of maximum two attempts and the overall ceiling of three semesters to complete the qualifying examination requirement shall always remain in force.

    5. A student may choose a Ph.D. thesis advisor after the successful fulfillment of the qualifying examination requirement. Until that time, the Head, Department of Mathematics shall be the guide for all official purposes.

    6. In case a student fails the qualifying examination requirement, then he/she may be offered the option of transferring to M.Phil. programme by continuing for about a semester so as to complete the requirements for the degree of M.Phil. degree.

    7. Notwithstanding the qualifying examination requirement, the course requirement stipulated by the Academic Office for the Ph.D. (or the M.Phil., as the case may be) programme will have to be satisfied. It may be noted that currently the course requirement for Ph.D. students consist of completing at least 34 course credits. However, students may register for up to two 500-level M.Sc. courses (for example, those that are relevant for the topic of qualifiers) to partially satisfy the credit requirement.

For students joining the Ph.D. program from July 2017 onwards, the following rules apply.

    1. Each student in the Ph.D. program of the Department of Mathematics is required to take Qualifying Examinations (QE) to be administered by the DPGC, Department of Mathematics.

    2. The QE shall consist of written examinations on seven subjects each of 100 marks, namely, (1) Algebra, (2) Analysis, (3) Geometry and Topology, (4) Differential Equations, (5) Probability, (6) Statistics and (7) Combinatorics and Theoretical Computer Science. A PhD candidate must score a minimum of 60 in at least two of these subjects to fulfil the QE requirements.

    3. The Q.E. will be conducted during 1-15 July and 15-31 December each year and the results will be declared by 21st July and 7th January, respectively.

    4. The student needs to pass the QE in two out of seven subjects (scoring a minimum of 60 marks out of 100 marks in each of the subjects) within three semesters of their joining the Ph.D. program. There are NO further restrictions on the number of attempts allowed to complete the QE requirement.

    5. A student may choose a Ph.D. thesis advisor after the successful fulfilment of the coursework and QE requirement. Until that time, the Head, Department of Mathematics shall be the guide for all official purposes.

    6. In case a student fails the QE requirement, then he/she may be offered the option of transferring to M.Phil. program by continuing for about a semester so as to complete the requirements for the degree of M.Phil.degree.

    7. Notwithstanding the QE requirement, the course requirement stipulated by the Academic Office for the Ph.D. (or the M.Phil., as the case may be) program will have to be satisfied. The Ph.D. students are required to complete the coursework with a minimum of 34 credits and a minimum 6.0 CPI. However, the students may register for up to two 500-level M.Sc. courses (for example, those that are relevant for the topic of qualifiers) to partially satisfy the credit requirement.

SYLLABUS FOR QUALIFYING EXAMINATIONS

For students joining the Ph.D. program from July 2017 onwards, the syllabus for qualifying examinations can be found here.

For students who have joined the Ph.D. program before July 2017, the syllabus is the following:

Topic (i): ALGEBRA

Groups, Simple groups and solvable groups, nilpotent groups, simplicity of alternating groups, composition series, Jordan-Holder Theorem. Semidirect products. Free groups, free abelian groups.

Rings, Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Poly-nomial rings over UFD's.

Fields, Characteristic and prime subfields, Field extensions, Finite, algebraic and finitely generated field extensions, Classical ruler and compass constructions, Splitting fields and normal extensions, algebraic closures. Finite fields, Cyclotomic fields, Separable and inseparable extensions.

Galois groups, Fundamental Theorem of Galois Theory, Composite extensions, Examples (including cyclotomic extensions and extensions of finite fields).

Norm, trace and discriminant.

Solvability by radicals, Galois' Theorem on solvability.

Cyclic extensions, Abelian extensions, Transcendental extensions.

Modules, submodules, quotient modules and module homomorphisms.

Generation of modules, direct sums and free modules. Tensor products of modules. Exact sequences, projective modules.

Tensor algebras, symmetric and exterior algebras.

Finitely generated modules over principal ideal domains, invariant factors, elementary divisors, rational canonical forms. Applications to finitely generated abelian groups and linear trans-formations.

Noetherian rings and modules, Hilbert basis theorem, Primary decomposition of ideals in noetherian rings.

Integral extensions, Going-up and Going-down theorems, Extension and contraction of prime ideals, Noether's Normalization Lemma, Hilbert's Nullstellensatz.

Localization of rings and modules. Primary decompositions of modules.

Topic (ii): ANALYSIS

Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theorem. Measure spaces.

Extension of measures from algebras to the generated sigma-algebras: Measurable sets; Lebesgue Measure and its properties.

Measurable functions and their properties; Integration and Convergence theorems.

Introduction to Lp-spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces.

Absolute continuity of measures, Radon-Nikodym theorem. Duals of Lp-spaces.

Product measure spaces, Fubini's theorem.

Fundamental Theorem of Calculus for Lebesgue Integrals.

Normed spaces. Continuity of linear maps. Hahn-Banach Extension and Separation Theorems. Banach spaces. Dual spaces and transposes.

Uniform Boundedness Principle and its applications. Closed Graph Theorem, Open Mapping Theorem and their applications. Spectrum of a bounded operator. Examples of compact operators on normed spaces.

Inner product spaces, Hilbert spaces. Orthonormal basis. Projection theorem and Riesz Representation Theorem for Hilbert spaces.

Topic (iii): GEOMETRY and TOPOLOGY

Functions on Euclidean spaces, continuity, differentiability; partial and directional derivatives, Chain Rule, Inverse Function Theorem, Implicit Function Theorem.

Riemann Integral of real-valued functions on Euclidean spaces, measure zero sets, Fubini's Theorem.

Partition of unity, change of variables.

Integration on chains, tensors, differential forms, Poincare Lemma, singular chains, integration on chains, Stokes' Theorem for integrals of differential forms on chains. (general version). Fundamental theorem of calculus.

Differentiable manifolds (as subspaces of Euclidean spaces), differentiable functions on manifolds, tangent spaces, vector fields, differential forms on manifolds, orientations, integration on manifolds, Stokes' Theorem on manifolds.

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts.

Basic constructions: cones, mapping cones, mapping cylinders, suspension.

Cell complexes, subcomplexes, CW pairs.

Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen's Theorem, Covering spaces, lifting properties, deck transformations. Universal coverings (existence theorem optional).

Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular Homology. Mayer-Vietoris Sequences. Long exact sequence of pairs and triples.

Homotopy invariance and excision (without proof).

Topic (iv): DIFFERENTIAL EQUATIONS

Methods for solving first order as well as second order equations, Power Series methods with properties of Bessel functions and Legendre polynomials.

Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuation of solutions and maximal interval of existence, continuous dependence.

Higher order linear differential equations and linear systems: fundamental solutions, Wronskian, variation of constants, matrix exponential solution, behaviour of solutions.

Two Dimensional Autonomous Systems and Phase Space Analysis: critical points, proper and improper nodes, spiral points and saddle points.

Asymptotic Behavior: stability (linearized stability and Lyapunov methods).

Boundary Value Problems for Second Order Equations: Green's function, Sturm comparision theorems and oscillations, eigenvalue problems

Cauchy Problems for First Order Hyperbolic Equations: method of characteristics, Monge cone.

Classification of Second Order Partial Differential Equations: normal forms and characteristics.

Initial and Boundary Value Problems: Lagrange-Green's identity and uniqueness by energy methods.

Stability theory, energy conservation and dispersion equation: mean value property, weak and strong maximum principle, Green's function, Poisson's formula, Dirichlet's principle, existence of solution using Perron's method (without proof).

Heat equation: initial value problem, fundamental solution, weak and strong maximum principle and uniqueness results.

Wave equation: uniqueness, D'Alembert's method, method of spherical means and Duhamel's principle.

Methods of separation of variables for heat, and wave equations.

Topic (v): PROBABILITY

Probability measure, probability space, construction of Lebesgue measure, extension theorems, limit of events, Borel-Cantelli lemma.

Random variables, Random vectors, distributions, multidimensional distributions, independence.

Expectation, change of variable theorem, convergence theorems.

Sequence of random variables, modes of convergence. Moment generating function and characteristics functions, inversion and uniqueness theorems, continuity theorems, Weak and strong laws of large number, central limit theorem.

Radon-Nikodym theorem, definition and properties of conditional expectation, conditional distributions.

Stochastic processes: Description and definition. Markov chains with finite and countably infinite state spaces. Classification of states, irreducibility, ergodicity. Basic limit theorems.

Markov processes with discrete and continu-ous state spaces. Poisson process, pure birth process, birth and death process. Brownian motion.

Applications to queueing models and reliability theory.

Basic theory and applications of renewal processes, stationary processes. Branching processes. Markov Renewal and semi-Markov processes, regenerative processes.

Topic (vi): STATISTICS

Distribution of functions of random variables, Order Statistics. Estimation - loss function, risk, minimum risk unbiased estimators, maximum likelihood estimation, method of moments, Bayes estimation.

Sufficient Statistics, completeness, Basu's Theorem, exponential families, invariance and maximal invariant statistics.

Testing of Hypotheses - parametric and non-parametric problems, examples with data analytic applications.

Confidence Intervals.

Simple and multiple linear regression models, estimation, tests and confidence regions. Check for normality assumption. Likelihood ratio test, confidence intervals and hypotheses tests; tests for distributional assumptions. Collinearity, outliers; analysis of residuals, Selecting the best fit, regression equation, transformation of response variables. Ridge's regression.

PhD Courses

The following courses are likely to be offered in each of the corresponding semesters.

First Semester

Course Code Name of the Course L T P C
MA 811 Algebra I 3 0 0 6
MA 813 Measure Theory 3 0 0 6
MA 815 Differential Topology 3 0 0 6
MA 817 Partial Differential Equations I 3 0 0 6
MA 833 Weak Convergence and Martingale Theory 3 0 0 6
MA 821 Theory of Estimation 3 0 0 6
MA 861 Combinatorics-I 3 0 0 6

Second Semester

Course Code Name of the Course L T P C
MA 812 Algebra II 3 0 0 6
MA 814 Complex Analysis 3 0 0 6
MA 816 Algebraic Topology 3 0 0 6
MA 818 Partial Differential Equations II 3 0 0 6
MA 820 Stochastic Processes 3 0 0 6
MA 822 Testing of Hypothesis 3 0 0 6
MA 824 Functional Analysis 3 0 0 6
MA 862 Combinatorics-II 3 0 0 6

    1. The credit requirements for students having M.Sc. or equivalent qualification shall be 34 to 46 credits.

    2. Credits acquired through PG level courses shall be 24 or more.

    3. Students may earn upto a maximum of 8 credits through seminars which should be spread over two semesters.

    4. Each student must credit at least 3 of the 13 courses MA 813 – 822 & MA 824.

PhD students are also required to take two courses on communication skills, viz, HS791 (Communication Skills I) offered by the HSS department and MA792 (Communication Skills II) offered by the Mathematics department

    a) These courses are compulsory for all Ph.D. students.

    b) Ph.D. students are normally required to clear the Communication skills course within the first two semesters.

    c) These courses are an addition to the minimum course credit requirement prescribed by the DPGCs/IDPCs.

    d) Students having a qualifying degree from IIT who have cleared the ‘Communication Skills’ course during their M.Tech. Programme are exempted from this requirement.

In addition to the above courses, some or all of the following courses may be offered subject to sufficient demand from the students and availability of faculty.

First Semester

Course Code Name of the Course L T P C
MA 841 Topics in Algebra I 3 0 0 6
MA 843 Topics in Analysis I 3 0 0 6
MA 845 Topics in Combinatorics I 3 0 0 6
MA 847 Topics in Geometry I 3 0 0 6
MA 849 Topics in Topology I 3 0 0 6
MA 851 Topics in Number Theory I 3 0 0 6
MA 853 Topics in Differential Equations I 3 0 0 6
MA 855 Topics in Numerical Analysis I 3 0 0 6
MA 857 Topics in Probability I 3 0 0 6
MA 859 Topics in Statistics I 3 0 0 6
MAS 801 Seminar 0 0 0 4

Second Semester

Course Code Name of the Course L T P C
MA 842 Topics in Algebra II 3 0 0 6
MA 844 Topics in Analysis II 3 0 0 6
MA 846 Topics in Combinatorics II 3 0 0 6
MA 848 Topics in Geometry II 3 0 0 6
MA 850 Topics in Topology II 3 0 0 6
MA 852 Topics in Number Theory II 3 0 0 6
MA 854 Topics in Differential Equations II 3 0 0 6
MA 856 Topics in Numerical Analysis II 3 0 0 6
MA 858 Topics in Probability II 3 0 0 6
MA 860 Topics in Statistics II 3 0 0 6
MAS 802 Seminar 0 0 0 4

PhD Course Contents

Departmental Courses

Note: Each course is of 6 credits with the structure of 3-0-0-6. A prerequisite for an even numbered course is exposure to the preceding odd numbered course, except in the case of MA 824 for which exposure to MA 819 shall be the prerequisite.

MA 811 Algebra I

    Review of field and Galois theory: solvable and radical extensions, Kummer theory, Galois cohomology and Hilbert's Theorem 90, Normal Basis theorem.

    Infinite Galois extensions: Krull topology, projective limits, profinite groups, Fundamental Theorem of Galois theory for infinite extensions.

    Review of integral ring extensions: integral Galois extensions, prime ideals in integral ring extensions, decomposition and inertia groups, ramification index and residue class degree, Frobenius map, Dedekind domains, unique factorisation of ideals.

    Categories and functors: definitions and examples. Functors and natural transformations, equivalence of categories,. Products and coproducts, the hom functor, representable functors, universals and adjoints. Direct and inverse limits. Free objects.

    Homological algebra: Additive and abelian categories, Complexes and homology, long exact sequences, homotopy, resolutions, derived functors, Ext, Tor, cohomology of groups, extensions of groups.

Text/References

    S. Lang, Algebra, 3rd Ed., Addison Wesley, 1993.

    N. Jacobson, Basic Algebra, Vol. 1 and 2, Hindustan Publishing Corporation, 1984.

MA 812 Algebra II

    Valuations and completions: definitions, polynomials in complete fields (Hensel's Lemma, Krasner's Lemma), finite dimensional extensions of complete fields, local fields, discrete valuations rings.

    Transcendental extensions: transcendence bases, separating transcendence bases, Luroth's theorem. Derivations.

    Artinian and Noetherian modules, Krull-Schmidt theorem, completely reducible modules, projective modules, Wedderburn-Artin Theorem for simple rings.

    Representations of finite groups: complete reducibility, characters, orthogonal relations, induced modules, Frobenius reciprocity, representations of the symmetric group.

Text/References:

    S. Lang, Algebra, 3rd Ed., Addison Wesley, 1993.

    N. Jacobson, Basic Algebra, Vol. 1 and 2, Hindustan Publishing Corporation, 1984.

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

    O. Zariski and P. Samuel, Commutative Algebra, Vol.1 and 2, Springer-Verlag, 1975.

MA 813 Measure Theory

    Review of measure theory: monotone convergence theorem, dominated convergence theorem, complete measures. Borel measures: Riesz representation theorem, Lebesgue measure on Rk, Lp spaces

    Complex measures: total variation, absolute continuity, Radon-Nikodym theorem, polar and Hahn decompositions, bounded linear functionals on Lp, generalised Riesz representation theorem.

    Differentiation: Maximal function, Lebesgue points, absolute continuity of functions, fundamental theorem of calculus, Jacobian of a differentiable transformation, change of variable formula.

    Product measures: Fubini's theorem, completion of product measures, convolutions, Fourier transform, Riemann-Lebesgue lemma, inversion theorem, Plancherel theorem, L1 as a Banach algebra.

    Content on a locally compact Hausdorff space, existence and uniqueness of the Haar measure on a locally compact group.

Text/References:

    K. Chandrasekharan, A Course on Topological Groups, Hindustan Book Agency, 1996.

    L. Nachbin, The Haar Integral, van Nostrand, 1965.

    I. K. Rana, An Introduction to Measure and Integration, 2nd Ed., American Mathematical Society, 2002.

    H. L. Royden, Real Analysis, 3rd Ed., Prentice Hall of India, 1988.

    W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

MA 814 Complex Analysis

    Review of basic complex analysis: Cauchy's theorem, Liouville's theorem, power series representation, open mapping theorem, calculus of residues.

    Harmonic functions, Poisson integral, Harnack's theorem, Schwarz reflection principle.

    Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof method,.

    Runge's theorem, Mittag-Leffler theorem, Weierstrass theorem, conformal equivalence, Riemann mapping theorem, characterisation of simply connected regions, Jensen's formula.

    Analytic continuation, monodromy theorem, little Picard theorem.

Text/References:

    L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1996.

    S. Lang, Complex Analysis, 4th Ed., Springer, 1999.

    D. H. Luecking and L. A. Rubel, Complex Analysis: A Functional Analysis Approach, Springer-Verlag, 1984.

    R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable, Birkh�user, 2001.

    R. Remmert, Theory of Complex Functions, Springer (India), 2005.

    W. Rudin, Real and Complex Analysis, McGraw Hill, 1987.

MA 815 Differential Topology

    Review of differentiable manifolds, tangent and cotangent bundles, tensors.

    DeRham complex, Poincare's Lemma, Mayer-Vietoris sequences, cohomology with compact supports, degree of a map, Poincare duality.

    Vector bundles, cohomology with vertical compact supports, Thom isomorphism, twisted DeRham complex, Poincare duality for non-orientable manifolds.

Text/References:

    R. Bott and L. W. Tu , Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982.

    L. Conlon, Differentiable manifolds, 2nd Ed., Birkh�user, Boston, 2001.

    G. E Bredon, Topology and Geometry, Springer-Verlag, New York, 1997.

MA 816 Algebraic Topology

    Paths and homotopy, homotopy equivalence, contractibility, deformation retracts.

    Basic constructions: cones, mapping cones, mapping cylinders, suspension.

    Cell complexes, subcomplexes, CW pairs.

    Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen's Theorem. Covering spaces, lifting properties, deck transformations, universal coverings.

    Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular Homology. Mayer-Vietoris sequences. Long exact sequence of pairs and triples. Homotopy invariance and excision.

    Degree. Cellular Homology.

    Applications of homology: Jordan-Brouwer separation theorem, Invariance of dimension, Hopf's Theorem for commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point Theorem.

    Optional Topics: Outline of the theory of: cohomology groups, cup products, Kunneth formulas, Poincare duality.

Text/References:

    M.J. Greenberg and J. R. Harper, Algebraic Topology, Benjamin, 1981.

    W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.

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

    W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991.

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

    J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004.

    H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, 1980.

MA 817 Partial Differential Equations I

    Distribution Theory and Sobolev Spaces: Distributional derivatives, Definitions and elementary properties of Sobolev Spaces, Approximations by smooth functions, Traces, Imbedding Theorems (without proof), Rellich-Kondrachov Compactness Theorem.

    Second Order Linear Elliptic Equations: Weak Solutions, Lax-Milgram Theorem, Existence and Regularity Results, Maximum Principles, Eigenvalue Problems.

    Second Order Linear Parabolic Equations: Existence of weak solutions and Regularity Results, Maximum Principles.

    Second Order Linear Hyperbolic Equations: Existence of weak solutions and Regularity Results, Maximum Principles, Propagation of Disturbance

Text/References:

    S. Kesavan, Topics in Functional Analysis and Applications, New Age International Pvt. Ltd., 1989.

    L C. Evans, Partial Differential Equation, American Mathematical Society, 1998.

    M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 2004.

    G. B. Folland, Introduction to Partial Differential Equations, 2nd Ed., Prentice-Hall of India, 1995.

    R. C. McOwen, Partial Differential Equations: Methods and Applications, 2nd Ed., Pearson Education, Inc., 2003.

MA 818 Partial Differential Equations II

    Nonlinear First-Order Scalar Equations: Method of Characteristics, Weak Solutions and Uniqueness for Hamilton-Jacobi Equations, Scalar Conservation Laws: shocks and entropy condition, weak solutions and uniqueness, and long time behavior.

    Calculus of Variations: Euler-Lagrange Equation, Second Variations, Existence of Minimizers: Coercivity, Lower-Semicontinuity, Convexity, and Constrained Minimization Problems.

    Hamilton-Jacobi Equations: Viscosity Solutions, Uniqueness, Applications to Control Theory and Dynamic Programming.

    System of Conservation Laws: Theory of Shock Waves, Traveling Waves, Entropy Criteria, Riemann Problem, Glimm Existence Result for System of Two Conservation Laws.

Text/References:

    L C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

    M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004.

    M. Defermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000.

    B. Dacorogna, Direct Methods in Calculus of Variation, Springer 1989.

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

    J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1993.

MA 833 Weak Convergence and Martingale Theory

    Review of conditional expectation : Conditional expectation and conditional probability, regular conditional distributions, disintegration, conditional independence.

    Martingales and Stopping times : Stopping times, random time change, martingale property, optional sampling theorem, maximum and upcrossing inequalities, martingale convergence theorem.

    Gaussian processes and Brownian motion: Symmetries of Gaussian distribution, existence and path properties of Brownian motion, law of iterated logarithm.

    Weak convergence in metric spaces with special reference to C[0, 1], Martingale central limit theorem.

Text/References:

    P. Billingsley, Convergence of probability measures, Wiley, 1999.

    K. R. Parthasarathy, Probability measures on metric spaces, Academic press, 1967.

    V S. Borkar, Probability theory : an advanced course, Springer, NewYork, 1995.

    A. W. Van-der-Vaart and J. A. Wellner, Weak convergence and empirical processes: With applications to Statistics, Springer-Verlag, 1996.

    D. Williams, Probability with martingales, Cambridge Mathematical textbooks, 1991.

MA 820 Stochastic Processes

    Review of discrete time Markov chains: Markov property and transition kernels, invariant distributions, recurrence and transience, ergodic behaviour of irreducible chains.

    Stationary processes and ergodic theory: Stationarity, invariance and ergodicity, discrete and continuous time ergodic theorems.

    Poisson and pure jump Markov processes: Random measures and Poisson point processes, mixed Poisson and binomial processes, independence and symmetry criteria, Markov transition and rate kernels, embedded Markov chains and explosion.

    Levy processes: Levy processes and subordinators, stable processes, infinitely divisible distributions.

    Markov processes: Transition and contraction operators, ratio ergodic theorem, space-time invariance, Harris recurrence and transience, existence and uniqueness of invariant measures.

Text/References:

    O. Kallenberg, Foundations of Modern Probability, 2nd Ed., Springer, 2000.

    R. B. Ash and M. F. Gardner, Topics in Stochastic Processes, Academic Press, 1975.

    D. W. Stroock, Markov processes from K. Ito's Perspective, Annals of Mathematics Studies, No. 155, Princeton University Press, 2003.

MA 821 Theory of Estimation

    Elements of decision theory such as complete class theorem, admissibility of Bayes rule, Minmax Theorem

    Review of sufficiency, consistency and efficiency. UMVU estimators and their properties. Application to normal and exponential one and two sample problems. Information inequality(multiple parameter case) Equivariance, Invariance. Application to location and scale families.

    MRE estimation.

    Bayes and minimax estimation for exponential families. Admissibility of estimators, Blyth's ratio method, Karlin's sufficient conditions.

    Pitman's estimator and its properties, Simultaneous estimation. Stein's phenomenon, Shrinkage estimation.

Text/References:

    J. Berger, Statistical decision theory, Springer-Verlag, 1980.

    T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, 1967.

    E. L. Lehmann, Theory of Statistical Inference, Wiley, 1983.

    S. Zacks, The Theory of Statistical Inference, Wiley, 1971.

    Jun Shao, Mathematical Statistics, 2nd Ed., Springer, 2003.

MA 822 Testing of Hypothesis

    UMP tests, Neymann-Pearson fundamental lemma, Distributions with ML ratio, Confidence bounds, Generalization of the fundamental lemma.

    Least favourable distributions, applications to normal distribution.

    Similarity and completeness, UMP unbiased two-sided tests, Applications to exponential families, Fisher-Beherns problem, Unbiased confidence sets. Most powerful permutation and invariant tests, Admissibility of tests, Chi-tests and invariance, The Hunt-Stein theorem and its applications.

Text/References:

    T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, 1967.

    L. Le. Cam, Asymptotic in Statistics, Springer-Verlag, 1990.

    E. L. Lehmann, Testing Statistical Hypotheses, Wiley, 1986.

    Jun Shao, Mathematical Statistics, 2nd Ed., Springer, 2003.

MA 824 Functional Analysis

    Review of normed linear spaces, Hahn-Banach theorems, uniform boundedness principle, open mapping theorem, closed graph theorem, Riesz representation theorem on Hilbert spaces.

    Weak and weak* convergence, reflexivity in the setting of normed linear spaces.

    Compact operators, Sturm-Liouville problems.

    Spectral projections, spectral decomposition theorem, spectral theorem for a bounded normal operator, unbounded operators, spectral theorem for an unbounded normal operator.

Text/References:

    M. Ahues, A. Largillier and B. V. Limaye, Spectral Computations for Bounded Operators, Chapman & Hall/CRC, 2001.

    J. B. Conway, Functional Analysis, 2nd Ed., Springer-Verlag, 1990.

    S. Lang, Complex Analysis, 4th Ed., Springer, 1999.

    B. V. Limaye, Functional Analysis, 2nd Ed., New Age International Publishers, 1996.

    F. Riesz and B. SzNagy, Functional Analysis, Dover Publications, 1990.

    W. Rudin, Functional Analysis, Tata McGraw Hill, 1974.

    K. Yosida, Functional Analysis, 5th Ed., Narosa, 1979.

MA 841 Topics in Algebra I

A selection of topics from the following:

    Regular sequences, grade and depth. Projective dimension, Auslander-Buchsbaum formula. Koszul complex. Rank of modules. Buchsbaum-Eisenbud acyclicity criterion. Graded rings and modules. Basic properties of graded modules: associated primes, dimension etc.

    Hensel's Lemma, Newton' Theorem and Weierstrass Preparation Theorem.

    Chevalley's Theorem on invariants of a finite pseudo-reflection group acting on the polynomial ring.

    The Jacobian criterion for regularity. Divisor class group of a noetherian normal domain and its properties under ring extensions etc. Applications to unique factorization.

    Cohen-Macaulay rings. Homological characterization of regular local rings.

    Injective hulls, Matlis Duality. Local cohomology. Basic properties. Invariance under flat and finite base changes. Canonical module: Existence and basic properties. Local duality and applications. Canonical module of graded rings.

Text/References:

    S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hackensack, NJ, 2006.

    W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition, Cambridge University Press, 1998

    H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

MA 842 Topics in Algebra II

A selection of topics from the following:

    Cohen-Macaulay rings and modules, Canonical Module, Gorenstein rings.

    Hilbert functions and multiplicities, Macaulay's Theorem

    Stanley-Reisner rings, shellability.

    Semigroup rings and rings of invariants

    Determinantal rings, Straightening law.

    Big Cohen-Macaulay modules, Hochster's finiteness theorem.

Text/References:

    S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hackensack, NJ, 2006.

    W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition, Cambridge University Press, 1998

    H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

MA 843 Topics in Analysis I

A selection of topics from the following:

    Singular Integrals (Calderon-Zygmund theory), the Kakeya problem, the Uncertainty Principle, the almost everywhere convergence of Fourier series, multilinear operators between Lp spaces.

    Pseudodifferential operators, Index theorems.

    Advanced complex analysis in one variable: Nevanlina theory, the existence of quasi-conformal maps, iterated polynomial maps, complex dynamics, compact Riemann surfaces, the Corona theorem.

    Holomorphic functions in several complex variables: elementary properties of functions of several complex variables, analytic continuation, subharmonic functions, Hartog's theorem, automorphisms of bounded domains.

Text/References:

    R.C. Gunning, Introduction to holomorphic functions of several variables. Vol. I. Function theory, Wadsworth & Brooks/Cole, 1990.

    A.W. Knapp, Advanced real analysis, Birkhauser, 2005.

    S. Lang and W. Cherry, Topics in Nevanlinna theory, Springer-Verlag, 1990.

    R. Narasimhan, Several complex variables, University of Chicago Press, 1995.

    E.M. Stein, Harmonic Analysis: Real Variable Methods,Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.

    S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups, Birkhauser, 2004.

MA 844 Topics in Analysis II

A selection of topics from the following:

    Fourier Series and Fourier Transforms: Orthonormal Sequences in Inner Product Spaces, Fourier Series, Riemann-Lebesgue Lemma, Convergence/Divergence of Fourier Series, Fejer Theory, Fourier Transform, Inversion Theorem, Approximate Identities, Plancherel Theorem

    Hp spaces: Harmonic and Subharmonic Functions, Hp spaces, Nevanlinna Class of Functions, Boundary Values, Non-tangential Limits, F. and M. Riesz Theorem, Inner Functions, Outer Functions, Factorization Theorems, Beurling's Theorem

    Banach Algebras: Examples of Banach Algebras, Spectrum, Gelfand Representation, C*-Algebras, Positive Linear Functionals, Gelfand-Naimark Representation

    Elements of Operator Theory: Hilbert Space Operators, Parts of Spectrum, Orthogonal Projections, Invariant Subspaces, Reducing Subspaces, Shifts, Decompositions of Operators

    Perturbation Theory for Linear Operators: Analyticity of the resolvent operator, spectral projection and the weighted mean of the eigenvalues, The method of majorizing series, Spectral Decomposition Theorem.

    Spectral Approximation: Norm and nu- convergence, Iterative refinement methods such as the Rayleigh-Schrodinger series and methods based on the fixed point techniques, error estimates.

    Approximate solutions of Operator Equations: Galerkin, Iterated Galerkin and Nystrom methods, Condition Numbers, Two Grid Methods.

Text/References:

    M. Ahues, A. Largillier, B.V. Limaye, Spectral Computation for bounded operators, Chapman & Hall/CRC, 2001.

    K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.

    G. Bachman, L. Narici and E. Beckenstein, Fourier and Wavelet Analysis, Springer-Verlag, 2000.

    S. K. Berberian, Lectures in Functional Analysis and Operator Theory, Narosa Publishing House, 1979.

    F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, 1983.

    J.B. Conway, A Course in Functional Analysis, Springer-Verlag, 1985.

    P.L. Duren, Theory of Hp spaces, Dover Publications, 2000.

    W. Hackbusch, Integral Equations: Theory and Numerical Treatment, Birkhauser, 1995.

    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995.

    R. Kress, Linear Integral Equations, Second Edition, Springer-Verlag, 1999.

    P. Koosis, Introduction to Hp spaces, 2nd Edition, Cambridge University Press, 1999.

    C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhauser, 1997.

    G.J. Murphy, C*-Algebras and Operator Theory, Academic Press Inc., 1990.

    W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

    W. Rudin, Functional Analysis, McGraw Hill, 1991.

    A. Vretblad, Fourier Analysis and its Applications, Springer-Verlag, 2005.

MA 845 Topics in Combinatorics I

A selection of topics from the following:

    Basic Combinatorial Objects : Sets, multisets, partitions of sets, partitions of numbers, finite vector spaces, permutations, graphs etc.

    Basic Counting Coefficients: The twelve fold way, binomial, q-binomial and the Stirling coefficients, permutation statistics, etc.

    Sieve Methods : Principle of inclusion-exclusion, permutations with restricted positions, Sign-reversing involutions, determinants etc.

    Combinatorial reciprocity.

    Theory of Symmetric functions.

Text/References:

    C. Berge, Principles of Combinatorics, Academic Press, 1972.

    I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition, Oxford University Press, 1995.

    R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986.

MA 846 Topics in Combinatorics II

A selection of topics from the following:

    Partially ordered sets, Mobius inversion.

    Rational generating functions: P-partitions and linear Diophantine equations.

    Polya theory and representation theory of the symmetric group.

    Combinatorial algorithms, and symmetric functions.

    Generating functions: Single and multivariable Lagrange inversion.

    Young tableaux and plane partitions

Text/References:

    M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979.

    I. G. Macdonald, Symmetric functions and Hall polynomials. Second edition, Oxford University Press, New York, 1995.

    B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, Wadsworth & Brooks/Cole, 1991.

    R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986.

    R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, 1999.

MA 847 Topics in Geometry I

A selection of topics from the following:

    Review of the theory of curves and surfaces in the Euclidean 3-space.

    Differentiable manifolds, and Riemannian structures. Connections, and curvature tensor.

    The theorems of Bonnet-Meyers and Hadamard. Manifolds of constant curvature.

Text/References:

    J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer-Verlag, New York, 1997.

    W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, 2002.

    M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

    S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.

    J. Milnor, Morse Theory, Princeton University Press, 1963.

MA 848 Topics in Geometry II

A selection of topics from the following:

    Affine and projective varieties, rational maps, nonsingularity.

    Algebraic Curves, Riemann Roch Theorem.

    Sheaves and Schemes. Basic properties. Divisors and Differentials.

    Cohomology of sheaves, Serre Duality Theorem.

Text/References:

    S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, RI, 1990.

    D. Eisenbud and J. Harris, The Geometry of Schemes, Springer-Verlag, 2000.

    R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

    I. R. Shafarevich, Basic Algebraic Geometry, Vol. 1 and 2, Second edition, Springer-Verlag, 1994.

MA 849 Topics in Topology I

A selection of topics from the following:

    CW complexes, Homotopy groups, Cellular Approximation.

    Whitehead's theorem, Hurewicz theorem.

    Excision, Fibre bundles, Long exact sequences.

    Postnikov Towers, Obstruction Theory.

    Stable homotopy groups. Spectral Sequences, Serre Class of abelian groups.

Text/References:

    B. Gray, Homotopy Theory, Academic Press, 1975.

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

    G. W. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1978.

    P. Hilton, Homotopy Theory and Duality, Gordon and Beach Sc. Publishers, 1965.

    N. Steenrod, The Topology of Fibre Bundles, 7th reprint, Princeton University Press, 1999.

    R. M. Switzer, Algebraic topology: Homotopy and Homology, Springer Verlag, 2002.

MA 850 Topics in Topology II

A selection of topics from the following:

    Basics of Topological groups, Lie group.

    Group actions, homogeneous spaces examples.

    G-spaces, existence of slice and tubes

    Covering homotopy theorem, Classification of G-Spaces.

    Finite group actions, homology spheres

    G-coverings, Cech theory

    Locally smooth actions, orbit types, principal orbits

    Actions of tori.

    Cohomology structure of fixed point sets, Z_p-actions projective spaces and product of spheres.

Text/References:

    G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press 1972.

    T. Br�cker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York, 1985.

    W. Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, 1975.

MA 851 Topics in Number Theory I

A selection of topics from the following:

    Algebraic number theory, abelian and non-abelian reciprocity laws, the Langlands programme, automorphic forms and representations.

    The arithmetic of algebraic groups.

    Arithmetic algebraic geometry: counting rational points of varieties over finite fields

    Galois representations and galois cohomology.

    Additive number theory: partitions, compositions, Goldbach problem.

Text/References:

    S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New York, 1994.

    D. Bump, Automorphic forms and representations, Cambridge University Press, Cambridge, 1997.

    H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.

    H. Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.

MA 852 Topics in Number Theory II

A selection of topics from the following:

    Harmonic analysis on Lie groups, L-functions, l-adic representations and motives.

    Diophantine equations and the applications of K-theory to number theory.

    Analytic number theory and transcendental methods.

    Applications of ergodic theory to number theory.

Text/References:

    S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New York, 1994.

    D. Bump, Automorphic forms and representations, Cambridge University Press, Cambridge, 1997.

    H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.

    H. Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.

MA 853 Topics in Differential Equations I

A selection of topics from the following:

    Schauder theory, regularity for second order elliptic equations. Nonlinear analysis and its applications to nonlinear PDEs: Fixed point methods, variational methods, monotone iteration, degree theory.

    Evolution equations: Existence via semigroup theory

    Nonlinear Hyperbolic systems: Theory of well posedness, compensated compactness,

    Young measures; propagation of oscillations, weakly nonlinear geometric optics.

Text/References:

    D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1984.

    D. Serre, Systems of Conservation Laws, Vols. 1, 2, Cambridge University Press, 2000.

    L. Evans, Weak Convergence Methods for Nonlinear PDEs, CBMS Regional Conference series in Math., American Mathematical Society, Providence RI, 1990

    A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, 1978.

    M. Struwe, Variational Methods: Applications to nonlinear PDEs and Hamiltonian systems, Springer-Verlag, 1990.

MA 854 Topics in Differential Equations II

A selection of topics from the following:

    Diffeomorphisms and flows: Elementary dynamics of diffeomorphisms, flows and differential equations, conjugacy, equivalence of flows, Sternberg's theorem on smooth conjugacy (statement only), Hamiltonian flows and Poincare maps.

    Local properties of flows and diffeomorphisms: Hyperbolic fixed points, Hartman-Grobman theorems for maps and flows, Normal forms for vector fields, Centre manifolds.

    Structural stability and hyperbolicity: Structural stability for linear systems, Flows on 2-dimensional manifolds, Peixoto's characterisation of structural stability on unit disc, Anosov and Horseshoe diffeomorphisms, Homoclinic points, Melnikov function.

    Bifurcations and Perturbations: Saddle-node and Hopf bifurcations, Andronov-Hopf bifurcation, The logistic map, Arnold's circle map; Perturbation theory: Melnikov's method for the study of perturbation of completely integrable systems.

    Floquet theory and Hill's equation and some of its applications.

    Two dimensional systems: Poincare-Bendixon theorem, Index of planar vector fields and the Poincare Hopf index theorem for two dimensional manifolds.

    Van der Pol's equation, Duffing's equation, Lorenz's equation.

    First integrals and functional independence of first integrals, notion of complete integrability, Jacobi multipliers, Liouville's theorem on preservation of phase volume, Jacobi's last multiplier theorem and its applications.

Text/References:

    D. K. Arrowsmith, C. M. Place: An Introduction to Dynamical Systems, Cambridge University Press, 1990.

    C. Chicone, Ordinary Differential Equations. Springer-Verlag, 1999.

    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 2002.

    P.Glendinning, Stability, instability and chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, 1994.

    J. Palis and W. C. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, 1982.

    R. Grimshaw, Nonlinear Ordinary Differential Equations. CRC press, 1991.

    N.A. Magnitskii and S.V. Sidorov, New Methods for Chaotic Dynamics, World Scientific, 2006.

    L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001.

MA 855 Topics in Numerical Analysis I

A selection of topics from the following:

    Review of finite difference methods for elliptic, parabolic and hyperbolic problems. Stability, consistency and convergence theory.

    Finite difference schemes for scalar conservation laws (Lax-Friedrichs, Upwind, Lax-Wendroff, etc.), Conservative schemes and their numerical flux functions, Consistency, Lax-Wendroff Theorem, CFL Condition, Nonlinear Stability and TVD property, Monotone Difference schemes, Numerical entropy condition, Convergence result.

    Finite difference Schemes for one-dimensional system of conservation laws, approximate Riemann solvers, Godunov's method, High resolution methods, Multidimensional approaches.

    Large Scale Scientific Computing: Classical Iterative Methods for solving Linear systems, Large Sparse Linear Systems, Storage Schemes, GMRES algorithm, Preconditioned Conjugate Gradient method and Multi-grid method, Newton's Method and some of its variations for solving nonlinear systems.

Text/References:

    Axelsson, O. Iterative Solution Methods, Cambridge University Press, 1994.

    Briggs, W. L., Henson, V. E. and McCormick, S. F. A Multigrid tutorial, SIAM, 2000.

    Godlewski, E. and Raviart, P. –A. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1995.

    Kroner, D. Numerical Schemes for Conservation Laws. John Wiley, 1997.

    LeVeque, R. J. Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

    LeVeque, R. J. Numerical Methods for Conservation Laws. Birkhauser, 1992.

    Quarteroni, A. and Valli, A. Numerical Approximation of Partial Differential Equations, Springer, 1997.

    Ueberrhuber, C. W. Numerical Computation: Methods, Software and Analysis, Springer-Verlag, 1997.

MA 856 Topics in Numerical Analysis II

A selection of topics from the following:

    Mixed Finite Element Methods: Examples of mixed variational formulations- primal, dual formulations; abstract mixed formulations, discrete mixed formulations, existence-uniqueness of solutions, convergence analysis, implementation procedures.

    Adaptive FEM: A study of -Explicit A posteriori error estimators, Implicit A posteriori estimators, Recovery based error estimators, Goal Oriented adaptive mesh refinement for second order elliptic boundary value problems.

    Discontinuous Galerkin Methods for second order elliptic boundary value problems: Global element methods, Symmetric Interior Penalty Method, Discontinuous hp- Galerkin Method, Non-symmetric interior penalty method: Consistency, approximation properties, existence and uniqueness of solutions, error estimates, implementation procedures.

    FEM for parabolic problems: The standard Galerkin method, semi-discretization in space. discretization in space and time, the discontinuous Galerkin Method, a mixed method, implementation procedures.

    Elements of Multigrid Methods: Multigrid Components - Interpolation, restriction Coarse-grid correction, V, W, and FMG cycles, Implementation, Convergence analysis, Performance diagnostics.

Text/References:

    Z. Chen, Finite Element Methods And Their Applications, Springer-Verlag, New York, 2005.

    S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd Edition, Springer-Verlag, New York, 2002.

    M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley and Sons, 2000.

    V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 2nd Edition, Springer-Verlag, Berlin, 2006.

MA 858 Topics in Probability II

A selection of topics from the following:

    Stochastic optimal control: compactness of laws, dynamic programming principle.

    Malliavin calculus and applications to finance: Wiener-Ito chaos expansion, Shorohod integral, Integration by parts formula, Clark- Ocone formula and application to finance.

Text/References:

    V.S. Borkar, Optimal control of diffusion processes, Longman Scientific and Technical, Harlow (copublished by John Wiley), 1989.

    D. Nualart, The Malliavin calculus and related topics, Springer-Verlag, 1995.

MA 859 Topics in Statistics I

A selection of topics from the following:

    Univariate Stochastic Orders-hazard rate order, likelihood ratio order, mean residual rate order. Univariate variability orders- convex order, dispersive order, peakedness order. Univariate monotone convex and related orders. Multivariate stochastic orders. Multivariate variability and related orders. Statistical Inference for stochastic ordering. Applications in reliability theory, biology, economics and scheduling.

Text/References:

    J. George Shanthikumar and Moshe Shaked (1994) Stochastic Orders and their Applications, Academic press.

    C.D. Lai and M. Xie (2006) Stochastic Ageing and Dependence for Reliability, Springer Verlag.

MA 860 Topics in Statistics II

A selection of topics from the following:

    Inference in Semi-parametric models: Models with infinite imensional parameters, Efficient estimation and the delta method, Score and information operators, Estimating equations, Maximum Likelihood estimation, Testing.

    Generalized linear models: Components of a GLM, estimation techniques, diagnostics, continuous response models, Binomial response models, Poisson response models, overdispersion, multivariate GLMs, quasi likelihoods, generalized estimating equations, generalized linear mixed models, programming in R and SAS.

Text/References:

    A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press, 2000.

    U. Grenander, Abstract Inference, John Wiley, 1981.

    P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd Edition, Chapman and Hall/CRC, 1994.

    L. Fahrmeir and G. Tutz, Multivariate Statistical Modeling based on Generalized Linear Models, 2nd Edition, Springer-Verlag, 1994.

    R. H. Myers, D. C. Montgomery and G. Geoffrey Vining, Generalized Linear Models with applications in Engineering and Sciences, Wiley-Interscience, 2001.