Curriculum of M. Phil., Mathematics

After revision 2017-2018

DDEEPPAARRTTMMEENNTT OOFF MMAATTHHEEMMAATTIICCSS

BBHHAARRAATTHHIIDDAASSAANN UUNNIIVVEERRSSIITTYY

TIRUCHIRAPPALLI – 620 024

Department of Mathematics

Curriculum Structure

New Syllabus

List of Subjects with codes

Total Credits-

Internal marks-40 External marks-60

* COURSES (CC)

Code Title of the Course Lecture Hours

Tutorial Hours

Practical Hours

Credits Prerequisite (Exposure)

17MP01CC Research Methodology 4 2 0 4 Nil

17MP02CC Algebra and Analysis 4 2 0 4 Nil

17MP03CC Paper on the Topic of Research

(To be framed by the Guide) 4 2 0 4 Nil

17MP04CC Teaching and Learning Skills

(Common Paper) 4 2 0 4 Nil

For each Course other than the Project

Continuous Internal Assessment (CIA) – 40 Marks

End Semester Examination (ESE) – 60 Marks

Total – 100 Marks

ESE Duration – 3 Hours.

For Dissertation

See Annexure (University Regulations)

Question paper pattern and CIA components

10 questions compulsory 10 x 01 = 10 ( 2 from each unit)

5 questions 05 x 04 = 20 (either or, one from each unit)

3 questions from 5 03 x 10 = 30 ( one question from each unit)

Total 60

CIA components

Tests - 30 (2 tests from 3)

Seminar – 5

Assignment - 5

Programme Outcomes

M.Phil. Granduands are well equipped with Research & Development Competences

expressive of their Creative Knowledge, Inventive Skill, Resolute Attitude and Innovative

Pursuits in their chosen fields.

M.Phil. Graduants Collate information from a variety of sources and Enrich a coherent

understanding of the subject concerned pertaining to Novel investigation on the problems

in everyday life.

COURSES

RESEARCH METHODOLOGY

Course Code 17MP01CC

Objectives:

To learn the concept of simply connect spaces and to prove the Riemann mapping

theorem.

To introduce algebraic invariant, homology groups for simplicial complexes.

To demonstrates the power of topological methods in dealing with problems

involving shape and position of continuous mappings.

UNIT – I

Riemann Mapping Theorem: Preservation of angles – Linear

fractional transformations – Normal families - Riemann Mapping Theorem.

UNIT –II

Fundamental Group: Revision on point set topology - Homotopy – Fundamental

group.

UNIT –III

Covering spaces: Covering map - Covering homotopy theorem – Regular covering

spaces.

UNIT –IV

Simplicial complexes: Geometry of Simplicial complexes – Barycentric

subdivisions

UNIT – V

Simplicial complexes: Simplicial approximation theory – Fundamental group

of a Simplicial complex.

UNIT – VI (Advanced topics only for discussion)

Current Contours:

Homology theory

TEXT BOOK(S):

[1] V. Karunakaran, Complex Analysis 2 edn, Narosa, New Delhi, 2005.

[2] I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and

Geometry, Springer Verlag, 2004.

Unit I - Chapter

Unit II - Chapter 3 (Section: 3.1 to 3.2 )

Unit III - Chapter 3 (Section: 3.3 )

Unit IV - Chapter 4 (Section: 4.1 to 4.2 )

Unit V - Chapter 4 (Section: 4.3 to 4.4 )

REFERENCES

[1] W. Rudin, Real and Complex Analysis, 3rd

edition, McGraw Hill International,

1986.

[2] James R. Munkres, Topology (2nd Edition), Prentice Hall of India, Pvt. Ltd.,

New Delhi, 2004.

Course outcomes : Students will be able to

L T P C

4 2 0 4

Understand the properties of conformal mapping and linear fractional

transformations

Comprehend the important Riemann mapping theorem in Complex analysis

Discuss on the concept of homotopy and homotopy equivalence of topological

spaces.

Compute the fundamental groups of standard topological spaces.

Learn thoroughly covering homotopy theorem

Understand simplicial complexes and its barycentric subdivisions.

Use simplicial approximations to find the fundamental group of simplicial

complexes.

Appreciate and deduce the important Brouwer’s fixed point theorem.

Algebra and

analysis

Course Code 17MP02CC

Objectives:

To understand the concept of Modules as representation objects for rings.

To learn the notion of Noetherian rings as a important concept in ring theory and

the role it plays in simplifying the ideal structure of rings.

To impart the knowledge of how linear functionals on spaces of continuous function

on a locally compact space are related to measures.

To learn the theory of Fourier transforms in a rigorous setup.

To get a practice in the study of solving the system of linear differential equation

using the concepts from linear algebra.

UNIT I

MODULES: Basic definitions – Group of homomorphisms – Direct products and

sums of modules – Free modules – Vector spaces – The dual space and dual module.

UNIT II

NOETHERIAN RINGS: Basic criteria – Associated primes – Primary

decomposition - Nakayama’s lemma

UNIT III

RIESZ REPRESENTATION THEOREM: Topological preliminaries - Riesz

representation theorem – Regularity properties of Borel measures –Lebesgue measure –

continuity properties of measurable functions

UNIT IV

FOURIER TRANSFORSMS: Formal properties – Inversion theorem – The

Plancherel theorem – Banach Algebra L1

UNIT V

DIFFERENTIAL EQUATIONS: Uncoupled Linear systems – Diagonalization –

Exponentials of operators – Fundamental theorem for Linear systems – Linear Systems in

R2 – Complex Eigen values – Multiple Eigen values – Jordan forms – Stability theory –

Non-homogeneous linear systems.

L T P C

4 2 0 4

Unit – VI (Advanced topics only for discussion)

Current Contours:

Modules over PID, Distribution spaces.

TEXT BOOKS

[1] Serge Lang, “Algebra”, Springer - Verlag, Revised Third Edition, 2002.

Unit – I - Chapter III: Sections 1 to 6

Unit – II - Chapter X: Sections 1 to 4.

[2] W. Rudin, Real and Complex Analysis, 3rd

edition, McGraw Hill International,

1986.

Unit III – Chapter 2

Unit IV – Chapter 9

[3] L. Perko, Differential Equations and Dynamical systems, Springer-Verlag, First

Indian Reprint, 2004.

Unit V – Chapter 1: Sections 1.1 – 1.10

REFERENCES

[1] C. Musili, Rings and Modules, 2nd

edition, Narosa, 1994.

[2] P.B. Bhattacharya et al., Basic Abstract Algebra, 2nd

edition, Cambridge University

Press, 1995.

[3] Serge Lang, Complex Analysis, Addison Wesley, 1977.

[4] C.D. Aliprantis and O.Burkinshaw, Priniciples of Real Analysis 2edn, Academic

Press, Inc. New York, 1990.

[5] E.A Coddington and N. Levinson, Theory of Ordinary differential equations , Tata

McGraw Hill, New Delhi, 1972.

Course outcomes :

Students will be able to

Understand the power and elegance of modules structure for instance one

understands the vector spaces as a special type of modules.

Realize Noetherian rings are generalization of skew polynomials.

Understand and discuss the concepts involved in the proof of Nakayama’s lemma

Throughly study Riesz representation theorem and its consequences which imparts

knowledge and skill in abstract measure theory and to apply them in diverse areas.

Comprehend Lucin’s theorem and use it for establishing functional analytic

properties of various function spaces

Rigorously study the Fourier transforms and how it helps to get the analytic tricks

required to solves problems emanating from various applications.

Get well versed in diagonalization and Jordan form in linear algebra.

Solve system of differential equations as an application of algebraic concepts.

GUIDE PAPER

ADVANCED FUNCTIONAL ANALYSIS

Course Code: 17MP03CC:01

Objectives:

To demonstrate the Banach fixed point theorem is important as a source of existence

and uniqueness theorem in difference branches of analysis.

To realize the concept of compact operator, weak topology, and adjoint operator.

To learn the relation between distribution derivative and the distribution generated

by classical derivative.

UNIT – I,

Banach Fixed point theorem - Applications of Banach’s Theorem to Linear

Equations - Applications of Banach Theorem to Differential Equations Applications

- Applications of Banach Theorem to Integral Equations.

UNIT II,

Compact Linear Operators on Normed Spaces – Further properties of Compact

Linear Operators – Spectral Properties of Compact Linear Operator on Normed

Linear Spaces -

UNIT III

Operator Equations Involving Compact Linear Operators – Theorem of Fredholm

Type – Fredholm Alternative – Weak and Weak* Topology – Banach Alaoglu

Theorem.

UNIT IV

Distribution: Introduction - Test Function and Distributions - Some operation with

Distributions - Supports and Singular Support of Distributions - Convolution of

Function and Distributions.

L T P C

4 2 0 4

UNIT V

Convolution of Distributions – Fundamental solutions – The Fourier Transform –

Schwartz spaces – The Fourier Inversion Formula – Tempered Distributions.

Unit – VI (Advanced topics only for discussion)

Current Contours: Locally convex spaces. Topological Vector spaces.

TEXT BOOK

[1] Erwin Kreyszig, Introductory Functional Analysis with Applications, University of

Winndsor, Wiley Classics library Edition published 1989.

[2] Bella Bollobas, Linear Analysis , Cambridge University press,Second Edition

2004.

[3] S. Kesavan, Topics in Functional Analysis and Its Applications, TIFR Bangalore,

India, Wiley Eastern Limited.

UNIT I Chapter 8 [1]

UNIT II Chapter 8 [1]

UNIT III Chapter 8 [1],Chapter 8 [2]

UNIT IV Chapter 1 (Section 1.1 – 1.6) [3]

UNIT V Chapter 1 (Section 1.7 – 1.11) [3].

Course Outcomes:

Students will be able to

Apple the existence and uniqueness of solution of the problems in differential

equations, linear algebraic equations and integral equations by using the Banach

fixed point theorem.

To understand the theory of compact operators and utilize its importance

Comprehend spectral theorem on compact operators.

To prove the Banach Alaoglu theorem by using the concept of weak* topology.

Realize the importance of Schwartz theory of distributions.

Get the working knowledge on various distributions such as dirac, tempered etc..

Workout on fundamental solutions of plenty of distributions.

Comprehend Fourier inversion formula and work on tempered distributions.

ADVANCED NUMERICAL ANALYSIS

Course Code 17MP03CC:02

Objectives:

To learn numerical analysis is concerned with obtaining approximate solutions

while maintaining reasonable bounds on errors.

To numerically obtain the roots of an equation using several methods these Muller

method and Regula-Falsi method.

UNIT – I

Transcendental and polynomial equations: Iteration methods based on second

degree equation - Rate of convergence - iterative methods – Methods for finding

complex roots – iterative methods : Birge-Vieta method, Bairstow’s method,

Graeffe’s root squaring method

UNIT – II

System of Algebraic linear equations: Direct methods - Gauss Jordan Elimination

Method – Triangularization method – Cholesky method – partition method. Error

Analysis – Iteration methods : Jacobi iteration method – Gauss - Seidal iteration

method – SOR method. Jacobi method for symmetric matrices – Power method –

Inverse power method.

UNIT – III

Interpolation and Approximation : Hermite Interpolations – Piecewise and Spline

Interpolation – Appproximation – Least Square Approximation..

L T P C

4 2 0 4

UNIT – IV

Differentiation and Integration : Numerical Differentiation – Numerical

Integration – Methods based on Interpolation.

UNIT - V

Ordinary Differential Equations :- Multi – step method – Predictor – Corrector

method – Boundary value problem – Initial value methods – Shooting method –

Finite Difference method..

Unit – VI (Advanced topics only for discussion)

Current Contours: Numerical methods for partial differential equations -

Eigen value problems – Finite Element Methods.

TEXT BOOK(S):

[1] M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific

and Engineering Computation, III Edn. Wiley Eastern Ltd., 1993.

Unit I – Chapter 2 – 2.4 to 2.8

Unit II - Chapter 3- 3.2– 3.5

Unit III – Chapter 4 4.4 – 4.6, 4.8 to 4.9

Unit IV – Chapter 5 – 5.2, 5.6, 5.7

Unit V – Chapter 6 – 6.4, 6.5, 6.8, 6.9, 6.10

REFERENCES

[1] Kendall E. Atkinson, An Introduction to Numerical Analysis, II Edn., John

Wiley & Sons, 1988.

[2] M.K. Jain, Numerical Solution of Differential Equations, II Edn., New Age

International Pvt Ltd., 1983.

[3] Samuel. D. Conte, Carl. De Boor, Elementary Numerical Analysis, Mc

Course outcomes:

Students will be able to

Understand the theoretical and practical aspects on the use of numerical methods.

Analyze the errors obtained in the numerical solutions of problems.

Demonstrate the use of interpolation methods to find intermediate values in the

given graphical and/or tabulated data.

Numerically solve lot of practical problems where exact solution is unknown.

Determine approximate solutions of system of linear algebraic equations using the

method of matrix decomposition.

Understand the idea of interpolation namely the deviation of the given function

from the approximating polynomial.

Evaluate the derivative of the given function by approximating polynomial.

Study in detail the numerical methods involved in shooting method, finite difference

method.

ALGEBRAIC CODING THEORY

Course Code 17MP03CC:03

Objectives:

To learn how codes in mathematics are used for error correction and data

transmission.

To comprehend the algebraic structure of linear codes viewed as a vector space over

a finite field.

UNIT – I

Linear Code: Introduction - Linear codes, Encoding, Decoding - Check Matrices

and Dual Code

L T P C

4 2 0 4

UNIT II

Linear Code continued: Classification by Isometry – Semilinear Isometry Classes

of Linear codes – The Weight Enumerator

UNIT III

Bounds and Modifications: Combinatorial Bounds for the Parameters - New

Codes from Old and Minimum distance – Further Modifications and Constructions.

UNIT IV

Reed Muller Codes – MDS Codes

UNIT V

Cyclic Codes: Cyclic Codes as Group Algebra Codes – Polynomila Representaion of

Cyclic Codes – BCH Codes and Reed-Solomon Codes

Unit – VI (Advanced topics only for discussion)

Current Contours: Graph Codes – Identifying codes.

TEXT BOOK

[1] Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert

and Alfred Wassermann, Error-Correcting Linear Codes, Classification by Isometry

and Applications, Springer-Verlag berlin Heidelberg 2006.

UNIT I Chapter 1 (Section 1.1 – 1.3)

UNIT II Chapter 1 (Section 1.4 – 1.6)

UNIT III Chapter 2 (Section 2.1 – 2.3)

UNIT IV Chapter 2 (Section 2.4 & 2.5)

UNIT V Chapter 4 (Section 4.1 – 4.3)

REFERENCE(S)

.

1. F. J Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes,

North Holland Publishing Company 1977.

2. D.G. Hoffman et al, Coding Theory: The Essentials, Marcel Dekker, Inc, 1991

3. S. Ling and C. Xing, Coding Theory : A first course, Cambridge University press,

2004.

Course Outcomes:

Coding theory demonstrates how mathematics is used in real life applications.

Students get thorough idea about channel, noise, encoding, decoding etc and also

error detection and error correction which are involved in a data communication

One can have a clear cut idea about parameters of codes and the aims of

constructing codes with minimum length and maximum number of codewords and

maximum distance.

To learn how finite fields, vector space, number theory and probability theory are

applied in Coding Theory.

To acquire the knowledge of efficient decoding schemes of various linear codes.

To understand various bounds involved in coding theory and how it is used for

proving non-existence of codes under certain situation.

To practice how to construct new codes from the existing codes and learn more

about the parameters of the newly obtained codes.

To get expertise in some important codes like Hamming code, Golay code and

Reed-Muller code and thier applications to information theory.

FIXED POINT THEORY

Course Code 17MP03CC:04

Objectives:

To understand the concept of fixed point theorem for various spaces.

L T P C

4 2 0 4

To learn the concept of uniformly convex banach spaces and Tarsiki’s fixed point

theorem.

UNIT I

Banach’s contraction principle – Further extensions- Caristi – Ekeland principle -

Equivalance of Caristi- principles.

UNIT II

Tarsiki’s Fixed point theorem - Hyperconvex spaces – Properties – fixed point

theorems – intersection of hyper convex spaces – Isbell’s convex hull.

UNIT III

Uniformly convex Banach spaces – Fixed point theorem of Browder, Gohde and

Kirk. Reflexive Banach spaces –Normal structure- Fixed point theorems.

UNIT IV

Generalized Banach Fixed-point theorem- Upper and lower semi continuity of

multivalued maps –Generalized Schauder Fixed point theorem – Variational

Inequalities and the Browder Fixed-Point theorem – Extremal Principle –

Applications to Game Theory – Michael’s selection theorem

UNIT V

Fixed point theorem for continuous functions- Brouwer's theorem -Schauder's

theorem - applications - Hairy ball theorem - pancake problems- Kyfan's best

approximation theorem.

Unit – VI (Advanced topics only for discussion)

Current Contours: Generalizations of Brouwer’s fixed point theorem.

Best proximity pairs.

TEXT BOOK(S)

1. E. Zeidler, Nonlinear Functional Analysis and its applications, Vol. I Springer –

Verlag New york (1986)

2. M. A. Khamsi & W. A. Kirk, An introduction of Metric spaces and Fixed point

theory, John Wiley & sons (2001).

UNIT – I - Chapter 3 (3.1 - 3.4) from [2]

UNIT – II - Chapter 4 from [2]

UNIT – III - Chapter 10 (10.1 -10.3) from[1] and chapter 5 (5.1 -5.4) from [2]

UNIT – IV - Chapter 9 from [1]

UNIT – V - Chapter 2 from [1]

REFERENCES

1. D.R. Smart, Fixed point theory, Cambridge University Press, (1974).

2. V.I. Istratescu, Fixed point theory, D. Reidel Publsihing Company, Boston (1979).

Course Outcomes:

Students will be able to

Appreciate how the study of fixed point theory helps to solve problems which are

theoretical as well as practical

Realize contraction, contractive maps have elegant results on the existence and

uniqueness of fixed points.

Study the theory of non-expansive fixed point theorems and understand the

geometry of the spaces involved.

Understand the depth of mathematical concepts required to give the proof Brouwer’s

fixed point theorem due to Milnor.

Appreciate the generalizations of Brouwer’s fixed point theorem, viz., Schauder and

the use of it in analysis and differential equations.

Comprehend the fixed point theory on multivalued maps and see the interconnection

with single valued cases.

Thoroughly understand the idea behind Michael’s selection theorem.

Understand Kyfan’s best approximation theorem and its consequences.