Courses for M.Sc. in Mathematics

Course Structure for M.Sc. in Mathematics

Semester I (Total Credits: 20)	Semester II (Total Credits: 20)
Linear Algebra (4)	Algebra I (4)
Elementary Number Theory (4)	Measure Theory (4)
Real Analysis (4)	Complex Analysis (4)
Basic Topology (4)	Computational Mathematics (4)
Ordinary Differential Equations (4)	Functional Analysis (4)
Total Credits at the end of first year = 40
Semester III (Total Credits: 20)	Semester IV (Total Credits: 20)
Algebra II (4)	Elective I (4)
Partial Differential Equations (4)	Elective II (4)
Probability and Statistics (4)	Elective III (4)
Project I (8)	Project II (8)
Total Credits at the end of second year: 40 + 40 = 80

List of Elective Courses

Code	Course Name	Code	Course Name
PM 501	Number Theory	PM 506	Advanced Algebra
PM 502	Differential Topology	PM 507	Algebraic Topology
PM 503	Harmonic Analysis	PM 508	Banach and Operator Algebras
PM 504	Analytic Number Theory	PM 509	Numerical Analysis
PM 505	Proofs	PM 510	Computational Fluid Dynamics

Details of the courses

Linear Algebra (PM4XX) 4 Credits

Introduction: This course is addressed to the first year students pursuing M.Sc. in Mathematics at SPS. It is a standard course in which various kinds of decompositions of the given finite dimensional vector space and different matrix representations of their operators are studied. It gets more interesting when this theory is specialized to the inner product spaces. Finally a very basic introduction to bilinear forms is given in the course. The course

aims partially at preparing the students to be able to study subjects like Representation Theory, Lie algebras, Functional Analysis, etc.

Prerequisites: Material covered in a standard undergraduate Linear Algebra course.

Polynomial rings: A quick revision of polynomial ring in one variable over a field (results without proof). (1 hour)
A quick review of basic linear algebra: Important definitions, results from chapters 1 through 5 of [1]. (1 week)
Fundamental polynomials associated to operators: Annihilating polynomial, minimal polynomial, characteristic polynomial of an operator. (2 weeks)
Diagonalization: Direct sum decomposition, invariant subspaces, primary decomposition, diagonalization, triangulation of one operator and that of a family of operators. (2-3 weeks)
D-N Decomposition: Decomposition into diagonalizable and nilpotent operators (1-2 weeks)
Canonical Forms: Cyclic decomposition theorem, rational canonical form, Jordan canonical form. (3-4 weeks)
Inner product spaces: Inner product spaces, orthogonality, Gram-Schmidt theorem, orthogonal projections, adjoint of an operator, self-adjoint, normal, unitary operators, spectral theorems. (3 weeks)
Bilinear forms: Symmetric and skew-symmetric forms, groups preserving bilinear forms. (1-2 weeks)

Suggested Texts:

K. Hoffmann and R. Kunze, Linear Algebra, 2nd edition,Prentice Hall Inc., 1971.
M. Artin, Algebra,Prentice Hall International, 1990.
D. S. Dummit and R. M. Foote, Abstract Algebra,3rd edition, Wiley, 2003.
S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 5th edition, Pearson Education, 2022.
S. Lang, Linear Algebra, Undergraduate Texts in Mathematics, Springer, 2004.
P. R. Halmos, Linear Algebra Problem Book, Mathematical Association of America, 1995.
N. Jacobson, Basic Algebra,Vol. 1, Hindustan Publishing Corporation, 1984.
S. Axler, Linear Algebra Done Right, Undergraduate Texts in Mathematics, Springer, 2014.

Elementary Number Theory (PM4XX) 4 Credits

Course Learning Objectives: This course introduces the students to number theory and prepares them for pursuing more specialized and advanced topics in number theory.

Prerequisites: Mathematics taught in the secondary school (10+2)

Divisibility: Basic properties of divisibility, greatest common divisor, the Euclidean algorithm, primes, unique factorization, infinitude of primes, divergence of the sum of reciprocals of primes.
Congruences: Basic properties of congruences, congruences of Fermat, Euler and Wilson, linear congruences, Chinese remainder theorem, polynomial congruences, Hensel’s lemma, primitive roots, RSA cryptosystems.
Quadratic residues: Quadratic residues and quadratic nonresidues, Legendre symbol and Gauss’s quadratic reciprocity law, Jacobi symbol and the reciprocity law.
Arithmetic functions: Some examples of arithmetic functions, Dirichlet product, multiplicative and completely multiplicative arithmetic functions.
Some Diophantine equations: Linear Diophantine equations, Pythagorean triplets, insolvability of some Diophantine equations such as x⁴ + y⁴ = z².
Simple continued fractions: Continued fraction expansion of real numbers, Dirichlet’s theorem on Diophantine approximation, rational approximations of real numbers by convergents in the continued fraction expansion, Hurwitz’s theorem, periodic continued fractions, Pell’s equation.
Distribution of prime numbers: Statements of Dirichlet’s theorem on primes in arithmetic progressions and the Prime Number Theorem, Chebyshev’s functions, some elementary results on the distribution of prime numbers, Bertrand’s postulate.

Suggested Texts:

I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, John Wiley & Sons, 1991.
G. E. Andrews, Number Theory, Dover, 1994.
A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984.
G. A. Jones and J. M. Jones, Elementary Number Theory, Springer, 1998.

Supplementary References:

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
H. Davenport, The Higher Arithmetic, 8th edition, Cambridge University Press, 2008.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition (revised by D. R. Heath-Brown and J. H. Silverman), Oxford University Press, 2008.
E. Landau, Elementary Number Theory, Chelsea, 1958.
W. J. LeVeque, Fundamentals of Number Theory, Dover, 1996.
T. Nagell, Introduction to Number Theory, Chelsea, 1981.
H. Rademacher, Lectures in Elementary Number Theory, Blaisdell, 1964.
I. M. Vinogradov, Elements of Number Theory, Dover, 1954.

Algebra I (PM401) 4 Credits

Introduction: It is wished that the students appreciate the beauty and challenges in classification of finite abelian groups of small orders. Moreover, their contrast with

infinite non-abelian groups can be observed. This course equips a student to learn representation theory of groups without much difficulty. Both commutative and non-commutative rings are studied once again with analogy and contrast in mind.

Prerequisites: Material covered in a standard undergraduate Algebra course.

Part I: Group Theory

A quick review of basic group theory, revision of results in permutation groups. (2 hours)
Group actions, example of linear groups such as GL(n), SL(n), O(n), SO(n), U(n), SU(n); counting formula, class equation, Sylow’s theorems (3-4 weeks)
Generators and relations, free groups, semidirect product of groups, structure of finite abelian groups, classification of finite groups of small orders (2-3 weeks)

Part II: Ring Theory

A brief review of ring theory: Basic definitions (rings, subrings, ideals, prime ideals, maximal ideals) and results in the theory of rings. (1 week)
Examples of rings (matrix rings, group rings, quaternions, rings of holomorphic functions, rings of convergent sequences, a few number rings), properties of ideals, integral domains and their rings of fractions, radical ideal, nilradical, Jacobson radical, ideal quotient, Chinese remainder theorem (2-3 weeks)
Concept of divisibility, gcd and its properties, prime elements, irreducible elements. (1 week)
Euclidean domains, principal ideal domains (PIDs), unique factorization domains (UFDs) with special emphasis on polynomial rings, Gauss’s lemma, Eisenstein’s criterion of irreducibility of polynomials. (4-5 weeks)

Suggested Texts:

I. N. Hersein, Topics in Algebra, Wiley Student edition, 2006.
M. Artin, Algebra, Prentice Hall International, 1990.
T. W. Hungerford, Algebra, Springer, 1980.
D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd edition, Wiley, 2003.
N. Jacobson, Basic Algebra, Vol. 1, Hindustan Publishing Corporation, 1984.
S. Lang, Algebra, 3rd edition, Graduate Texts in Mathematics 211, Springer, 2005.

Complex Analysis (PM402) 4 Credits

Prerequisites: Basic knowledge of Real Analysis

Quick Review of Complex numbers: Basic operations, conjugate, modulus, argument, exponential function, roots
Holomorphic functions: Continuity, derivative, holomorphic functions, Cauchy-Riemann differential equations, harmonic functions
Elementary functions: Polynomial and rational functions, exponential function, logarithm, trigonometric and hyperbolic functions
Complex integration: Paths and contours, integration, estimation theorem, Cauchy's integral formula, Cauchy's theorem, Liouville's theorem, fundamental theorem of algebra, maximum modulus principle, Schwarz's lemma
Series: (absolute and uniform) Convergence of series, power series, Taylor series, Laurent series, the identity principle
Zeros, singularities and residues: Classification of singularities, orders of poles and zeros, winding number, meromorphic functions, Cauchy's residue theorem, argument principle
Mappings: Linear fractional transformations, conformal mappings
Application of complex integration: Computation of indefinite integrals
Additional Topics: Branch points, doubly periodic functions, construction of sine, cosine as an inverse of a multi-valued function, Riemann mapping theorem, Dirichlet problem, analytic continuation, multivariable complex analysis

Main Text Book:

L. V. Ahlfors, Complex analysis, An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.
J. B. Conway, Functions of one complex variable, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995.

Supplementary References:

W. Rudin, Real and complex analysis, Third edition, McGraw-Hill Book Co., New York, 1987.
E.M. Stein, R. Sharkarchi, Complex analysis, Princeton Lectures in Analysis, 2, Princeton University Press, Princeton, NJ, 2003.
E. Goursat, A Course in Mathematical Analysis, Functions of a complex variable, Part I of Vol. II, Ginn and Company, 1916.

Real Analysis (PM403) 4 Credits

Prerequisites: Basic knowledge of Real Analysis and Linear Algebra

Quick review of basic Real Analysis: Construction of real numbers, order on real numbers and the least upper bound property, convergence of sequence and series, power series, multiplication of series, absolute and conditional convergence, rearrangements (with proof of Riemann's Theorem). Continuity, uniform continuity, compactness and connectedness in metric spaces. Differentiation: L'Hospital's rule, derivatives of higher orders, Taylor's theorem, differentiation of vector-valued functions
The Riemann-Stieltjes Integral: Definition and existence of the integral, properties of the integral, integration and differentiation, integration of vector-valued functions, rectifiable curves
Sequences and Series of Functions: Pointwise and uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, equicontinuity, Arzela-Ascoli theorem, Stone-Weierstrass theorem
Calculus of Several Variables: Differentiation of functions of several real variables (directional derivatives, partial derivatives, differentiability and the total derivative, chain rule, Jacobian, higher derivatives, interchange of the order of differentiation, Taylor's theorem), inverse function theorem, implicit function theorem, rank theorem, differentiation of integrals, derivatives of higher order
Additional Topics: Integration of differential forms: Integration, primitive mappings, partition of unity, change of variables, differential forms, Stokes' theorem, closed and exact forms. Some special functions: Power series, exponential and logarithmic functions, trigonometric functions, Gamma function, Fourier series

Main text book:

W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw Hill Book Company, New York, 1976.

Supplementary References:

T. M. Apostol, Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.
T. Tao, Analysis I and II, Third editions, Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2006.
M. Spivak, Calculus on Manifolds: A modern approach to classical theorems of advanced Calculus, West View Press, 27^th printing, 1998.
K. Janich, Vector Analysis, Undergraduate Texts in Mathematics, Springer, 2001.
S. Lang, Undergraduate Analysis, Second edition, Springer, 2005.
H. L. Royden and P. M. Fitzpatrick, Real Analysis, Fourth Edition, Pearson Education, Inc., 2010.

Basic Topology (PM404) 4 Credits

Prerequisites: Basic knowledge of Real Analysis and Metric Spaces

Familiarity with Set Theory: Countable and uncountable sets, axiom of choice and its variants.
Topological Spaces and continuous functions: Topology, basis, sub-basis, Hausdorff and regular spaces, order topology, subspace topology, limit points, continuous functions, homeomorphisms, product topology and metric topology.
Quotient Topology: Quotient map, quotient topology, quotient space.
Nets: Subnets, convergence of nets
Connectedness and Compactness: Connectedness, path-connectedness, compactness, comparison with compactness in metric spaces via nets, local compactness and one-point compactification
Countability and Separation Axioms: First and second countability, separability, normality, complete regularity, Urysohn's lemma, Tietze extension theorem, Tychonoff theorem and Stone-Cech compactification
Additional Topics: Urysohn Metrization theorem, local finiteness, Nagata-Smirnov metrization theorem, paracompactness and Smirnov metrization theorem

Main Text Books:

J. R. Munkres, Topology, Second Edition, Pearson, 2000.
G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139, Springer, 1993.

Supplementary References:

C. O. Christenson and W. L. Voxman, Aspects of Topology, Second edition, B. C. S. Associates, 1998.
K. Janich, Topology, Undergraduate Text in Mathematics, Springer, 1984.
J. L. Kelley, General Topology, Graduate Text in Mathematics, Springer, 1975.
G. F. Simmons, Topology and Modern Analysis, Tata McGraw-Hill, 2004.
J. Dugundji, Topology, McGraw-Hill Inc., 1988.

Algebra II (PM405) 4 Credits

Introduction: The primary contents are field theory and Galois theory. A very brief introduction to modules is given towards the end. This provides sufficient background to pursue more advanced topics such as Algebraic Number Theory, Representation Theory

and Algebraic Geometry.

Prerequisites: Linear Algebra and Algebra I.

Revision of polynomial rings (1 hour)
Basic definition of field extension, degree, characteristic, algebraic extensions, basic results (2-3 hours)
Existence and uniqueness (up to isomorphism) of the splitting field of a polynomial, existence and uniqueness (up to isomorphism) of the algebraic closure of a field (1 week)
Cyclotomic fields (1 week)
Embeddings of an algebraic extension of the base field into a fixed algebraic closure of and over the base field, results on extending the embeddings to a bigger domain, their effect on the polynomial rings (3 weeks)
Separability and number of embeddings (2 weeks)
Normal extensions and automorphisms (1 week)
Finite fields, main theorem of finite fields (2 weeks)
Galois extensions, fundamental theorem of Galois theory, computing various Galois groups (2-3 weeks)
Transcendental extensions, Luroth’s theorem (2-3 hours)
Introduction to module theory (5-6 hours)

Suggested Texts:

I. N. Hersein, Topics in Algebra, Wiley Student edition, 2006.
M. Artin, Algebra, Prentice Hall International, 1990.
T. W. Hungerford, Algebra, Springer, 1980.
D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd edition, Wiley, 2003.
N. Jacobson, Basic Algebra, Vol. 1, Hindustan Publishing Corporation, 1984.
S. Lang, Algebra, 3rd edition, Graduate Texts in Mathematics 211, Springer, 2005.

Supplementary References:

S. Lang, Algebra, Graduate Texts in Mathematics, 211, revised third edition, Springer-Verlag, New York, 2002.
E. Artin, Edited and supplemented with a section on applications by Arthur N. Milgram, Second edition, with additions and revisions, Fifth reprinting, Notre Dame Mathematical Lectures, No. 2, University of Notre Dame Press, South Bend, Ind. 1959.
N.S Gopalkrishnan, University Algebra, Second edition, New Age International, New Delhi, 1986.
N.S Gopalkrishnan, Commutative Algebra, Oxonian Press Pvt. Ltd., New Delhi, 1984.

Measure Theory (PM406) 4 Credits

Prerequisites: Real Analysis, Basic Topology

Quick review of Riemann Integration
Lebesgue Measure on ℝ: Outer measure, outer regularity of outer measure, Lebesgue measure, regularity of Lebesgue measure, non-measurable sets
Lebesgue Integral: Simple functions, almost everywhere property, measurable functions, integrable functions, approximation of integrable functions by step and continuous functions
Convergence of sequence of functions: Pointwise, uniform, Egorov's theorem, Lusin's theorem
Abstract measure spaces: Sigma algebras and measurable spaces, measures and measure spaces, completeness of a measure, measurable functions and their integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, modes of convergence
Product measure: product sigma algebra, sigma-finite measure spaces, existence of product measures, Tonelli's theorem, Fubini's theorem
Additional Topics: Lebesgue differentiation theorem, almost everywhere differentiability, absolute continuity, Caratheodory's Extension theorem for outer measures, Hahn-Kolmogorov extension theorem for pre-measures, Lebesgue-Stieltjes measure, Radon measure

Main Text book:

T. Tao, An Introduction to Measure Theory, GTM 126, American Mathematical Society, 2011.

Supplementary References:

G. B. Folland, Real Analysis: Modern Techniques and their Applications, expanded and revised edition, John Wiley and Sons, 2013.
W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.

Functional Analysis (PM407) 4 Credits

Prerequisites: Real Analysis, Basic Topology

Hilbert spaces: Inner product spaces, Hilbert spaces, orthogonality, Riesz representation Theorem, orthonormal sets, orthogonalization, unconditional sum, orthonormal bases, isomorphisms of Hilbert spaces, separable Hilbert spaces, direct sums of Hilbert spaces
Operators on Hilbert spaces: Examples, adjoint of an operator, invertible operators, self-adjoint operators, unitary operators, isometries, projections, compact operators
Banach Spaces: Normed spaces, equivalence of norms, some inequalities, Banach spaces, finite dimensional spaces, quotient and products of normed spaces, bounded linear operators and functionals
Dual Spaces: Hahn-Banach theorem, dual of a quotient space and a subspace, reflexive spaces
Category Theorems: Baire category theorem, open mapping theorem, closed graph theorem, principle of uniform boundedness
Operators on Banach spaces: Adjoint of an operator, annihilators, compact operators
Additional Topics: Weak topology, weak-* topology, Banach-Alaoglu theorem, Goldstine's theorem, reflexivity in terms of weak topology, separable Banach spaces

Main Text Books:

John B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Second edition 1990, corrected fourth printing, Springer, 1994.
S. Kesavan, Functional Analysis, Texts and Readings in Mathematics (TRIM series), 52, Corrected reprint, Hindustan Publishing Agency, 2017.

Supplementary References:

G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004.
W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.
W. Rudin, Functional Analysis, McGraw-Hill Education, Third Edition, 1986.
G. K. Paderson, Analysis Now, Graduate Texts in Mathematics, 118, Springer, 2012.
H. L. Royden and P. M. Fitzpatrick, Real Analysis, Fourth Edition, Pearson Education, Inc., 2010

Probability and Statistics (PM409) 4 Credits

Prerequisites: Real Analysis, basic knowledge of combinatorics

Probability and random variables: Repeated experiments and empirical definition of probability. Sample space, events. Probability as a set function on a $\sigma$-algebra. Random variables, distribution functions and probability density functions. Expectation value, variance and higher moments. Moment generating functions, inequalities of Markov and Chebyshev.
Conditional probability and independence: Conditional probability, marginal distributions and conditional distributions. Covariance and correlation, stochastic independence.
Some probability distributions: Binomial, Poisson and normal distributions. Properties of their moments.
Distributions of functions of random variables: Sampling. Transformations of random variables, Student's t and F distributions. Distributions of mean and variance of a sample. Expectations of functions of random variables.
Limiting distributions: Stochastic convergence of random variables. Weak and strong laws of large numbers (without proofs). Central limit theorem.
Additional topics:} Hypothesis testing - Examples and definitions. Uniformly most powerful tests. Likelihood ratio tests. Statistical significance.

Main Text Books:

R. V. Hogg and A. T. Craig, Introduction to Mathematical Statistics, Fourth edition, McMillan Publishing Company, 1978.
S. Ross, A First Course in Probability, 8th Edition, Prentice Hall/Pearson, 2010.

Supplementary References:

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Third edition, Wiley, 2008.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Second edition, Wiley, 2008.

Computational Mathematics (PM410) 4 Credits

Prerequisites:} Algebra I, Real Analysis, Complex Analysis, Linear Algebra, additionally some knowledge of elementary number theory and ODEs will help, but is not essential.

Goal: The aim is to introduce the students to algorithmic way of learning, to teach them the importance of computations and to the use of computers for implementation of a few algorithms. We will be using a few programming tools such as C++, MATLAB/SciLab, SAGE to implement a few well-known algorithms from Calculus, Number Theory, Linear Algebra, Algebra, Graph Theory, Discrete Mathematics and possibly Differential Equations and Statistics depending on the preparation and inclination of the students.

This course will have a practical component and labwork.

Algorithms: Introduction to algorithms with a few standard examples
Brief Introduction to Programming:} Introduction to programming languages and computational systems such as C++, SAGE, MATLAB/SciLab
Algorithms in Calculus: Newton-Raphson iteration method for finding real root, numerical integration
Algorithms in Linear Algebra: Solving systems of linear equations, diagonalization
Algorithms in Differential Equations: Solution of ordinary differential equations, Runge-Kutta
Algorithms in Number theory: Sieve of Eratosthenes, primality tests, Euclidean algorithm, greatest common divisor, solution to Pell's equation using continued fractions
Algorithms in Graph Theory and Discrete Mathematics: Kruskal's algorithm, finding Eulerian cycles, sorting and searching algorithms
Additional topics: Computation of Galois groups, Grobner bases

Main References:

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein, Introduction to Algorithms, Third edition, MIT Press, 2009.
S. Pemmaraju and S. Skiena, Computational Discrete Mathematics, First Reissue edition, Cambridge University Press, 2009.
Henry Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993.
SAGE http://www.sagemath.org/, open-source mathematics software system

Supplementary References:

D. F. Holt, B. Eick, E. A. O'Brien, Handbook of Computational Group Theory, Chapman and Hall/CRC Press, 2005.
F. Villegas, Experimental Number Theory, Oxford Graduate Texts in Mathematics, Book 13, 2007.
D. E. Knuth, The Art of Computer Programming Volumes 1 to 4, Addison-Wesley Professional, First edition, 2011.
M. Petkovsek, H. Wilf and D. Zeilberger, A = B, A. K. Peters/CRC Press, 1996.

Ordinary Differential Equations (PM411) 4 Credits

Prerequisites: Calculus, Linear Algebra, Real Analysis

Quick Review of some basic methods of solving first order ODEs: Motivation, Order and degree of ODEs. Method of separation of variables. Exact Differential Equations. Integrating Factors.
Existence, Uniqueness and Continuity Theorems for first order ODEs: Picard’s existence and uniqueness Theorem, Picard’s successive approximation method, Continuity of solutions with respect to initial conditions, Gronwal’s inequality.
Second order linear ODEs: Vector space of space of solutions. Wronskian and linear independence of solutions. Linear ODE with constant coeffecients. Variation of parameters. Method of undetermined coefficients. Sturm Separation Theorem. Sturm Comparison Theorem.
Solution in series of second order ODEs: Ordinary and singular points. Power series solution at an ordinary point. Legendre’s equation. Solutions at a regular singular point using the Frobenius method. Bessel’s equation.
Systems of first order ODEs: System of first order ODEs versus n-th order ODE. Existence and uniqueness Theorem for system of first order ODEs. Existence and uniqueness Theorem for n-th order ODEs. Picard’s succesive approximation method. Homogeneous linear systems of first order ODEs. Fundamental matrix and solution matrix. Nonhomogeneous linear systems. Linear systems with constant coefficients.
Boundary-value problems and self-adjoint eigenvalue problems: Two point boundary value problems, Green’s functions. Sturm-Liouville systems. Eigenvalues and eigenfunctions.
Additional topics:

Stability Analysis: Linear systems, Stability for linear systems with constant coefficients, Stability of linear plane systems
Laplace transform : Properties of the Laplace transform, Convolution Theorem, Step function, Impulse function.
Cauchy-Peano Existence Theorem: Arzela-Ascoli Theorem. Existence of solution of ODEs not satisfying the Lipshitz condition.

Suggested texts:

Coddington, E., An Introduction to Ordinary Differential Equations, Dover Publications, 2012. (Original: Prentice-Hall, 1961.)
Coddington, E., and Levinson N., Theory of Ordinary Differential Equations, Tata-McGraw-Hill, 1990.
Myint-U, T., Ordinary Differential Equations, North-Holland, New York, 1978.
Rabenstein, A. L., Introduction to Ordinary Differential Equations, Elsevier Science, 2014.
Ross, S. L., Introduction of Ordinary Differential Equations, 4 th Ed., John Wiley and Sons, 2007.
Simmons, G. F., Differential Equations with Applications and Historical Notes, CRC Press, 2017.

Partial Differential Equations (PM412) 4 Credits

Prerequisites: Calculus, Linear Algebra, Complex Analysis, Ordinary Differential Equations

Motivation: PDE in natural science. First order PDE, examples.
Second order PDE: Classification and reduction to canonical forms. Well-posed problem. Characteristics. Green's function.
Laplace equation (elliptic): Boundary value problem, Dirichlet and Neumann boundary conditions. Harmonic functions. Mean value theorem. Solution by separation of variables.
Heat equation (parabolic): Initial and boundary value problem. Solution by separation of variables. Duhamel's principle.
Wave equation (hyperbolic): D'Alembert's solution. Cauchy problem, existence and uniqueness of solutions. Solution by separation of variables.
Additional topics: Fourier transform method. Laplace and Mellin transforms.

Main Text Book:

I. Sneddon, Elements of Partial Differential Equations, Dover reprint, Dover, 2006.

Supplementary References:

L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, 2010.
J. Fritz, Partial Differential Equations, Fourth edition, Springer, 1991.
E. L. Ince, Ordinary Differential Equations, reprint edition, Dover, 1956.
V. I. Arnold, Lectures on Partial Differential Equations, Third edition, Springer, 2006.
T. Amaranath, An Elementary Course in Partial Differential Equations, Second edition, Narosa, 2014.

Project I (PM4XX) 8 Credits

Prerequisites: Reasonably good understanding of M.Sc. first year courses; especially those

Project II (PM4XX) 8 Credits

Prerequisites: Project I

Project II can be a continuation of Project I. The student can study some advanced topics and do some literature survey. The student may also read some research papers. It is expected that the student will meet the supervisor regularly and present the material that he/she has learnt and keep his/her supervisor updated with his/her progress.

The student is also expected to write an M.Sc. thesis on the topics covered in the project, which he/she will have to present to a panel of examiners at the end of the term.

Grading scheme: The student’s performance will be evaluated based on the presentations (made during the meetings with the supervisor), the thesis and the final presentation.

Number Theory (PM501) 4 Credits

Prerequisites: Algebra I, Algebra II, Real Analysis, Complex Analysis

Goal: To provide an introduction to Number Theory that is beyond an undergraduate elementary number theory course and having algebraic, algebraic geometric and also analytic components. This course will cover a few topics which will illustrate that number theory uses tools from all the various disciplines of mathematics.

Unique factorization and applications: ℤ, k[x], unique factorization in a principal ideal domain, study of ℤ[i], ℤ[ω] and ℤ[1+-52]
Congruences, structure of (ℤ/nℤ)^*
Quadratic reciprocity, Gauss and Jacobi Sums
Equations over finite fields, Hasse-Davenport relation, zeta function as a generating function of number of solutions
Riemann zeta function. Definition of Dirichlet L-functions attached to a character with the possibility of deeper study of L-function attached to a character of order 2
Diophantine equations of genus 0 over the rationals: Pythagorean triplets, Pell's equation
Irrationality and transcendence of e and π
Additional Topics: Brief introduction to algebraic number theory, a very brief introduction to elliptic curves, the group law on an elliptic curve possibly without proof of the associativity property, a few important concrete examples and computations, connection to congruence number problem and Fermat's last theorem

Main Text Book:

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer-Verlag, 1990.

Supplementary References:

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth Edition (edited by R. Heath-Brown, J. Silverman, and A. Wiles), Oxford University Press, 2008.
I. Niven, H. S. Zuckerman, H. L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, John Wiley and sons, Inc., 1991.
N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, second edition, Springer, 1993.
J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, second edition, Springer-Verlag New York, 2009.
J. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Second edition, Springer International Publishing, 2015.
J.-P. Serre, A course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag New York, 1973.
F. Villegas, Experimental Number Theory, Oxford Graduate Texts in Mathematics, Book 13, 2007.

Differential Topology (PM502) 4 Credits

Prerequisites: Linear Algebra, Topology, Real Analysis

Quick review of multivariable calculus: Implicit and inverse function theorems
Quick review of separation axioms and paracompactness: First and second countability, separability, normality, complete regularity, Urysohn lemma and paracompactness
Topological manifolds: Definition and some basic properties
Smooth manifolds: Smooth atlas, smooth structure, smooth manifolds
Smooth maps: Smooth maps, diffeomorphisms, bump functions, partitions of unity
Tangent space and the differential: Derivations, tangent space, tangent vectors to curves, differential of a smooth map
Vector fields: Tangent bundle, vector fields on manifolds, orientation on a manifold
Immersions, submersions and embeddings: Implicit and inverse function theorem for manifolds, submersions, immersions
Submanifolds: Embedded submanifolds and their tangent spaces, regular and critical points and values, level sets, immersed submanifolds
Additional Topics: Smooth manifolds with boundary and associated definitions, , submanifolds of manifolds with boundary, embedded submanifolds and their tangent spaces, Lie brackets, Sard's theorem, differential forms and integration on manifolds, Stokes' theorem

Main Text Books:

J. M. Lee, Introduction to Smooth Manifolds, GTM, Springer, 2006.
K. Janich, Vector Analysis, Undergraduate Texts in Mathematics, Springer, 2001.

Supplementary References:

M. Spivak, Calculus on Manifolds: A modern approach to classical theorems of advanced Calculus, West View Press, 27^th printing, 1998.
V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, 1974.
F. W. Warner, Foundations of Differential Manifolds and Lie Groups, Graduate Texts in Mathematics, 94, First edition, Springer, 1983.
W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw Hill Book Company, New York, 1976.
G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139, Springer, 1993.
A. A. Kosinski, Differential Manifolds, Dover Publications Inc., 2007.
J. R. Munkres, Elementary Differential Topology, Revised Edition, Annals of Mathematics Studies (AM-54), Princeton University Press, 1967.

Harmonic Analysis (PM503) 4 Credits

Prerequisites: Real Analysis, Complex Analysis, Measure Theory

Fourier series: Fourier development of periodic functions, examples, Dirichlet and Fejer kernels, convergence of Fourier series, Gibb's phenomenon, Parseval's equation
Review of measure theory and integration: Lebesgue measure, Lebesgue integral, L^p space
Fourier transform: Motivation and definition, examples, Fourier inversion formula, uniform continuity and Riemann-Lebesgue lemma, Plancherel's theorem, Poisson summation formula, convolution theorem, differentiation of Fourier transforms, Hermite functions, Laplace transform
Applications: Filtering, differential equations, central limit theorem
Additional Topics: Spherical harmonic analysis, harmonic analysis on topological groups, Pontryagin duality

Main Text Book:

E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.

Supplementary References:

Y. Katznelson, An introduction to harmonic analysis, Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
G. B. Folland, Fourier analysis and its applications. The Wadsworth and Brooks/Cole Mathematics Series, Wadsworth and Brooks/Cole Advanced Books and Software, 1992.
L. Grafakos, Classical Fourier analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014.

Analytic Number Theory (PM504) 4 Credits

Prerequisites: Complex Analysis

Arithmetic functions and Dirichlet series: The ring of arithmetic functions, Dirichlet series, important arithmetic functions, average estimates
Characters: Group characters, Dirichlet characters, detection of residue classes, Gauss sums
Prime number distribution: Infinitude of primes, Chebyshev's bounds, Riemann zeta function, Perron's formula, prime number theorem, Dirichlet L-functions, primes in arithmetic progressions
Circle method: General set up, ternary Goldbach problem, partitions
Sieve methods: Selberg's sieve, large sieve, estimates for twin primes, estimates for twins of almost-primes
Additional Topics: Modular forms, exponential sums

Main Text Book:

T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.

Supplementary References:

H. L. Montgomery, R.C. Vaughan, Multiplicative number theory I - Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007.
H. Davenport, Multiplicative number theory, Third edition, Revised and with a preface by H. L. Montgomery, Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 2000.
R. C. Vaughan, The Hardy-Littlewood method, Second edition, Cambridge University Press, 1997.
A. C. Cojocaru, M. R. Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, 66, Cambridge University Press, Cambridge, 2006.
S. W. Graham, G. Kolesnik, van der Corput's method of exponential sums, London Mathematical Society Lecture Note Series, 126, Cambridge University Press, Cambridge, 1991.
N. Koblitz, Introduction to elliptic curves and modular forms, Second edition, Graduate Texts in Mathematics, 97, Springer-Verlag, New York, 1993.

Proofs (PM505) 4 Credits

Prerequisites: Algebra I, Real Analysis, Complex Analysis, Basic Topology

Goal: To discuss the concept of a proof or what would constitute as a piece of an evidence. To introduce, explain and discuss proofs in mathematics, computer science and other disciplines, brief discussion about kinds of proofs using examples, The concept/definition of zero knowledge proof in computer science, computer assisted proofs, formal proof. Another important goal of this course will be to teach the students how to write mathematics and proofs (using examples and possibly small projects as assignments).

What is a proof or a piece of evidence: Proofs in different walks of life - in politics, in social sciences, in history, as per the judiciary, criminology, in biology, medical sciences, weather prediction, in physics and in mathematics - illustrated using examples in brief.
Proofs in mathematics:
- Language of proofs. Introduction to logic
- Methods of proving: Numerical, computational, by induction (various kinds), by contradiction (reductio ad absurdum), contrapositive, types of proofs - direct, indirect, constructive, non-constructive
- How to disprove: how to construct counterexamples?
Proofs in computer science: The concept of zero-knowledge proof in brief using couple of the following or similar examples (a) Using the game called ``where is Waldo''? (b) Graph colouring problem and it's solution by Wigderson et al (c) ``How to explain zero knowledge protocols to your children''. The concept of a proof certificate - e.g. Pratt's certificate that certifies that a specific large number is a prime number (based on Lucas's theorem). Size and complexity of a proof.
Computer Assisted Proofs: One or two examples from the below will be discussed in brief: four colour theorem, classification of finite groups, sphere packing, existence of Lorentz attractor
Additional Topics: Role of proofs in mathematics: Discussions around the main points of the bulletin of AMS article of W. Thurston, errors in mathematical research papers, role of speculation, conjectures, and questions in the progress of mathematics. Concept of a formal proof

Main References:

S. Krantz, The proof is in the pudding - The changing nature of mathematical proof, Springer-Verlag, 2011.
Franklin and Daoud, Proofs in Mathematics - An Introduction, Quakers Hill Press, 1996/Kew Books, 2011.
M. Aigner and G. Ziegler, Proofs from THE BOOK, 4th edition, Springer-Verlag, 2009.
M. Petkovsek, H. Wilf and D. Zeilberger, A = B, A K Peters/CRC Press, 1996.
T. C. Hales, Formal proof, Notices of the AMS, Vol 55, Number 11, 1370-1380 (and references therein).
L. Lamport, How to Write a 21^st century proof, J. Fixed Point Theory App. Vol. 11, Issue 1 (2012), pp 43-63.
B. Mazur, The faces of evidence (in Mathematics)}, Notes for the presentation and discussion at Museion, February 5, 2014.

Supplementary References:

A. Jaffe and F. Quinn, ``Theoretical Mathematics'': Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the AMS, Vol. 29, No. 1, July 1993, Pages 1-13.
W. Thurston, On proof and progress in Mathematics, Bull. of the AMS, 30 (1994), 161-177.
T. C. Hales, Jordan's proof of the Jordan Curve Theorem, STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23), 2007.
B. Mazur, Announcement of a joint undergraduate course (Law, Harvard) taught by Noah Feldman and co-taught by B. Mazur: ``Nature of Evidence''.
B. Mazur, Shadows of Evidence - An essay on science, 2013.

Advanced Algebra (PM506) 4 Credits

Prerequisites: Algebra I, Algebra II

Commutative Algebra: Projective and injective modules, projective and injective resolutions, functors, chain complex, exact sequences, higher derived functors, functoriality
Group cohomology: Ext and Tor, group cohomology, group extensions, H¹, H²
Central simple algebras: Simple modules, Schur's lemma, semisimple modules, central simple algebras, Wedderburn's decomposition theorem, tensor operation
Brauer group:} Brauer group of a field, relation of Brauer group to Galois cohomomology of the field

Main Text Books:

N. Jacobson, Basic Algebra Vol. I and II, Second edition, W. H. Freeman and Company, 1989.
T. Y. Lam, A First Course in Non-Commutative Rings, Second edition, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 2001.

Supplementary References:

N. S. Gopalkrishnan, Commutative Algebra, Oxonian Press Pvt. Ltd., New Delhi, 1984.

Algebraic Topology (PM507) 4 Credits

Prerequisites: Basic knowledge of Topology and Group Theory

The Fundamental group: Homotopy of paths. Definition of the fundamental group, covering spaces, the fundamental group of a circle, retractions, Brouwer fixed point theorem for a disc
Deformation Retracts and Homotopy Type
Fundamental groups of n-spheres and some surfaces
The Seifert-van Kampen Theorem: Free products of groups, free groups, Seifert-van Kampen theorem, fundamental groups of wedge of circles and of tori
Classification of Covering spaces: Covering spaces, equivalence of covering spaces, universal covering space, covering transformations, existence of covering spaces
Singular Homology: Singular complex, singular homology groups, homotopy axiom, Hurewicz theorem relating the fundamental and homology groups
Additional Topics: Brief idea about simplicial homology, reduced homology groups, homology of spheres

Main Text Books:

J. R. Munkres, Topology, Second Edition, Pearson, 2000.
J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988.

Supplementary References:

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139, Springer, 1993.
F. H. Croom, Basic concepts of Algebraic Topology, Undergraduate Texts in Mathematics, Springer, 1978.
Anant R. Shastri, Basic Algebraic Topology, First edition, Chapman and Hall/CRC, 2013.

Banach and Operator Algebras (PM508) 4 Credits

Prerequisites: Real analysis, Basic Topology and Functional Analysis

Banach algebras: Banach algebras, ideals and quotients, invertible elements, spectrum and spectral radius, Spectral mapping theorem, Gelfand-Mazur theorem, commutative Banach algebras and their Gelfand representations, holomorphic functional calculus, quotients, Stone-Weierstrass theorem
C*-algebras: Banach *-algebras and C*-algebras, multiplier algebra, unitization, Gelfand-Naimark representation of commutative C*-algebras, functional calculus, spectral mapping theorem, positive elements of C*-algebras
Operators on Hilbert spaces: Spectrum and other properties of normal, self adjoint, projection and unitary operators, partial isometry, polar decomposition, finite-rank and compact operators, diagonalization, Hilbert-Schmidt operators, trace-class operators
The Spectral Theorem: Spectral measures, spectral theorem for normal operators
Gelfand-Naimark Representation: Ideals in C*-algebras, approximate units, quotients, positive linear functionals, Gelfand-Naimark representation of C*-algebras
Additional Topics: von Neumann algebras - Strong and weak operator topologies, commutants, von Neumann algebras, double commutant theorem, polar decomposition, projections, Calkin algebra, pre-dual of a von Neumann algebra, Kaplansky density Theorem, Abelian von Neumann algebras

Main Text Book:

G. J. Murphy, $C^*$-algebras and Operator Theory, Academic Press Inc., 1990.

Supplementary References:

W. Arveson, A course on Spectral Theory, GTM, Springer, 2002.
R. G. Douglas, Banach algebra techniques in Operator Theory, Second Edition, GTM, Springer, 1998.
J. Dixmier, C*-algebras, North-Holland Publishing Company, 1977.
E. Kaniuth, A course on commutative Banach algebras, Graduate Texts in Mathematics, Springer, 2009.
M. Takesaki, Theory of Operator Algebras I, Springer, 2002.

Numerical Analysis (PM509) 4 Credits

Prerequisites: Basic analysis and algebra.

Quick review of basic numerical computations and errors: Floating point systems, Exact and approximate numbers, Rounding of numbers, Significant digits, accuracy, and various types of errors encountered in computations, order, and rate of convergence. Roots of algebraic and transcendental equations: Direct and Iterative methods with convergence criteria and error analysis.
Solution of a system of equations: Direct methods: Pivoting, Gauss elimination with pivoting, Tridiagonal systems, Thomas algorithm, Cholesky factorizations, SOR methods. Eigen-value problems: Power method, Givens method, Jacobi eigenvalue method, and Householder methods.
Numerical solutions of IVPs and BVPs: Single-step and multistep methods: Euler’s method, Runge-Kutta methods, Adam-Bashforth method, Shooting, and finite difference methods.
Solution of BVPs by Variational Methods: Introduction, Variational principle, Ritz Method, Functional for a differential equation, Galerkin’s Method, Collocation Method.
Solution of BVPs by Finite Element Method: Introduction, Ritz finite element method, Galerkin finite element method, Rayleigh-Ritz Method, and application on BVPs.

Suggested References:

Main Textbooks:

Richard L. Burden and J. Douglas Faires: Numerical Analysis, Ninth Edition, Brooks/Cole, Cengage Learning, 2011.
Kendall E. Atkinson: An introduction to numerical analysis, Second edition, John Wiley and Sons, 2008.
G. D. Smith: Numerical solution of partial differential equations: Finite Difference Methods, Third edition, Oxford University Press, 1985.
D. M. Causon, C. G. Mingham, and L. Own: Introductory Finite Volume Methods for Partial Differential Equations, Springer, 2009.

Supplementary References:

M. K. Jain, S. R. K. Iyengar, and R. K. Jain: Numerical Methods for Scientific and Engineering Computation. New Age International, 2012.
R. S. Gupta, Elements of Numerical Analysis, Macmillan India Ltd., 2009.

Computational Fluid Dynamics (PM510) 4 Credits

Prerequisites: Basic Calculus and Analysis.

Unit I: Concept of fluids, physical properties of fluids, continuum hypothesis, density, specific weight, specific volume, kinematics of fluids.

Unit II: Eulerian and Lagrangian method of description of flows, equivalence of Eulerian and Lagrangian methods, streamline, path line, streak lines, stream function¸ vortex lines, rotational and irrotational motion.

Unit III: General theory of stream function, complex-potential, stress, rate of strain, source, sink, and doublets, equation of conservation of mass (continuity equation), equation of conservation of momentum, Euler's equation of motion, Bernoulli’s equation.

Unit IV: Mathematical description of physical phenomenon, finite control volume, Reynolds-transport theorem, Navier Stokes equation, equation of moments of momentum, equation of energy.

Unit V: Computational techniques to solve the flow models: Non-dimensionalization, Grid formulation and appropriate transformations, Lax-Wendroff Technique, MacCormack's Technique, Quasi-linearization, Spectral methods, Finite difference Method, Finite volume Method and bvp4c, bvp5c.

Practicals: Practicals will be performed on the MATLAB software.

Spectral Method to solve the boundary value problem.

Finite difference Method to solve the boundary value problem.

Bvp4c and bvp5c methods to solve the boundary value problem.

Suggested References:

Main Text Books:

R. H. Pletcher, J. C. Tannehill and D. A. Anderson, Computational Fluid Mechanics and Heat Transfer, CRC Press, Taylor and Francis, 2013.
J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, 1995.
S. V. Patankar, Numerical Heat Transfer and Fluid Flow, CRC Press, Taylor and Francis, Indian Edition, 2017.

Supplementary References:

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Second Edition, SIAM, 2004.
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, 2013.
H. K. Versteeg, and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Second Edition, Pearson, 2008.
S K Som, Gautam Biswas, S Chakraborty, Introduction to Fluid Mechanics and Fluid Machines, McGraw Hill Education, 3rd Edition, 2017.
C. A. J. Fletcher, Computational Techniques for Fluid Dynamics: Fundamental and General Techniques: 001, Springer Verlag, 2nd ed. 1998.

Courses for M.Sc. in Mathematics