Department of Mathematics | Indian Institute Of Technology Madras , Chennai

We at the department of mathematics extend a warm welcome to all those who wish to pursue research in mathematics. You may find a list of current areas of active research under the Research tab. Click on any area to see the faculty member working in that area with their specializations. A news item will appear in the main page when the advertisement for the next admission is released. The online application portal can be accessed here.

Selection Process

Applicants should have a Master's degree in Mathematics with a good academic record and having a valid GATE score or UGC-JRF/NET/CSIR-JRF/NBHM/INSPIRE or equivalent qualification for the current year. In addition final Year M.Sc Mathematics students from IITs (entry through JAM) with CGPA 8.0 and above are exempted from qualifying in National level examinations. Students who get more than 8.0 CGPA in M.Sc in Mathematics department of IIT Madras can be admitted directly to the Ph.D programme in with interview at departmental level.

Syllabus for the Test/Interview

Candidates are required to answer questions from at least three of the following general areas.

MA5310 Linear Algebra

Course Details:

Unit 1. Vector spaces, subspaces, basis and dimension, coordinates, algebra of linear transformations, isomorphisms, representation of linear transformations by matrices, linear functionals and annihilators, double dual and transpose of a linear transformation. (14 lectures)
Unit 2. Characteristic values and characteristic polynomials, diagonalizabletransformations, annihilating polynomials, Cayley-Hamilton theorem, invariant subspaces and triangular form, simultaneous triangularization and diagonalization, direct sum decompositions, invariant direct sums, primary decompostion theorem. (14 lectures)
Unit 3. Inner product spaces, adjoints, unitary and normal transformations, spectral Theorem, Jordan canonical form. (12 lectures)

Course References:

Text Books:
1. K. Hoffman and R. Kunze: Linear Algebra, 2nd Edition, Prentice Hall of India, 2005
2. S. Axler: Linear Algebra Done Right, 2nd Edition, Springer UTM, 1997.
Reference Books:
1. P. Halmos, Finite dimensional vectors paces, Springer, 1974.
2. Peter D. Lax, Linear algebra, Wiley student edition, 1997.
3. E.B. Vinberg, A course in algebra, Graduate text in Mathematics, volume 56, AMS, 2003.
4. M. Thamban Nair and Arindama singh, Linear algebra, Springer, 2018.

MA5320 Algebra 1

Course Details:

Group Theory:
Review of basic Group Theory, Direct sum and direct product of groups, Group Actions, Cayley's Theorem, The Class Equation, Sylow's Theorems (15 lectures)
Ring Theory:
Review of basic Ring Theory, Properties of Ideals, Prime and Maximal Ideals, Two-sided ideals and Quotient Rings, Chinese Reminder Theorem, Euclidean Domain, Euclidean Algorithm, Principal Ideal Domain, Euclidean Domain is a Principal Ideal Domain, UFD, PID implies UFD. (20 lectures)
Field Theory:
Field extensions, basic properties of degree of field extensions. (5 lectures)

Course References:

Text Books:
1. D. S. Dummit and R. M. Foote: Abstract Algebra, 2nd Edition, John-Wiley, 1999.
2. S. Lang: Algebra 3rd Edition, Addison-Wesley, 1999.
Reference Books:
1. J.A. Gallian: Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.
2. M. Artin: Algebra, Prentice Hall inc 1994.
3. I.N. Herstein: Topics in Algebra, John-Wiley, 1995.
4. T. A. Hungerford: Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, 1980.

MA5330 Real Analysis

Course Details:

Unit I - Metric spaces
Review of the real number system, Basic concepts/ definitions and examples, continuous functions, completeness, Baire category theorem, contraction mapping theorem, connectedness, compactness, HeineBorel theorem. (14 lectures)
Unit II - The Riemann-Stieltjes integral
The Riemann-Stieltjes integral and its properties, integrals of continuous and monotone functions, fundamental theorems of calculus, integration by parts, change of variables formula. (13 lectures)
Unit III - Uniform convergence
Sequences and series of functions, Weierstrass M-test, uniform convergence and its relation to continuity, differentiation and integration, Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem. (13 lectures)

Course References:

Text Books:
1. W. Rudin - Principles of mathematical analysis, Mcgraw-Hill 1976.
2. Terence Tao - Analysis I and II, with Chapter 11 from volume I and Chapters 1,2 and 3 from Volume II, trim series 37 and 38, Hindustan Book Agency.
Reference Books:
1. C. C. Pugh, Real Mathematical Analysis, Springer, 2002.
2. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.
3. G. F. Simmons, Topology and Modern Analysis, Kreiger, 2003.

MA5360 Complex Analysis

Course Details:

Unit I: Topology of the Complex plane, limits, continuity, Complex functions – Introduction to the complex exponential, the complex logarithm and trigonometric functions, Linear fractional transformations. (9 Lectures).
Unit II: Differentiability, Cauchy-Riemann equations, Sufficient conditions for differentiability, holomorphic functions, harmonic functions, harmonic conjugates, branches of the logarithm. (9 Lectures)
Unit III: The complex integral, Cauchy's theorem, The Cauchy integral formula, Morera's theorem, Liouville's theorem, Maximum Modulus Principle, Schwarz lemma, Open mapping theorem, Convergence of series of complex numbers, Power series, Radius of convergence, Differentiation of power series, absolute and uniform convergence of a power series, Taylor’s Series, Laurent’s Series, Singularities, Classification of singularities, Cauchy’s Residue theorem, Behaviour of a function near singularities including the CasoratiWeierstrass theorem, evaluation of Real integrals, Argument principle, Rouche’s theorem. (22 Lectures).

Course References:

Text Books:
1. S. Ponnusamy and H. Silverman, Complex Variables with Applications, Birkhauser, Boston, 2006.
2. D. Sarason, Complex Function Theory, Hindustan Book Agency, 2008.
Reference Books:
1. L. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.
2. T.W. Gamelin, Complex Analysis, Springer-Verlag, 2001.
3. J.B. Conway, Functions of one Complex Variable, 2nd edition, Springer-Verlag, 1978.
4. R.E. Greene and S.G. Krantz, Function theory of One Complex variable, AMS, 2006.
5. J.W. Brown and R.V. Churchill, Complex Variables and Applications, McGraw Hill, 2008.

MA5400 Probability Theory

Course Details:

Probability: Axiomatic Definitions, Three Definitions of Probability, Independent Events, Arithmetical Density, Counting and Probability.
Random Variables, Distributions and Expectations, Discrete and Continuous Random Variables, Borel Fields and General Random Variables.
Conditioning and Independence, Sequential Sampling, Mean Variance and Transforms.
Poisson Process, Law of Large Numbers, Central Limit Theorem, Stirling's Formula and De-Moivre's Theorem.
Introduction to Random Walk, Markov Chain and Martingale

Course References:

Text Books:
1. K. L. Chung and F. AitSahlia, Elementary probability theory, fourth edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2003.
Reference Books:
1. Feller, William. An Introduction to Probability Theory and its Applications. Vol. I and II. New York, NY: Wiley, 1968-1971.
2. Jacod, Jean; Protter, Philip. Probability essentials. Second edition. Universitext. Springer-Verlag, Berlin, 2003.
3. Walsh John B. Knowing the Odds: An Introduction to Probability. American Mathematical Soc. 2012.
4. Dudley, R. M. Real Analysis and Probability. Cambridge, UK: Cambridge University Press, 2002.

MA5390 Ordinary Differential Equations

Course Details:

Existence-Uniqueness for systems:
Picard’s theorem, Non-local existence theorem. (6 lectures)
Second Order Equations:
General solution of homogeneous equations, Non-homogeneous equations, Wronskian, Method of variation of parameters, Sturm comparison theorem, Sturm separation theorem, Boundary value problems, Green's functions, Sturm-Liouville problems. (15 lectures)
Series Solution of Second Order Linear Equations:
ordinary points, regular singular points, Legendre polynomials and properties, Bessel functions and properties. (15 lectures)
Systems of Differential Equations:
Algebraic properties of solutions of linear systems, The eigenvalueeigenvector method of finding solutions, Complex eigenvalues, Equal eigenvalues, Fundamental matrix solutions, Matrix exponential. (4 lectures)

Course References:

Text Books:
1. E.A. Coddington, An Introduction to Ordinary Differential Equations, PHI Learning 1999.
2. Tyn Myint U, Ordinary Differential Equations, Elesvier North- Holland, 1978.
Reference Books:
1. M. Braun, Differential Equations and Their Applications, 3rd Ed., Springer-Verlag, 1983.
2. G.F. Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed.,McGrawHill, 1991.
3. P.J. Collins, Differential and Integral Equations, Oxford University Press, 2006.
4. W.E.Boyce and R.C. Di-Prima, Elementary Differential Equations and Boundary Value Problems, John Wiely & Sons, 2001.
5. R.P. Agarwal and D. O'Regan, An Introduction to Ordinary Differential Equations, Springer- Verlag, 2008.

MA5920 Partial Differential Equations

Course Details:

First order partial differential equations: Linear, quasi-linear equations-Method of characteristics, Lagrange method. (6 lectures)
Second order partial differential equations: Classification and Canonical forms of equations in two independent variables, One dimensional wave equation- D'Alembert's solution, Reflection method for halfline, Inhomogeneous wave equation, Fourier Method. (13 lectures)
One dimensional diffusion equation: Maximum Minimum principle for the diffusion equation, Diffusion equation on the whole line, Diffusion on the half-line, inhomogeneous equation on the whole line, Fourier method. (13 lectures)
Laplace equation: Maximum -Minimum principle, Uniqueness of solutions; Solutions of Laplace equation in Cartesian and polar coordinates-Rectangular regions, circular regions, annular regions; Poison integral formula. (8 lectures)

Course References:

Text Books:
1. Ioannis P Stavroulakis and Stepan A Tersian, Partial differential equations- an introduction with mathematica and maple, world - Scientific, Singapore, 1999.
2. I. N. Sneddon, Elements of partial differential equations, McGraw-Hill, New York,1986.
Reference Books:
1. Jeffery Cooper, Introduction to partial differential equations with matlab, Birkhauser, 1998.
2. Clive R Chester, Techniques in partial differential equations, McGraw-Hill, 1971.
3. W. E. Williams, Partial differential equations, Clarendon Press, Oxford, 1980.
4. Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhauser.
5. R.P. Agarwal and D. O'Regan, Ordinary and Partial Differential Equations, Springer- Verlag.

MA5380 Topology

Course Details:

Topological Spaces: open sets, closed sets, neighbourhoods, bases, subbases, limit points, closures, interiors, continuous functions, homeomorphisms.
Examples of topological spaces: subspace topology, product topology,metric topology, order topology, topological groups.
Quotient topology, examples of quotient topology: construction of cylinder, cone, suspension, Moebius band, torus, orbit spaces.
Connectedness and Compactness: Connected spaces, Components, Local connectedness, Compact spaces, Local compactness, Tychnoff Theorem. Separation Axioms, Urysohn lemma, Urysohn Metrization theorem,Tietze Extension theorem, One-point compactification, paracompactness and partition of unity.

Course References:

Text Books:
1. J. Dugundji, Topology, UBS, 1999.
2. M. A. Armstrong, Basic Topology, Springer, 2005.
Reference Books:
1. J. R. Munkres, Topology: A First Course, Prentice Hall, 1975.
2. G. F. Simmons: Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 1963.

MA5350 Discrete Mathematics

Course Details:

Logic: Connectives, quantifiers, validity, satisfiability, consequences, equivalence, logical laws,
deductions, conjunctive and disjunctive normal forms of truth functions.
Set Theory: Relations and functions , cardinality, Cantor-Schroder- Bernste in theorem, finite and
infinitesets, countable and uncountablesets, continuum hypothesis, axiom of choice, well
ordering principle, Zorn's lemma.
Graph Theory: Relations and digraphs, simple graphs , paths and cycles , connectedness , trees ,
Hamiltonian and Eulerian graphs , planar graphs
Combinatorics: Introduction to Combinatorics, Pigeonhole principle, Permutations and Combinations, Generating functions and combinations, Binomial coefficients. Bijective combinatorics.

Course References:

Text books:
1. R. R. Stoll, Set Theory and Logic, Dover Publications Inc., New York, 1979.
2. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer-Verlag, New York, 2008.
3: Miklos Bona, A walk through combinatorics, ‎ World Scientific , 2011.

References:

1. A. Singh, Logics for Computer Science, PHI Learning, New Delhi, 2003.
2. J. R. Munkres, Topology, PHI Learning, New Delhi, 2001.
3. P. R. Halmos, Naive Set Theory, Springer-Verlag, New York, 1974.

S No.	Course No.	Course Name
1	MA5310	Linear Algebra
2	MA5320	Algebra 1
3	MA5330	Real Analysis
4	MA5360	Complex Analysis
5	MA5400	Probability Theory
6	MA5390	Ordinary Differential Equations
7	MA5920	Partial Differential Equations
8	MA5380	Topology
9	MA5350	Discrete Mathematics

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Syllabus for the Test/Interview