Table of Contents
 1 Studying at the School Of Mathematics
 2 Before Exam:
 3 During Exam:
 4 Combined Syllabus & Reference books For M.Sc & Ph.D.
 5 Classical Algebra:
 6 Abstract Algebra:
 7 General :
 8 Linear Algebra:
 9 Complex Analysis:
 10 Calculus and Real Analysis:
 11 Topology:
 12 Differential Equations:
 13 Coordinate geometry:
 14 Some Important Links of Our Websites
 15 For any Mathematical Doubts –Click here
Studying at the School Of Mathematics
The Tata Institute is now a Deemed University and runs a Graduate Programme leading to the award of a Ph.D. degree. Bright young students aspiring for a career in research in mathematics are invited to apply for Research Scholarships in the School of Mathematics of the Institute. With its distinguished faculty engaged in highquality research in a broad spectrum of areas, the School provides a stimulating environment for pursuing doctoral studies.
Research Scholars who are admitted to the Ph.D. program are paid a scholarship of Rs. 25000/ per month, increased to Rs. 28000/ after two years, upon successful completion of the graduate school requirements.
Research Scholars who are admitted to the Integrated Ph.D. program, are paid a scholarship of Rs. 16000 per month for the first year, increased to Rs. 25000 after one year, and further increased to Rs. 28000, upon successful completion of the graduate school requirements.
There is, in addition, an annual contingency education grant.All Research Scholars are also eligible for hostel accommodation at nominal charges.
The Research Scholarships are tenable for a period up to five years for Ph.D. students and for a period of up to six years for Integrated Ph.D. students, subject to annual review of performance. An additional year with reduced scholarship (Rs. 14000 per month) may be sanctioned in exceptional cases.
During the first year, introductory courses are given in basic areas of mathematics, and the second year is allocated for preparatory studies for pursuing doctoral work.
Applications for the program are invited through the announcement of the scholarships in all major newspapers. The last date of receiving applications for admission in any academic year is around the middle of October, in the previous year. The admission procedure involves a written test in December, held at various designated centers across the country, followed by interviews of selected candidates, in Mumbai.
Before Exam:
 As you, all know the syllabus is huge so start preparing few months before the exam.
 Don’t get messed up with the topics out of syllabus. Stick strictly to your subject’s syllabus.
 A meticulous study is a correct approach that can help you resolve all your doubts and attain lucid perception of various theories, concepts and their uses regarding the syllabus.
 Try to solve as many as mock test papers & previous year’s papers, especially 12 months before the exam, as it will give you a clear idea about the difficulty level, pattern, & how to allow minimum time to solve a particular type of questions. This will help you know yourself, your capabilities & the field in which you are lagging.
 You should give at least 6 to 7 hours daily to your studies.
 The best way to prepare for the exam is preparing a timetable and then sticking to it throughout the preparation tenure.
 Refer to standard books only, and prepare handwritten notes.
 Don’t take any stress, eat well & try to sleep at a fixed time and for at least 7 to 8 hours a day during your preparation.
During Exam:
 Always remember “Time is the key to success”. Learn to manage your time. Try to solve a question with maximum speed, allotting a very little time to it.
 Don’t spend too much time on a particular question, it’s better to move on and solve other questions.
 Try to attempt section first which you can do very fast.
 Don’t try to attempt the questions which you don’t know or don’t know to solve. Keep negative marking policy always in your mind.
 Don’t forget to take proper sleep & rest on the previous night of the exam.
Combined Syllabus & Reference books For M.Sc & Ph.D.
Entrance Exam in Mathematics
This is useful for those preparing for M.Sc & Ph.D. Entrance in Mathematics in IMSC, TIFR, IISC, ISI, CMI, IIT, IISER, NISER, UOHYD, RKMVU. This is duly compiled to cover all the entrances given above.
Classical Algebra:
De Moivre’s theorem, the relation between roots and coefficient of nth degree equation, the solution to the cubic and biquadratic equation, the transformation of equations. Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations, Binomial theorem. Theory of equations. Inequalities. Elementary set theory. Functions and relations. Elementary number theory: Divisibility, Congruences, Primality.
Reference Books:
1.Combinatorics – Krishnamurthy
2.Higher Algebra – Bernald•Child.
Abstract Algebra:
Groups, homomorphisms, cosets, Lagrange’s Theorem, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields,
algebraic extensions, finite fields.
Reference Books
1.Algebra by Artin
2.Topics in Algebra, I. N. Herstein, J. Wiley
3.Abstract Algebra, D. S. Dummit, and R. M. Foote, J. Wiley
ReviewAbstract Algebra book of D.S. Dummit and Foote
General :
Elementary Combinatorics, Binomial Theorem, Elementary Probability Theory, Logarithms, Progressions.
Reference Books:
An Introduction to ProbabilityFeller
.
Linear Algebra:
Vector Space, subspace and its properties, linear independence and dependence of vectors, matrices, the rank of a matrix, reduction to normal forms, linear homogeneous and nonhomogenous equations, Rank, the inverse of a matrix. systems of linear equations. Linear transformations, eigenvalues, and eigenvectors. CayleyHamilton theorem, symmetric, skewsymmetric and orthogonal matrices.
Reference Books:
1.Linear Algebra – Rao•Bhimasankaram
2.Linear Algebra – Hoffman•Kunze
3.Higher Algebra – S.K.Mapa
Complex Analysis:
Holomorphic functions, CauchyRiemann equations, integration, zeroes of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville’s theorem, poles and singularities, residues and contour integration, conformal maps, Rouche’s theorem, Morera’s theorem.
Reference Books:
Complex AnalysisL.V.Ahlfors
Calculus and Real Analysis:
(a) Real Line: Limits, continuity, di_erentiablity, Reimann integration, sequences, series, limsup, lim inf, pointwise and uniform convergence, uniform continuity, Taylor expansions,
(b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives, total derivative, Jacobian, gradient, line integrals, surface integrals, vector _elds, curl, divergence, Stoke’s theorem
(c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation.
Reference Books:
1.Mathematical Analysis, T. M. Apostol, Narosa.
2.Introduction To Real Analysis – Bartle•Sherbert
3.Principles of Mathematical AnalysisRudin.
Topology:
General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation. Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness
Reference Books
1.Topology of Metric Spaces by Kumaresan
2.Principles of Topology by Fred H. Croom
3.TopologyJames Munkres
Differential Equations:
Ordinary differential equations of the first order of the form y’ = f(x,y). Linear differential equations of the second order with constant coefficients. Linear, homogenous, separable equations, first order higher degree equations, algebraic properties of solutions, linear homogeneous equations with constant coefficients.
Reference Books:
Introduction to Differential Equation – Ghosh & Maity.
Coordinate geometry:
Straight lines, circles, parabolas, ellipses, and hyperbolas. ThreeDimensional Geometry.
Reference Books:
Coordinate Geometry – S.L.Loney
Some Important Links of Our Websites
 What is TIFR
 How to Study TIFR Maths?
 Reference Book For TIFR Maths
 Why IIT JAM Maths
 How To Study Jam Maths
 What’s New with JAM 2018 ?
 Suggested Books For IIt Jam Maths

For any Mathematical Doubts –Click here