QE in Algebra: Syllabus

Field extensions: Algebraic extensions, algebraic closure, normal extensions,

separable extensions, finite fields, inseparable extensions. (Dummit-Foote:
Chapter 13, Lang: Chapter V)

Galois theory: Galois extensions, linear independence of characters, norm,

trace and discriminants, Hilbert theorem 90, cyclic extensions, solvable and
radical extensions, Kummer theory, algebraic independence of
homomorphisms, the normal basis theorem. (Dummit-Foote: Chapter 14,
Lang: Chapter VI)

Ring extensions: Integral extensions, ​integral Galois extensions, prime ideals in

integral ring extensions, decomposition and inertia groups, ramification index
and residue class degree, Frobenius map. (​Dummit-Foote: Chapter 16, ​Lang:
Chapter VII)

Modules over a PID and its applications. (Dummit-Foote: Chapter 12, Lang:
Chapter III.7)

Noetherian modules and rings: Primary decomposition, Nakayama's lemma,

filtered and graded modules, the Hilbert polynomial, Artinian modules and
rings. (Dummit-Foote: Chapter 15, Lang: Chapter X)

Semisimple and simple rings: Semisimple modules, Jacobson density theorem,

semisimple and simple rings, Wedderburn-Artin structure theorems, Jacobson
radical. (Dummit-Foote: Chapter 18, Lang: Chapter XVII)

Representations of finite groups: Basic definitions, characters, class functions,

orthogonality relations, induced representations and induced characters,
Frobenius reciprocity, decomposition of the regular representation.
(Dummit-Foote: Chapter 18, Lang: Chapter XVIII)

References:

1. David S. Dummit, Richard M. Foote: ​Abstract Algebra​, second edition,

John Wiley and sons, Inc., 2005.
2. Nathan Jacobson: ​Basic algebra​, Vol. I-II, second edition, Dover
publications, Inc., 2009.
3. Serge Lang: ​Algebra​, GTM 211, third edition, Springer-Verlag, 2004.
 Syllabus for Qualifying Examination in ANALYSIS

Measure Theory

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.

Functional Analysis

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. Spectral theorem for compact self-adjoint operators.
Hilbert spaces. Orthonormal basis.
Projection theorem and Riesz Representation Theorem for Hilbert spaces.

Complex Analysis

Analytic Functions, Harmonic functions, Cauchy-Goursat Theorem, Conformal mappings.

Taylor and Laurent series, Isolated singularities and residues, Zeroes and poles.
Maximum Modulus Principle, Argument Principle, Rouche’s theorem.
Liouvilles Theorem, Morera’s Theorem.

REFERENCES

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

2. B. V. Limaye, Functional Analysis, 3rd Ed., New Age International Publishers, 2014.

3. J. B. Conway, Functions of one complex variable, 2nd Edition, Narosa, New Delhi, 1978.
 Suggested Chapters/Sections from the reference books

Measure Theory

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

Chapter Sections
3 1, 2, 3, 5, 6
4 1, 2, 3, 4
5 1, 2, 3, 4
6 1, 2, 3, 5
11 1, 2, 3, 4, 5, 6
12 1, 2, 4

Functional Analysis

2. B. V. Limaye, Functional Analysis, 3rd Ed., New Age International Publishers, 2014.

Chapter Sections
II 5, 6, 7, 8
III 9, 10, 11, 12
IV 13, 14
VI 21, 22, 24
VII 28: Lemma 28.4, Theorem 28.5

Complex Analysis

3. J. B. Conway, Functions of one complex variable, 2nd Edition, Narosa, New Delhi, 1978.

Chapter Sections
III 1, 2, 3
IV 1, 2, 3, 4, 5
V 1, 2, 3
VI 1
X 1, 2
Qualifiers in Combinatorics and Theoretical Computer Science

References:
Extremal Combinatorics with Applications in Computer Science - Jukna
Chapters 4,5,6,7,8,10,13,16.

Enumerative Combinatorics - Stanley vols 1 and 2 Chapters 1,2,3,5

Extremal Combinatorics: Pigeonhole principle (Jukna Chap 4), Matchings and

SDRs (Jukna Chap 5), Sunflower Lemmas (Jukna Chap 6), Intersecting Families
(Jukna Chap 7), Chains and antichains (Jukna Chap 8), Density theorems (Jukna
Chap 10), Linear Algebra method (Jukna Chap 13), Polynomial method (Jukna
Chap 16).

Enumerative Combinatorics: Ordinary Generating functions (Stanley v1 - Chaps

1,2)
Exponential Generating Functions (Stanley v2 - Chap 5),
Mobius Inversion on Posets (Stanley v1 - Chap 3),
Trees, Composition of generating functions and Exponential Formula (Stanley v2 -
Chap 5).

References:
Extremal Combinatorics with Applications in Computer Science - Jukna

Enumerative Combinatorics - Stanley vols 1 and 2 (Chapters 1,2,3 and 5)

 PH.D DE QUALIFIERS

1. Proposed syllabus for the PDE-qualifier

1.1. Introduction to PDEs.
(1) First order PDEs ([3, Sections 2.1, 2.2, 2.3, 2.4, 2.5]): Transport equation, quasilinear
equations, Method of characteristics.
(2) Laplace equation ([1, Section 2.2]: Fundamental solution, Mean-value properties, prop-
erties of harmonic functions, Green’s functions, Energy method, Poisson’s equation.
(3) Heat equation ([1, Section 2.3]): Fundamental solution, Mean-value properties, Duhamel’s
principle, energy methods.
(4) Wave equation ([1, Section 2.4]): D’Alembert’s formula, method of spherical means,
Duhamel’s principle, energy methods, causality.
1.2. Linear Elliptic Partial Differential Equations. ([1, Chapter 6] and [2, Chapter 3])
(1) Notion of weak solutions, Lax-Milgram theorem and its applications.
(2) Regularity of solutions to Dirichlet problem on a bounded domain.
(3) Maximum principle, Harnack’s inequality.
(4) Eigenvalue problems.
1.3. Linear Evolution Partial Differential Equations. ([1, Sections 7.1 and 7.2])
(1) Existence of solutions, apriori bounds, regularity, and maximum principle for second
order parabolic equations.
(2) Existence of solutions, regularity, propagation of disturbance for second order hyperbolic
equations.

References
[1] L. C. Evans, Partial differential equations, Volume 19 of Graduate Studies in Mathematics. American Math-
ematical Society, Providence, RI, second edition, 2010.
[2] S. Kesavan, Topics in functional analysis and applications, John Wiley & Sons, Inc., New York, 1989.
[3] Y. Pinchover and J. Rubinstein, An introduction to partial differential equations, Cambridge University
Press, Cambridge, 2005.

1
 Syllabus – Probability Qualifier

Part- I : Probability

Probability Space, Probability mesaures, construction of Lebesque measure, Extension theorem

[Chapater 2]
Random variables and Random vectors, distributions, multi distributions, independence
Expectation of a random variable, Change of variable theorem, Sequence of random variables,
convegence theorems. [Chapter 3]

Convergence almost surely, in probability, in law, convergence in distribution, limit of events,

Borel -Cantelli lemma [Chapter 4, 4.1-4.4]

Moment genertaing functions, Characteristic functions, Uniqueness theorem,

Inversion theorem, continuity theorem [Chapter 6]

Weak law of large numbers, strong law of large numbers, central limit theorem,
[Chpater 5, Chpter 7: 7.1-7.3]

Radon Nikodym theorem, Condition expectation definition, existence and

its properties. [Chpater 9: 9.1]

Reference: K L Chung, A course in probability theory, 3rd edition, Academic Press, San Dieago,
2001.

Part -II : Stochastic Processes

Discrete time Markov chains, Markov property, transition kernels, invariant distributions,
recurence, transients, ergodic behavior of irreducible chains.
[ Chapter 2, 1-7, Chapter 3, 1-5]

Homogeneous and non-homogeneous Poisson processes, [Chapter 4, 1-3]

Martingales, sub and super martingales, stopping times. [Chapter 6, 1-2]

Karlin, S. And Taylor, H.M., A first course in stochastic processes, 2nd Edition,
Academic Press, NewYork, 1975.
Syllabus Statistics Qualifier (Applicable to Students admitted to the PhD
program in the Department of Mathematics, IITB, from Spring 2022
onwards)

Inference:

1) Parametric models, exponential and location-scale family, Sufficiency, Minimal

Sufficiency, Complete Statistic, Decision Rule, Loss Function and Risk, Point
estimators, consistency, asymptotic bias, variance and MSE, asymptotic
inference.[Chapter 2 of Shao 2003]

2) UMVUE, U-statistics, Asymptotic Unbiased estimator, V-statistics [Chapter 3 of

Shao 2003]

3) Bayes Decision and Bayes estimators, Invariance, Minimaxity and admissibility,

MLE and efficient estimation method. [Chapter 4 of Shao 2003]

4) The NP Lemma, monotone likelihood ratio, UMP test for one sided and two sided

hypothesis, UMP Unbiased test, UMP invariant test, likelihood ratio test, chi-squared
test, Sign, permutation and rank test, Kolmogorov- Smirnov and Cramer-von Mises
test and asymptotic test [Chapter 6 of Shao 2003]

Main Text: Mathematical Statistics, Jun Shao, 2nd Ed.,Springer, 2003.

Additional Texts:

1) Theoretical Statistics D.R. Cox, D.V. Hinkley CRC Press, 1974.

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

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

4) Theory of Statistics, Mark J. Schervish, Springer, 1995.

Regression and Statistical Modelling:

1) Full rank model (Chapters 3 and 4 of Searle 1971)

2) One-way classification model (Section 6.2 of Searle 1971)
3) Two-way Crossed Classification model (Chapter 7: Sections 7.1,7.2 of Searle
1971)

Main Text: Linear Models by S.R. Searle (1971) Wiley & Sons

Additional Texts: 1) Linear Model Methodology by A. I. Khuri (2009) CRC Press

Existing Syllabus for Statistics Qualifier: (Applicable to Students admitted to
the PhD program in the Department of Mathematics, IITB, prior to Spring
2022)

Inference:
Chapter 6: Principles of data reduction Chapter 7: Point Estimation
Chapter 8: Hypothesis testing
Chapter 9: Interval estimation
Chapter 10: Asymptotic evaluations

Main Text: Statistical Inference by Casella and Berger

Regression

Chapter 1 Introduction
Chapter 2 Multiple regression
Chapter 3 Tests and confidence regions Chapter 4 Indicator Variable
Chapter 5: The normality Assumption
Chapter 6: Unequal variances
Chapter 7; Correlated errors
Chapter 8: Outliers and influential observations Chapter 10 Multicollinearity
Chapter 11 Variable selection

Main Text: Regression Analysis by Sen and Srivastava

-------------------------------------------------------------------------------
*Topic: Geometry and Topology*

Differentiable manifolds, differentiable functions, tangent spaces, inverse function

theorem, local immersion and local submersion theorems, vector fields, differential
forms, de Rham cohomology, orientation, integration on manifolds, Stokes
theorem. Statement and applications (without proof) of Poincare duality for
differential forms.

References for the above: Chapters 1-4 and first three sections of Chapter5 of Bott
and Tu.

Fundamental groups, Van-Kampen theorem, examples of fundamental groups of

projective spaces, circle, tori, surfaces. Covering spaces, deck transformations,
lifting theorem, universal cover, G- coverings, Galois correspondence for covering
spaces.

References for the above: Chapters 11 to 14 in Fulton

Cones, mapping cylinder, suspensions. Singular homology (homotopy invariance,

excision, long exact sequence of pairs, Mayer-Vietoris sequence), examples of
computing singular homology. Statement and applications (without proof) of
Kunneth formula for singular homology.

References for the above: (a)Section 1 of Chapter 2 in Hatcher (for everything

except Kunneth and Mayer-Vietoris sequence), or (b) Sections 8 to 17 in
Greenberg and Harper, and section 29 for Kunneth formula.

Books:

1. Bredon, Glen E. : Topology and geometry. Corrected third printing of the 1993
original. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997.
xiv+557 pp. ISBN: 0-387-97926-3 55-01 (54-01 57-01)

2. Fulton, William : Algebraic topology. A first course. Graduate Texts in

Mathematics, 153. Springer-Verlag, New York, 1995. xviii+430 pp. ISBN:0-387-
94326-9; 0-387-94327-7 55-01 (30-01 57-01)
3. Vick, James W. : Homology theory. An introduction to algebraic topology.
Second edition. Graduate Texts in Mathematics, 145. Springer-Verlag, New York,
1994. xiv+242 pp. ISBN: 0-387-94126-6 55-01
(57-01)

4. Hatcher, Allen : Algebraic topology. Cambridge University Press, Cambridge,

2002. xii+544 pp. ISBN: 0-521-79160-X; 0-521-79540-0 55-01 (55-00)

5. Bott, Raoul; Tu, Loring W. : Differential forms in algebraic topology. Graduate

Texts in Mathematics, 82. *Springer-Verlag, New York-Berlin,* 1982. xiv+331
pp. ISBN: 0-387-90613-4

6. Greenberg, Marvin J.; Harper, John R. : A first course. Mathematics Lecture

Note Series, 58. *Benjamin/Cummings Publishing Co., Inc., Advanced Book
Program, Reading, Mass.,* 1981. xi+311 pp. (loose errata). ISBN:0-8053-3558-7;
0-8053-3557-9