MATHEMATICS

I Semester
Lec Tut SS Lab DS AL TC Grading Credits
Course No. Course Title
Hr Hr Hr Hr Hr Hr System (AL/3)
MTH 101 Calculus of One Variable 3 1 4.5 0 0 8.5 4 O to F 3
II Semester
Lec Tut SS Lab DS AL TC Grading Credits
Course No. Course Title
Hr Hr Hr Hr Hr Hr System (AL/3)
MTH 102 Linear Algebra 3 1 4.5 0 0 8.5 4 O to F 3
III Semester
Lec Tut SS Lab DS AL TC Grading Credits
Course No. Course Title
Hr Hr Hr Hr Hr Hr System (AL/3)
Mathematics
MTH201 Multivariable Calculus 3 1 4.5 0 0 8.5 4 O to F 3
DC
MTH203 Introduction to Groups and Symmetry 3 1 4.5 0 0 8.5 4 O to F 3
IV Semester
Course Lec Tut SS Lab DS AL TC Grading Credits
Course Title
No. Hr Hr Hr Hr Hr Hr System (AL/3)
Mathematics
MTH202 Probability and Statistics 3 1 4.5 0 0 8.5 4 O to F 3
DC
MTH204 Complex Variables 3 1 4.5 0 0 8.5 4 O to F 3
DC: Departmental Compulsory Course
V Semester
Course No. Course Title Lec Tut SS Hr Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr System
MTH 301 Group Theory 3 0 7.5 0 0 10.5 3 O to F 4
MTH 303 Real Analysis I 3 0 7.5 0 0 10.5 3 O to F 4
MTH 305 Elementary Number Theory 3 0 7.5 0 0 10.5 3 O to F 4
MTH *** Departmental Elective I 3 0 7.5 0 0 10.5 3 O to F 4
*** *** Open Elective I 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4
Total Credits 15 0 34.5/37.5 0 0 49.5/52.5 15 19/20

VI Semester
Course No. Course Title Lec Tut SS Hr Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr System
MTH 302 Rings and Modules 3 0 7.5 0 0 10.5 3 O to F 4
MTH 304 General Topology 3 0 7.5 0 0 10.5 3 O to F 4
MTH 306 Ordinary Differential Equations 3 0 7.5 0 0 10.5 3 O to F 4
MTH *** Departmental Elective II 3 0 7.5 0 0 10.5 3 O to F 4
*** *** Open Elective II 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4
Total Credits 15 0 34.5/37.5 0 0 49.5/52.5 15 19/20
VII Semester
Course No. Course Title Lec Tut SS Hr Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr System
MTH 401 Fields and Galois Theory 3 0 7.5 0 0 10.5 3 O to F 4
MTH 403 Real Analysis II 3 0 7.5 0 0 10.5 3 O to F 4
MTH 405 Partial Differential Equations 3 0 7.5 0 0 10.5 3 O to F 4
MTH 407 Complex Analysis I 3 0 7.5 0 0 10.5 3 O to F 4
*** *** Open Elective III 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4
Total Credits 15 0 34.5/37.5 0 0 49.5/52.5 15 19/20

VIII Semester
Course No. Course Title Lec Tut SS Hr Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr System
MTH 404 Measure and Integration 3 0 7.5 0 0 10.5 3 O to F 4
MTH 406 Differential Geometry of Curves 3 0 7.5 0 0 10.5 3 O to F 4
and Surfaces
MTH *** Departmental Elective III 3 0 7.5 0 0 10.5 3 O to F 4
*** *** Open Elective IV 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4
*** *** Open Elective V 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4
Total Credits 15 0 31.5/37.5 0 0 46.5/52.5 15 18/20
IX Semester
Course No. Course Title Lec Tut SS Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr Hr System
MTH 501 MS Thesis - - - - - 30 - O to F 12
MTH 503 Functional Analysis 3 0 7.5 0 0 10.5 3 O to F 4
MTH *** Departmental Elective V 3 0 7.5 0 0 10.5 3 O to F 4
HSS 503 Law Relating to Intellectual Property and 1 0 2.5 0 0 3.5 1 S/X 1
Patents
Total Credits 7 0 17.5 0 0 54.5 7 21

X Semester
Course No. Course Title Lec Tut SS Lab DS AL TC Grading Credits
Hr Hr Hr Hr Hr Hr System
MTH 501 MS Thesis - - - - - 30 - O to F 12
MTH *** Departmental Elective VI 3 0 7.5 0 0 10.5 3 O to F 4
MTH *** Departmental Elective VII 3 0 7.5 0 0 10.5 3 O to F 4
Total Credits 6 0 15 0 0 51 6 20
MTH 101: Calculus of One Variable (3)
Learning Objectives
This is a core mathematics course for first-semester BS-MS students. The course
introduces the basic concepts of differential and integral calculus of one real
variable with an emphasis on careful reasoning and understanding of the material.
Course Contents
• Introduction to the real number system: algebraic and order properties,
bounded sets, supremum and infimum, completeness property, integers and
rationals, absolute value and triangle inequality
• Sequences and series: convergence of a sequence, Cauchy's criterion, limit of
a sequence, supremum and infimum, absolute and conditional convergence of
an infinite series, tests of convergence, examples
• Limits and continuity: definitions, continuity and discontinuity of a function at
a point, left and right continuity, examples of continuous and discontinuous
functions, intermediate value theorem, boundedness of a continuous function
on a closed interval, uniform continuity
• Differentiation: definition and basic properties, Rolle's theorem, mean value
theorem, Leibnitz's theorem on successive differentiation, Taylor's theorem
• Integration: Riemann integral viewed as an area, partitions, upper and lower
integrals, basic properties of the Riemann integral, fundamental theorem of
calculus, integration by parts, applications.
Suggested Books
1. G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry, 9th
edition, Indian student edition, Addison-Wesley, 1998
2. G. B. Thomas, M. D. Weir, J. R. Hass, Thomas’ Calculus, 12th Edition,
Pearson, 2014.
3. T. M. Apostol, Calculus, Volumes 1 and 2, 2nd edition, Wiley Eastern,
1980
4. J. Stewart, Calculus: Early Transcendentals, Cengage Learning
MTH 102: Linear Algebra (3)
Learning Objectives
This course focuses on the elementary theory of matrices. Most of the concepts in
this theory find their origins in a systematic study of solutions of a given set of
finitely many linear equations in finitely many unknowns. Soon enough in this
study, one realizes that there is a broader mathematical framework in which all of
these concepts can be suitably defined and studied, with the results obtained
therein having a greater value. We introduce and study the rudiments of this
framework of vector spaces, and linear operators between any two of them. This
course ends with a discussion of the very important spectral theorem for
symmetric matrices.
Course Contents
• Matrices: Review of complex numbers, matrix operations, special matrices
(diagonal, triangular, symmetric, skew-symmetric, orthogonal, hermitian,
skew hermitian, unitary, normal), vectors in Rn and Cn, matrix equation Ax =
b, row-reduced echeln form, row space, column space, and rank of a matrix,
determinants, systems of linear equations.
• The space Rn: Linear independence and dependence, linear span, linear
subspaces.
• Finite-dimensional vector spaces: Bases and dimensions, linear
transformations, matrix of a linear transformation, Rank-nullity theorem.
• Inner product spaces: Orthonormal bases, Gram-Schmidt orthogonalization,
projections.
• Linear operators: Eigenvalues and eigenvectors of a linear operator,
characteristic polynomial, diagonalizability of a linear operator, eigenvalues
of the special matrices stated above, Spectral theorem for real symmetric
matrices, and its application to quadratic forms, positive definite matrices.
Suggested Books
1. H. Anton, Elementary linear algebra and applications, 8th edition, John
Wiley, 1995.
2. D. C. Lay, Linear algebra and its applications, 3rd Edition, Pearson, 2011.
 3. G. Strang, Linear algebra and its applications, 4th edition, Thomson,
2006.
4. S. Lang, Linear Algebra, 3rd Edition, Springer, 1987.
5. S. Kumaresan, Linear algebra - A Geometric Approach, Prentice Hall of
India, 2000.

MTH 201: Multivariable Calculus (3)

Learning Objectives
This course generalizes the various concepts and results pertaining to function of
one real variable to functions of several real variables. The material covered in
this course is used extensively in physical and engineering sciences. It also lays
the foundation for courses such as Real Analysis II and Differential Geometry of
Curves and Surfaces, which form an integral part of the mathematics core
curriculum
Course Contents
• Vector algebra: Vectors in R3, dot product of vectors, length of a vector,
orthogonality of vectors, cross product of vectors.
• Geometry in R3: Lines, planes, and quadric surfaces.
• Vector-valued functions: Continuity and differentiability of vector-valued
functions of real variable, curves in R3, tangent vectors.
• Multivariable functions: Limits and continuity, partial derivatives, gradient,
directional derivatives, maxima, minima and saddle points, Lagrange
multipliers.
• Integration: Double and triple integrals, change of coordinates, vector fields,
line integrals, surface integrals, statements of the Green’s, Divergence, and
Stokes’ theorems, and their applications.
Suggested Books
1. G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry, 9th
edition, Indian student edition, Addison-Wesley, 1998
2. G. B. Thomas, M. D. Weir, J. R. Hass, Thomas’ Calculus, 12th Edition,
Pearson, 2014.
 3. J. Stewart, Calculus, 7th Edition, Cengage Learning, 2012.
4. J. E. Marsden and A. Tromba, Vector Calculus, W.H. Freeman &
Company, 2004

MTH 202: Probability and Statistics (3)

Learning Objectives
This course aims at providing an introduction to the theory of probability and
statistics. No prior knowledge in this subject is required. However, basic
knowledge in Calculus and Linear Algebra would be required. The course will
cover basic probability, probability distributions, random variables, sampling
distributions, and statistical inference (estimation and hypothesis testing).
Course Contents
• Probability: Classical, relative frequency and axiomatic definitions of
probability, addition rule and conditional probability, multiplication rule, total
probability, Bayes Theorem and independence, problems
• Combinatorial analysis: Permutations, Combinations, partitions
• Random Variables: Discrete, continuous and mixed random variables,
probability mass, probability density and cumulative distribution functions,
mathematical expectation, moments, probability and moment generating
function, median and quantiles, Markov inequality, Chebyshev’s inequality,
problems
• Special Distributions, Joint Distributions: Joint, marginal and conditional
distributions, product moments, independence of random variables, bivariate
normal distribution
• Multivariate distributions: Properties, distributions of sums and quotients of
random variables
• Sampling Distributions: The Central Limit Theorem, distributions of the
sample mean and the sample variance for a normal population, Chi-Square, t
and F distributions
• Descriptive Statistics: Graphical representation, Summarization and tabulation
of data
• Estimation: Unbiasedness, consistency, the method of moments and the
method of maximum likelihood estimation, confidence intervals for
parameters in one sample and two sample problems of normal populations,
confidence intervals for proportions
• Testing of Hypotheses: Null and alternative hypotheses, the critical and
acceptance regions, two types of error, power of the test, Neyman-Pearson
Fundamental Lemma, tests for one sample and two sample problems for
normal populations, tests for proportions, Chi-square goodness of fit test and
its applications
Suggested Textbooks
1. P. Hoel, S. Port, C. Stone, Introduction to Probability Theory, 1st Edition,
Brooks Cole, 1972
2. P. Hoel, Introduction to Mathematical Statistics, 5th Edition, Wiley, 1984
3. V. Rohatgi, A. Saleh, Introduction to Probability Theory and
Statistics, 2nd Edition, Wiley, 2000
References
1. Richard Isaac, The Pleasures of Probability, Undergraduate Texts in
Mathematics, Springer, 1995
2. J. Schiller, R. Srinivasan, M. Spiegel, Schaum’s Outline of Probability and
Statistics, 4th Edition, McGraw Hill Education, 2013
3. W.Feller, An Introduction to Probability Theory and Its
rd
Applications, Volume 1, 3 Edition, Wiley, 1968
4. S.M. Ross, A First Course in Probability, 6th Edition, Prentice Hall

MTH 203: Introduction to Groups and Symmetry (3)

Learning Objectives
Symmetries in nature are a source of curiosity for various scientific fields. Often
these are the reasons of stability of various structures and patterns formed in
nature. The study of symmetries is naturally intertwined with the concept of
transformations of the corresponding objects. Group theory arose in nineteenth
century to formalize these ideas. This course is aimed at building these ideas as
we explicitly try to understand the nature of symmetries that occur in each
individual case and compute these in details.
Course Contents
• Examples of symmetries: Symmetries of equilateral triangle and square;
translations, rotations and reflections in the Euclidean plane.
• Definition of a group, subgroup, abelian group, group Z of integers, statement
of division algorithm, description of all subgroups of Z.
• Equivalence relations, group of congruence classes Zn, order of an element in
a group, definition of cyclic group, cyclicity of groups of prime order, group
of units in Zn.
• Definition of a homomorphism and normal subgroup, kernel and image of a
homomorphism, quotient group, isomorphism theorems (statement and
applications).
• Permutations of a finite set, permutation group Sn, cycle notation, length of a
cycle, transpositions, decomposing a permutations as a product of
transpositions, parity of a permutation, alternating group An as normal
subgroup, conjugacy in permutation groups, generating sets of Sn and An.
• Groups of real and complex matrices: general linear groups, determinant of a
matrix as a group homomorphism, special linear groups, complex matrices as
real matrix, orthogonal and special orthogonal groups, unitary and special
unitary groups.
• Two dimensional symmetries: group of symmetries of geometric objects in
Euclidean spaces, dihedral group as the group of symmetries of a regular
polygon, isometries of the Euclidean plane, a detailed account of the
classification of isometries: translations, rotations, reflections, glide
reflections; wallpaper symmetries, finite subgroups of SO(2).
• Three dimensional symmetries: platonic solids and their dual, symmetries of a
tetrahedron, symmetries of a cube and octahedron, symmetries of icosahedron
and dodecahedron, classification of finite subgroups of SO(3).
Suggested Books
1. Online notes: https://neil-strickland.staff.shef.ac.uk/courses/groups/
2. Online Notes Groups and Symmetry, Andrew Baker
http://www.maths.gla.ac.uk/~ajb/dvi-ps/2q-notes.pdf
 3. Groups and Symmetry (Undergraduate Texts in Mathematics), Mark A.
Armstrong, Springer, 1997
4. A First Course in Abstract Algebra (3rd Edition), Joseph J. Rotman,
Pearson, 2005
5. Algebra (2nd Edition), M. Artin, Pearson, 2010
6. Matrix groups for Undergraduates, Kristopher Tapp, AMS, 2005
7. Symmetry: A Mathematical Exploration, Kristopher Tapp, Springer, 2012
8. Online notes: http://www.math.columbia.edu/~bayer/F03/symmetry/

MTH 204: Complex Variables (3)

Learning Objectives
By extending the real number system to include √−1, one obtains the set of
complex numbers, which possesses an algebraic structure similar to real numbers.
This motivates us to study the calculus of functions of a complex variable.
Though a generalization, this calculus has a striking difference from the calculus
of a real variable, which leads to some surprising results. The course intends to
highlight some of these important results. This course is a precursor to MTH 407
Complex Analysis-I and the material covered is also used widely in physical and
engineering sciences.
Course Contents
• Review of complex numbers, functions of one complex variable, limits and
continuity, definition and examples of analytic functions, Cauchy-Riemann
equations, definition of a harmonic function, harmonic conjugates
• Representation of an analytic function as a power series, term by term
differentiation, elementary complex functions and comparison with real
counterparts
• Contour integration, statement of Goursat’s theorem, proof of Cauchy’s
theorem in a disc, Cauchy’s integral formulae
• Zero set of an analytic function, form of an analytic function in a
neighborhood of a zero, definition and examples of removable singularities,
poles, essential singularities respectively, Laurent series expansion of a
complex function
• Residues, residue theorem in a disc, evaluation of real integrals and improper
integrals
Suggested Books
1. R. V. Churchill and J. W. Brown, Complex variables and applications (7th
edition), McGraw-Hill, 2003
2. J. M. Howie, Complex Analysis, Springer-Verlag, 2004
3. E.M. Stein and R. Shakarchi, Complex Analysis, Overseas Press (India)
Pvt. Ltd. 2006
4. Murray R. Spiegel, Theory and Problems of Complex Variables,
Schaum’s Outline Series (McGraw-Hill), 2009

MTH 301: Group Theory (4)

Learning Objectives
This is an introductory course on Group theory. We will begin by studying the
basic concepts of subgroups, homomorphisms and quotient groups with many
examples. We then study group actions, and prove the Class equation and the
Sylow theorems. They are in turn used to prove the structure theorem for finite
abelian groups and to discuss the classification of groups of small order. We then
turn to solvability, prove the Jordan-Holder theorem, and discuss nilpotent groups
(if time permits).
Course Contents
• Definition of group, basic properties, examples (Dihedral, Symmetric, Groups
of Matrices, Quaternion Group, Cyclic, Abelian Groups)
• Homomorphisms, Isomorphisms, subgroups, subgroup generated by a set,
subgroups of cyclic groups
• Review of Equivalence relations, Cosets, Lagrange’s theorem, Normal
subgroup, Quotient Group, Examples, Isomorphism theorems,
Automorphisms
• Group actions, orbits, stabilizer, faithful and transitive actions, centralizer,
normalizer, Cayley’s theorem, Action of the group on cosets
• Conjugation, Class equation, Cauchy’s theorem, Applications to p-groups,
Conjugacy in Sn
• Sylow theorems, Simplicity of An and other applications
• Direct products, Structure of Finite abelian groups
• Semi-Direct products, Classification of groups of small order
• Normal series, Composition series, Solvable groups, Jordan-H¨older theorem,
Insolvability of S5
• Lower and upper central series, Nilpotent groups, Basic commutator
identities, Decomposition theorem of finite nilpotent groups (if time permits)
Suggested Books
1. I. N. Herstein, Topics in Algebra, 2nd Edition, Wiley, 2006
2. T. W. Hungerford, Algebra, Springer Verlag, 2005
3. M. Artin, Algebra, Prentice-Hall of India, 1994
4. D. S. Dummit, R. M. Foote, Abstract Algebra, 2nd Edition, Wiley
5. J. Rotman, A First Course in Abstract Algebra : With Applications,
Prentice Hall
6. J. Rotman, An Introduction to Theory of Groups, Springer GTM, 1999
7. H. Kurzweil, B. Stellmacher, The Theory of Finite Groups, Springer
Universitext, 2004
8. M. Suzuki, Group Theory I, Springer GMW 247

MTH 302: Rings and Modules (4)

Pre-requisites: MTH 301
Learning Objectives
This is an introductory course on Ring theory and Modules. We begin with the
basic definitions and examples of rings, and discuss ideals, quotient rings, and the
Chinese remainder theorem. We then discuss the important classes of
commutative rings, irreducibility in general and specifically in the context of
polynomial rings. We then introduce modules and some basic notions before
discussing generating sets and free modules. We then prove the structure theorem
for finitely generated modules over a PID and its applications. Finally, we discuss
tensor products and some homological algebra (if time permits).
Course Contents
• Definition of rings, Homomorphisms, basic examples (Polynomial ring,
Matrix ring, Group ring), Integral domain, field, Field of fractions of an
integral domain
• Ideals, Prime and Maximal ideals, Quotient Rings, Isomorphism theorems,
Chinese Remainder theorem, Applications
• Principal ideal domains, Irreducible elements, Unique factorization domains,
Euclidean domains, examples
• Polynomial rings, ideals in polynomial rings, Polynomial rings over fields,
Gauss’ Lemma, Polynomial rings over UFDs, Irreducibility criteria
• Definition of modules, submodules, The group of homomorphisms, Quotient
modules, Isomorphism theorems, Direct sums, Generating set, free modules,
Simple modules, vector spaces
• Free modules over a PID, Finitely generated modules over PIDs
• Applications to finitely generated abelian groups and Rational and Jordan
canonical forms
• (if time permits) Tensor product of modules, Exact sequences of modules,
Hom functor, Projective modules, Injective modules, Baer’s criterion
Suggested Books
1. D.S. Dummit, R.M. Foote, Abstract Algebra, 2nd Edition, Wiley
2. G. Birkhoff, S. McLane, Algebra (3rd Edition), AMS
3. S. Lang, Algebra (3rd Edition), Pears
4. C. Musili, Rings and Modules (2nd Edition), Narosa
5. M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra
(1st Indian Edition), Levant Books
6. N. Jacobson, Basic Algebra (Vols - I & II), Hindustan Book Agency
MTH 303: Real Analysis I (4)
Learning Objectives
This is an introductory course on analysis for BS-MS mathematics students. The
aim of this course is to introduce and develop basic analytic concepts of limit,
convergence, integration and differentiation.
Course Contents
• Real numbers: The algebraic and order properties of R, absolute value of real
numbers, the supremum and infimum properties, completeness of R, the
Archimedean property, intervals, open sets, closed sets, compact and
connected sets, Cantor set.
• Functions on R: Limit, Continuity and differentiability of real-valued
functions, monotone functions, uniformly continuous functions, continuity
and compactness, continuity and connectedness, functions of bounded
variation, total variation.
• Sequences and series of functions: Pointwise and uniform convergence of
sequences and series of functions, uniform convergence and its consequences,
space of continuous functions on a closed interval, equicontinuous families,
Arzela-Ascoli theorem, Weierstrass approximation theorem.
• Power series and special functions: Taylor’s theorem, power series, radius of
convergence, exponential, trigonometric and logarithmic functions.
• Riemann-Stieltjes integral: Definition and properties of Riemann-Stieltjes
integral, differentiation of the integral, fundamental theorem of calculus,
integration by parts.
Suggested Books
1. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw Hill,
1953.
th
2. R. G. Bartle and D. R. Sherbert, Introduction to real analysis, 4 edition,
Wiley, 2011.
nd
3. T. M. Apostol, Mathematical Analysis, 2 edition, Narosa Publishing,
1985.
4. R. R. Goldberg, Methods of Real Analysis, Oxford & Ibh, 2012.
MTH 304: General Topology (4)
Pre-requisites: MTH 303Real Analysis I
Learning Objectives
General topology or point-set topology, as it is otherwise known, lies at the
cornerstone of almost all areas in modern mathematics. This course covers all the
basic notions in topology, and prepares the student for advanced courses in
analysis, geometry, and topology.
Course Contents
• Topological spaces: Topology on a set and open sets; Examples of topological
spaces; Coarse and fine topologies; Basis and subbasis for a topology;
Subspace topology; Closed sets and limit points.
• Continuity: Continuous maps between topological spaces; Properties of
continuous maps; Open and closed maps; Homeomorphisms; Topological
embedding; Pasting Lemma.
• Product topology. The product topology on X×Y; The product and box
topologies for arbitrary products; Projection maps; Properties of the product
topology.
• Metrizable spaces: Metric on spaces; Uniform metric and topology;
Metrizability of the product topology; Sequence Lemma; Sequential definition
of continuity; Uniform limit Theorem.
• Quotient topology: Quotient maps; Open and closed maps; Saturated open
sets; Quotient spaces with examples; Properties of quotient spaces.
• Connectedness: Connected and path connected spaces with examples;
Connected and path components; Totally disconnected spaces;Locally
connected spaces; Properties of connected and path connected spaces.
• Compactness: Open covers for spaces; Compact spaces; Tube lemma and
compactness for finite products; Finite intersection property and the
Tychonoff’s theorem; Heine-Borel Theorem; Extreme value theorem;
Lebesque number lemma; Uniform continuity theorem; Limit point
compactness; Sequential compactness; Local compactness; One-point
compactification.
• Countability axioms: First and second countable spaces with examples;
Properties of first and second countable spaces; Dense subsets and
separability; Lindelöf spaces.
• Separation axioms: T1 and Hausdorff spaces with examples; Regular and
Normal spaces with examples; Properties of Hausdorff, regular and normal
spaces; Urysohn's Lemma; Completely regular spaces and their properties;
Urysohn metrization theorem; Tietze's extension theorem.
Suggested Books
1. J. R. Munkres, Topology (2nd Ed), Dorling Kindersley, 2006.
2. C. Adams and R. Franzosa, Introduction to topology, Pearson
3. Prentice Hall, 2008.
4. G. F. Simmons, Introduction to Topology and Modern Analysis,
5. Tata McGraw Hill, 2008.
6. B. Mendelson, Introduction of topology, Dover, 1990.
7. M. A. Armstrong, Basic topology, Springer International, 1983.
8. R. A. Conover, A first course in topology, Dover, 2003.
9. S. Kumaresan, Topology of metric spaces (2nd Ed), Narosa, 2011.

MTH 305: Elementary Number Theory (4)

Pre-requisites (recommended): MTH 101: Calculus of One Variable
Learning Objectives
The aim of this course is to develop a conceptual understanding of the elementary
theory of numbers and to expose the students to writing proper mathematical
proofs.
Course Contents
Foundations: Principle of mathematical induction (with emphasis on writing a
few basic proofs), binomial theorem, countable and uncountable sets, some basic
results on countability, countability of Z, Q and uncountability of R.
Divisibility: Basic properties, division algorithm, GCD, LCM, properties of GCD,
relation between GCD and LCM, Euclidean algorithm for finding GCD,
Pythagorean triples, linear Diophantine equations, fundamental theorem of
arithmetic, Euclid's lemma, existence of infinitely many primes.
Modular arithmetic: Basic properties of congruences, linear congruences, Chinese
remainder theorem, Fermat's little theorem, Wilson's theorem.
Number theoretic functions: Arithmetic functions (tau, sigma and Mobius) and
their properties (specifically multiplicative property of the functions tau, sigma
and the Mobius inversion formula), Euler's phi function and its properties, Euler's
Theorem, Fermat's little theorem as a corollary of Euler's theorem.
Quadratic reciprocity: Primitive roots (order of an integer modulo n, primitive
roots for primes), quadratic congruences, definition of quadratic residue,
Legendre symbol and its properties, quadratic reciprocity law.
Continued fractions: Finite continued fractions, approximation of rational
numbers by finite simple continued fractions, solution of linear Diophantine
equations using finite continued fractions, infinite continued fractions, unique
representation of irrationals as an infinite continued fraction, Pell's equation and
its solutions using continued fractions.
Suggested Books
Textbooks
1. David Burton, Elementary Number Theory, 7th edition, McGraw Hill
Education, 2012.
st
2. John Stillwell, Elements of Number Theory, 1 edition, Springer, 2003.

References
1. James Tattersall, Elementary Number Theory in Nine Chapters, 1st edition,
Cambridge University Press, 1999.
rd
2. Ya. Khinchin, Continued Fractions, 3 edition, Dover, 1997.
nd
3. Thomas Koshy, Elementary Number Theory with Applications, 2
edition, Elsevier, 2007.
MTH 306: Ordinary Differential Equations (4)
Pre-requisites: MTH 303 Real Analysis I
Learning Objectives
This is the first course in the theory of Differential Equations. The aim of the
course is to introduce students to the basic theory and problem-solving methods
for first order and second order Ordinary Differential Equations.
Course Contents
First-Order Linear equations: exact equations, orthogonal trajectories,
homogeneous equations, integrating factors, reduction of order
Second-order linear equations: equations with constant coefficients, method of
undetermined coefficients, variation of parameters, power series solutions, special
functions, applications
Higher-order linear equations
Some basic concepts of Fourier series
Quick review of elementary linear algebra, Picard’s existence and uniqueness
theorem, Sturm comparison theorem
Systems of first-order equations, homogeneous linear systems with constant
coefficients
Non-linear equations: critical points and stability, Liapunov’s direct
method, Poincare-Bendixson theory
Suggested Books
1. George F. Simmons & Steven Krantz, Differential equations, Paperback
edition, Tata-McGraw Hill 2009
2. G. Birkhoff & G. C. Rota, Ordinary differential equations, Paperback
edition, John Wiley &Sons, 1989
3. E. Coddington & N. Levinson, Theory of ordinary differential equations,
Paperback edition, Tata-McGrawa Hill, 2008
4. W. Hurewicz, Lectures on ordinary differential equations, Dover, New
York, 1999
MTH 307/417: Programming and Data Structures (4)
Learning Objectives
The main objective of the course is to introduce students to algorithmic and
logical thinking, and the fundamentals of computer programming. The course
includes some deeper aspects of the theory of computer science like effective data
storage and retrieval techniques, sorting techniques etc. Since the course does not
assume a prior knowledge in computer science, it will prove useful to students of
all disciplines. The course is particularly relevant to students pursuing applied or
computational sciences.
Course Contents
• Programming in a structured language such as C
• Data Structures: definition, operations, implementations and applications of
basic data structures
• Array, stack, queue, dequeue, priority queue, double linked list, orthogonal
list, binary tree and traversal algorithm, threaded binary tree, generalized list
• Binary search, Fibonacci search, binary search tree, height balance tree, heap,
B-tree, digital search tree, hashing techniques
Suggested Textbooks
1. Donald E. Knuth, The art of computer programming (five volumes, 0 - 4),
Addison Wesley
2. V. Aho, J. E. Hopcroft & J. E. Ullman, Data Structures & Algorithm,
Addison Wesley
3. W. Kernighan, D. M. Richie, The C Programming Language, Prentice
Hall

MTH 308/412: Combinatorics and Graph Theory (4)

Learning Objectives
Students will learn the basic combinatorial principles: inclusion-exclusion,
multinomial coefficients and other counting arguments. They will also learn the
combinatorial structures known as graphs and the associated concepts and ideas.
Course Contents
• Combinatorics: Elementary principles of combinatorics (permutations and
combinations), binomial coefficients, inclusion-exclusion principle,
generating functions, recurrence relation, pigeon-hole principle and Ramsey
theory
• Graph theory: definition, isomorphisms, degree sequences, connectivity, trees,
colourings, Eulerian graphs, directed graphs, network flows
Suggested Texts
1. R. A. Brualdi, Introductory Combinatorics (5th Ed.), Prentice Hall
2. F. Harary, Graph Theory, Westview Press
st
3. Bondy, U. S. R. Murty, Graph Theory (1 Ed.), Springer, GTM
4. S. M. Cioaba& M. Ram Murty, A First Course in Graph Theory, TRIM
Series, HBA

MTH 311: Advanced Linear Algebra (4)

Learning Objectives
This course reviews undergraduate linear algebra and proceeds to more advanced
topics. Its purpose is to provide a solid understanding of linear algebra of the sort
needed throughout graduate mathematics.
Course Contents
• Linear transformations and Determinants: The algebra of linear
transformations, Multi-linear functions, The Grassman Ring
• Elementary canonical forms: Characteristic Values, Annihilating Polynomials,
Invariant subspaces, Simultaneous Triangulation and Diagonalization, The
Primary Decomposition Theorem, S-N Decomposition, Canonical forms and
Differential Equations
• The Rational and Jordan Forms: Cyclic subspaces and annihilators, Cyclic
decompositions and the Rational form, The Jordan Form, Computation of
Invariant factors, Semi-simple operators
• Operators on Inner product spaces: Forms on Inner product spaces, Positive
forms, Spectral Theory, Unitary Operators, Normal Operators
• Bilinear forms: Bilinear forms, Symmetric bilinear forms, Skew-symmetric
 bilinear forms
Suggested Reading
1. K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, 1961
2. Serge Lang, Linear Algebra (2nd Edition), Addition-Wesley Publishing,
1971
3. M.W. Hirsch and S. Smale, Differential equations, dynamical systems and
linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press,
1974
nd
4. P. Halmos, Finite dimensional vector spaces (2 Edition), Undergraduate
texts in Mathematics, Springer-Verlag New York Inc., 1987
5. Serge Lang, Algebra, Graduate Texts in Mathematics (3rd Edition),
Springer-Verlag New York Inc., 2005

MTH 401: Fields and Galois Theory (4)

Pre-requisites: MTH 301
Learning Objectives
Field Extensions are studied in an attempt to find a formula for the roots of
polynomial equations, similar to the one that exists for a quadratic equation. The
Galois group is introduced as a way to capture the symmetry between these roots;
and the solvability of the Galois group determines if such a formula exists or not.
In the 19th century, Galois proved that a formula does not exist for a general 5th
degree equation. More importantly, the use of groups to study the symmetry of
other objects is a pervasive theme in Mathematics, and this is traditionally the first
place where one encounters it. The topics to be covered include irreducibility of
polynomials, Field Extensions, Normal and Separable Extensions, Solvable
Groups, and Solvability of polynomial equations by radicals, Finite fields, and
Cyclotomic fields.
Course Contents
• Polynomial rings, Gauss lemma, Irreducibility criteria
• Definition of a field and basic examples, Field extensions
• Algebraic extensions and algebraic closures
• Straight Edge and compass constructions (optional)
• Splitting fields, Separable and Inseparable extensions
• Cyclotomic polynomials, Galois extensions
• Fundamental theorem of Galois theory
• Composite and Simple extensions, Abelian extensions over Q
• Galois groups of polynomials, Solvability of groups, Solvability of
polynomials
• Computations of Galois groups over Q
Suggested Textbooks
1. Ian Stewart, Galois Theory (3rd Edition), Chapman & Hall/CRC Press
(2004)
nd
2. J. Rotman, Galois Theory (2 Edition), Springer (2005)
3. D. J. H. Garling, A Course in Galois Theory, Cambridge University Press
(1986)
4. D. S. Dummit and R. M. Foote, Abstract Algebra (2nd Edition), John
Wiley & Sons (1999)

MTH 403: Real Analysis II (4)

Pre-requisites: MTH 303
Learning Objectives
This course deals with the study of functions of several real variables and the
geometry associated with such functions. There are two parts to this course. The
first part deals with the study of differentiation and integration of such functions.
The second part is devoted to the statement and proof of the higher dimensional
version of the fundamental theorem of calculus, viz, Stoke's theorem (and its
companions). This is one of the standard courses in any mathematics curriculum.
It also serves as a first introduction to differential geometry and topology.
Course Contents
• Vector-valued functions, continuity, linear transformations, differentiation,
total derivative, chain rule
• Determinants, Jacobian, implicit function theorem, inverse function theorem,
rank theorem
• Partition of unity, Derivatives of higher order
• Riemann integration in Rn, differentiation of integrals, change of variables,
Fubini’s theorem
• Exterior algebra, simplices, chains of simplices, Stokes theorem, vector fields,
divergence of a vector field, Divergence theorem, closed and exact forms,
Poincare lemma
Suggested Reading
1. David Widder, Advanced Calculus, second edition, Dover, 1989
2. M. Spivak, Calculus on manifolds, fifth edition, Westview Press, 1971
3. J. Munkres, Analysis on manifolds, Westview Press, 1999.

MTH 404: Measure and Integration (4)

Pre-requisites: MTH 403 Real Analysis II
Learning Objectives
The concept of `measure' generalizes the notion of length, area, and volume and
Riemann integration of continuous functions, which have been studied in previous
courses. In this course we provide the students with a solid background on the
fundamentals of measure and integration theory and prepare them for advance
courses in analysis and related areas.
Course Contents
• Topology of the real line, Borel, Hausdorff and Lebesgue measures on the real
line, regularity properties, Cantor function
• σ-algebras, measure spaces, measurable functions, integrability, Fatou’s
lemma, Lebesgue’s monotone convergence theorem, Lebesgue’s dominated
convergence theorem, Egoroff’s theorem, Lusin’s theorem, the dual space of
C(X) for a compact, Hausdorff space, X
• Comparison with Riemann integral, improper integrals
• Lebesgue’s theorem on differentiation of monotonic functions, functions of
bounded variation, absolute continuity, differentiation of the integral, Vitali’s
covering lemma, fundamental theorem of calculus
• Holder’s, inequality, Minkowski’s inequality, convex functions, Jensen’s
inequality, Lp spaces, Riesz-Fischer theorem, dual of Lp spaces
Suggested Texts
1. W. Rudin, Real and Complex Analysis, third edition. Tata-McGraw Hill, 1987
2. H. Royden, Real Analysis, third edition, Prentice-Hall of India, 2008
3. R. Wheeden, A. Zygmund, Measure and Integral, Taylor and Francis, 1977
4. J. Kelley, T. Srinivasan, Measure and Integral, Volume I, Springer, 1987
5. Rana, An Introduction to Measures and Integration, Narosa Publishing House
6. E. Lieb, M. Loss, Analysis, Narosa Publishing House

MTH 405: Partial Differential Equations (4)

Pre-requisites: MTH 306 Ordinary Differential Equations
Learning Objectives
This is an introductory course in partial differential equations for students
majoring in mathematics. After discussing the solutions of first-order linear and
quasi-linear equations in considerable detail we introduce the Cauchy problem for
first and higher order equations and then briefly discuss the Cauchy-Kovalevski
existence theorem and Holmgren's uniqueness theorem. We follow this by a study
of second-order linear equations; here the goal is to understand the solutions of
the three prototypical equations, Laplace, Wave and the Heat equation, in the
classical set-up.
Course Contents
• First-order equations: linear and quasi-linear equations, general first-order
equation for a function of two variables, Cauchy problem, envelopes
• Higher-order equations: Cauchy problem, characteristic manifolds, real
analytic functions, Cauchy-Kovalevski theorem, Holmgren’s uniqueness
theorem
• Laplace equation: Green’s identity, Fundamental solutions, Poisson’s
equation, Maximum principle, Dirichlet problem, Green’s function, Poisson’s
formula
• Wave equation: spherical means, Hadamard’s method, Duhamel’s principle,
the general Cauchy problem
• Heat equation: initial-value problem, maximum principle, uniqueness,
regularity
Suggested Texts
1. F. John, Partial differential equations, 4th edition, Springer, 1982
2. G. B. Folland, Introduction to Partial differential equations, 2nd edition,
Princeton University Press, 1995
3. J. Rauch, Partial differential equations, Springer, GTM 128, 1991
4. L. Evans, Partial differential equations, American Mathematical Society
GSM series, 1998

MTH 406: Differential Geometry of Curves and Surfaces (4)

Pre-requisites: MTH 306 Ordinary Differential Equations
Learning Objectives
Curves and surfaces are the objects that are generalizations of the real line and the
Euclidean plane respectively. The structures of the local geometry is lifted to
these objects whereas, the global perspective changes. For example the
parameters like curvature and torsion are introduced which are of no relevance in
the real line and Euclidean plane. The course would mainly highlight these
perspectives and we would classify the curves in terms of these parameters.
Beyond these we would try to generalize them while we consider surfaces. Since
the curvatures are non-trivial we would need technical notions of distance and
metric over the surfaces and we will classify them using these notions. At the end,
we will turn to some intrinsic connection of the geometry and topology of
surfaces with these parameters (e.g. Gauss-Bonnet theorem).
Course Contents
• Curves: curves in space, tangent vector, arc length, curvature, torsion, Frenet
formulas
• Surfaces: parametrization, tangent plane, orientability, first fundamental form,
area, orientation, Gauss map, second fundamental form, Gauss curvature,
ruled and minimal surfaces
• Geodesics, isometries of surfaces, Gauss’ Theorema Egregium, Codazzi-
Mainardi equations
• Gauss-Bonnet theorem for compact surfaces
Suggested Textbooks
1. Pressley, Elementary Differential Geometry, Springer, Indian reprint, 2004
2. Manfredo do Carmo, Differential Geometry of Curves and Surfaces,
Prentice Hall, 1976
3. D. J. Struik, Lectures on Differential Geometry, Dover, 1988
4. Barrett O’Neill, Elementary Differential Geometry, Second edition,
Academic Press (Elsevier), 2006

MTH 407/607: Complex Analysis I (4)

Pre-requisites: MTH 303 Real Analysis I
Learning Objectives
The learning objective of this course include the definition of analyticity, the
Cauchy-Riemann equations and the concept of differentiability. Also to be learnt
are the theorems on entire functions, residue theorem and applications and finally
conformal mapping.
Course Contents
• Complex numbers: powers and roots, geometric representation, stereographic
projection
• Complex differentiability: limits, continuity and differentiability, Cauchy
Riemann equations, definition of a holomorphic function
• Elementary functions: sequences and series, complex exponential,
trigonometric, and hyperbolic functions, the logarithm function, complex
powers, Mobius transformations
• Complex integration: contour integrals, Cauchy's integral theorem in a disc,
Cauchy’s Integral Formula, Liouville’s theorem, Fundamental Theorem of
Algebra, Morera’s theorem, Schwarz reflection principle
• Series representation of analytic functions: Taylor series, power series, zeros
and singularities, Laurent decomposition, open mapping theorem, Maximum
Principle
• Residue theory: residue formula, calculation of certain improper integrals,
Riemann’s theorem on removable singularities, Casorati Weierstrass theorem,
the argument principle and Rouche's theorem
• Conformal mappings: conformal maps, Schwarz lemma and automorphisms
of the disk and the upper half plane
Suggested Books
Texts
1. Elias M. Stein, Rami Shakarchi, Complex Analysis (Princeton Lectures in
Analysis), Princeton University Press, 2003
2. Theodore W. Gamelin, Complex Analysis, Springer Verlag, 2001
3. John B. Conway, Functions of one Complex Variable I, Springer, 1978
4. E. Freitag and R.Busam, Complex Analysis, Springer, 2005
References
1. Lars Ahlfors, Complex Analysis. McGraw Hill, 1979
2. R. Remmert, Theory of Complex Functions. Springer Verlag, 1991
3. C. Caratheodory, Theory of Functions of a complex variable, AMS
Chelsea, 2001

MTH 408/522: Numerical Analysis (4)

Pre-requisites: MTH 303 Real Analysis I
Course Contents
• Round off errors and computer arithmetic
• Interpolation: Lagrange interpolation, divided differences, Hermite
interpolation, splines, numerical differentiation, Richardson extrapolation
• Numerical Integration: trapezoidal, Simpsons, Newton-Cotes, Gauss
quadrature, Romberg integration, multiple integrals
• Solution of linear algebraic equations: direct methods, Gauss elimination,
pivoting, matrix factorizations
• Iterative methods: matrix norms, Jacobi and Gauss-Siedel methods, relaxation
methods
• Computation of eigenvalues and eigenvectors: power method, householders
method, QR algorithm
• Numerical solutions of non-linear algebraic equations: bisection, secan and
Newton's, zeroes of polynomials
Suggested Textbooks
1. R. L. Burden, D. J. Faires, Numerical Analysis
2. E. K. Blum, Numerical Analysis and Computation, Theory and Practice,
Dover, 2010
3. S. D. Conte, C. De Boor, Elementary Numerical Analysis, third edition,
McGraw-Hill, 1980
4. D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics,
volumes 1 and 2, Dover, 1988

MTH 409: Optimization Techniques (4)

Pre-requisites: MTH 303 Real Analysis I
Learning Objectives
The optimization algorithms deal with optimizing several real-valued functions
with some constraints. We will study linear and non-linear programming
techniques. The linear programming deals with optimizing linear functions with
linear constraints using hyperplanes; whereas the non-linear programming deals
with optimizing functions with constraints, possibly non-linear. A lot of real-
world problems occurring in science and industry, which involves optimization,
will be discussed in this course.
Course Contents
• Maxima and minima, Lagrange multipliers method, formulation of
optimization problems, linear programming, non-linear programming, integer
programming problems
• Convex sets, separating hyperplanes theorem, simplex method, two phase
simplex method, duality theorem, zero-sum two-person games, branch and
bound method of integer linear programming
• Dynamic programming, Bellman’s principle of optimality
Suggested Books:
1. Katta G. Murty, Linear Programming, Revised edition, Wiley, 1983
 2. Griva, S. Nash, A. Sofer, Linear and Non-linear Optimization, second
edition, SIAM, 2008
3. M. Bazaraa, H. Sherali, C. Shetty, Non-linear Programming: Theory and
Algorithms, third edition, Wiley Inter-Science, 2006

MTH 410/514/620: Representation Theory (4)

Pre-requisites: MTH 301, MTH 302
Learning Objectives
The aim of the course is to introduce representations of finite groups, their
character theory, and some basic examples. Representation theory is used to study
groups in various settings, including Physics and Chemistry.
Course Contents
• Representations of groups, subrepresentations, Irreducible representations,
tensor product of representations, Maschke's theorem, Wedderburn
decomposition
• Characters of representations, Generalized characters, Schurs lemma,
Orthogonality, Regular representations, Decomposition theorems
• Representations of direct product of finite groups, Induced representations,
Reciprocity theorem
• Representations and characters of standard finite and infinite groups: cyclic
groups, dihedral groups, symmetric and alternating groups of small order etc.
• Applications of Representation Theory
Suggesting Books
1. J. P. Serre, Linear Representations of Finite groups (Graduate Texts in
Mathematics), 2nd Edition, Springer-Verlag New York Inc., 1977
2. W. Fulton and J. Harris, Representation Theory, A First Course, 2nd
Indian reprint, Springer India, 2007
3. G. James, M. Liebeck, Representations and Characters of Groups,
Cambridge University Press, 2001
4. M. Suzuki, Group Theory II, Springer-Verlag, 1983
MTH 411/511: Introduction to Lie Groups and Lie Algebras (4)
Pre-requisites: MTH 311: Advanced Linear Algebra, MTH 301: Group Theory
Learning Objectives
The proposed course aims at providing and introduction to Lie groups, Lie
algebras and their representations. The first part of the course focuses on matrix
Lie groups (closed subgroups of GL(n; C)) and Lie algebras. The second part of
the course deals with representations of semisimple Lie groups and Lie algebras.
We begin with SU(2) and SU(3), as these cases very well illustrate the ideas of
Cartan subalgebras, the roots, weights and the Wey1 group. We also look at
Semisimple Lie groups and Lie algebras in general towards the end.
Course Contents
• Matrix Lie Groups: Definition and examples; Lie group homomorphisms and
isomorphisms, Lie subgroups, polar decomposition.
• Lie algebras: matrix exponential and matrix logarithm, the Lie algebra of a
matrix Lie group, Lie subalgebras, complexification of a real Lie algebra,
Baker-Campbell-Hausdorff formula.
• Representation Theory: standard and adjoint representations, unitary
representations, irreducible representations of su(2), direct sum ad tensor
product of representations, dual representations, Schur’s Lemma.
• Semisimple Theory: Representations of SU(3), weights and roots, the WeyI
group. Semisimple Lie algebras, complete reducibility, Cartan subalgebras,
root systems.
Suggested Books:
1. Hall, Brian Lie Groups, Lie Algebras, and Representations. Graduate
Texts in Mathematics, Vol. 222, Springer Verlag, 2003.
2. Rossmann, Wulf. Lie Groups: An Introduction through Linear Groups.
Oxford Graduate Texts in Mathematics 5, Oxford University Press, 2002.
3. Humphreys, James E. Introduction to Lie Alogebras and Representation
Theory. Graduate Texts in Mathematics, Vol. 9, Springer, 1973.
4. Baker, Andrew. Matrix Groups: An Introduction to Lie Group Theory.
Springer Veriag, 2002.
MTH 503/614: Functional Analysis (4)
Pre-requisites: MTH 304 General Topology, MTH 404 Measure and Integration
Learning Objectives
Functional analysis is the branch of mathematics concerned with the study of
spaces of functions. This course is intended to introduce the student to the basic
concepts and theorems of functional analysis with special emphasis on Hilbert
and Banach Space Theory. This gives the basics for more advanced studies in
modern Functional Analysis, in particular in Operator Algebra Theory and
Banach Space Theory.
Course Contents
• Normed Linear spaces, Bounded Linear Operators, Banach Spaces, Finite
dimensional spaces, Quotient Spaces
• Hilbert spaces, Riesz Representation Theorem, Orthonormal sets, Bessel's
Inequality, Parseval's Identity, Fourier Series
• Dual Spaces, Dual of Lp spaces , Hahn-Banach Extension Theorem,
Applications
• Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness
Principle
• Weak and Weak-* topologies, Hahn-Banach Separation Theorem,
Alaoglu's Theorem, Reflexivity
• Compact Operators, Adjoint of an operator, Spectral theorem for Compact
Self-Adjoint operators
• (If time permits) Banach Algebras, Ideals and Quotients, Gelfand-Mazur
Theorem, Fredholm Alternative, Fredholm Operators, Atkinson's theorem
Suggested Books:
1. G.F. Simmons, Introduction to Topology and Modern Analysis, Tata
McGraw Hill, 2008
2. W. Rudin, Functional Analysis, McGraw Hill Book Company, 1973
3. J.B. Conway, A course in Functional analysis, GTM 96, Springer, 1990
4. F. Hirsch and G. Lacombe, Elements of Functional Analysis, GTM 192,
Springer
 5. S. Kesavan, Functional Analysis, TRIM 52, Hindustan Book Agency
6. Martin Schechter, Principles of Functional Analysis, Graduate Studies in
Mathematics, American Math. Soc. 2nd Ed
7. B.V. Limaye, Functional Analysis, New Age books, 2nd Ed

MTH 504/604: Complex Analysis II (4)

Pre-requisites: MTH 407: Complex Analysis I
Learning Objectives
The aim of this course is to introduce to some advanced topics of contemporary
complex analysis. The course is intended for the students who have done a first
course in complex analysis. The course will solidify the understandings of
complex analysis and will prepare the students to use the concepts learned in this
course to other areas of mathematics as well as in applied areas.
Course Contents
• Harmonic functions: mean value property, Schwarz reflection principle, the
Poisson kernel, Dirichlet problem
• Maximum modulus principle, Hadamard three-circle theorem, Phragmen-
Lindelof theorem, Rado’s theorem
• Approximations by rational functions: Runge’s theorem, Mittag-Leffler
theorem, simply connected regions
• Space of analytic functions, Hurwitz theorem, Montel’s theorem, space of
meromorphic functions, proof of Riemann mapping theorem, analytic
continuation along curves, statement of monodromy theorem
• Entire functions: infinite products, Weierstrass factorization theorem, gamma
and zeta functions, little and big Picard theorems
Suggested Books
Texts
1. Conway J.B., Functions of One Complex Variable, Springer-Verlag NY,
1978
2. Rudin W., Real and Complex Analysis, McGraw-Hill, 2006
3. Lang, S., Complex Analysis, Springer, 2003.
 4. Epstein B. and Hahn L-S., Classical Complex Analysis, Jones and Bartlett,
2011
References:
1. Carathodory C., Theory of functions of a complex variable, Vol. I and II,
Chelsea Pub Co, NY 1954
2. Remmert R., Classical topics in complex function theory, Springer 1997
3. Ahlfors L., Complex Analysis, Lars Ahlfors, McGraw-Hill, 1979.

MTH 505/623: Introduction to Ergodic Theory (4)

Pre-requisites: MTH 304 General Topology, MTH 404 Measure and Integration
Learning Objectives
The word 'ergodic' is an amalgamation of the two Greek words 'ergon' (work) and
'odos' (path). Ergodic theory deals with the study of qualitative properties of flow
(dynamical system) induced by actions of groups on spaces (measure spaces,
topological spaces or manifolds). This is an introductory course where several
fundamental examples of dynamical systems will be discussed. One of the main
theorems, the Birkhoff's Ergodic theorem, which relates the time average of the
flow with the space average, will be proved. This course will introduce necessary
techniques and tools to understand a dynamical system. A good understanding of
a first course in topology and measure theory is essential.
Course Contents
• Discrete Dynamical systems: definition and examples - maps on the circle, the
doubling map, shifts of finite type, toralautomorphisms.
• Topological and Symbolic dynamics: transitivity, minimality, topological
conjugacy and discrete spectrum, topological mixing, topological entropy,
topological dynamical properties of shift spaces, circle maps and rotation
number.
• Ergodic Theory: invariant measures and measure-preserving transformations,
ergodicity, recurrence and ergodic theorems (Poincare recurrence, Kac's
lemma, Von Neumann's ergodic theorem, Birkhoff's ergodic theorem),
applications of the ergodic theorem (continued fractions, Borel normal
numbers, Khintchine’s recurrence theorem), ergodic measures for continuous
 transformations and their existence, Weyl’s equidistribution theorem, mixing
and spectral properties.
• Information and entropy - topological, measure-theoretic, and their
relationship. Skew products, factors and natural extensions, induced
transformations, suspensions and towers. Topological pressure and the
variational principle, thermodynamic formalism and transfer operators,
applications of thermodynamic formalism: (i) Bowen's formula for Hausdorff
dimension, (ii) central limit theorems.
Suggested Books:
1. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New
York, 1982
2. M.G. Nadkarni, Basic Ergodic Theory, Second Edition, Hindustan Book
Agency, India
3. M. Brin and G. Stuck, Introduction to Dynamical Systems, CUP, 2002
4. M. Pollicott and M. Yuri, Dynamical systems and Ergodic theory, CUP,
1998
5. P. R. Halmos, Lectures on Ergodic Theory, Chelsea, New York, 1956
6. W. Parry, B. Bollobas, W. Fulton, Topics in Ergodic Theory, CUP, 2004
7. A.B. Katok and B. Hasselblatt, Introduction to the Modern Theory of
Dynamical Systems, Cambridge, 1995

MTH 506/610: Fourier Analysis on the Real Line (4)

Pre-requisites: MTH 404 Measure and Integration, MTH 503 Functional
Analysis: Normed linear spaces, completeness, Uniform boundedness principle,
MTH 405 Partial Differential Equations: Basic knowledge of Laplacian, Heat
and Wave equations
Learning Objectives
This is an introductory course on Fourier Analysis on the real line. Fourier
transform is a very useful tool to study various physical problems. In this course,
we intend to rigorously study the key concepts of the subject. We will also study
some applications of Fourier series (or Fourier integral) to partial differential
equations and some number-theoretic problems.
Course Contents
1. The vibrating string, derivation and solution to the wave equation, The
heat equation
2. Definition of Fourier series and Fourier coefficients, Uniqueness,
Convolutions, good kernels, Cesaro/Abel means, Poisson Kernel and
Dirichlet’s problem in the unit disc
3. Mean-square convergence of Fourier Series, Riemann-Lebesgue Lemma,
A continuous function with diverging Fourier Series
4. Applications of Fourier Series: The isoperimetric inequality, Weyl’s
equidistribution Theorem, A continuous nowhere-differentiable function,
The heat equation on the circle
5. Schwartz space*, Distributions*, The Fourier transform on R: Elementary
theory and definition, Fourier inversion, Plancherel formula, Poisson
summation formula, Paley-Weiner Theorem*, Heisenberg Uncertainty
principle, Heat kernels, Poisson Kernels
6. (If time permits/possible project topic) Definition of Fourier transform on
Rd, Definition of X-ray transform in R2 and Radon transform in R3,
Connection with Fourier Transform, Uniqueness
Suggested Books
Texts
1. E.M.Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton
Univ Press, 2003
2. (For topics marked with a *) W. Rudin, Functional Analysis, 2nd Ed, Tata
McGraw-Hill, 2006
References
1. J. Douandikoetxea, Fourier Analysis (Graduate Studies in Mathematics),
AMS, 2000
2. L. Grafakos, Classical Fourier Analysis (Graduate Texts in Mathematics),
2nd Ed, Springer, 2008
MTH 507: Introduction to Algebraic Topology (4)
Pre-requisites: MTH 301 Group Theory, MTH 304 General Topology
Learning Objectives
This is a first course in algebraic topology. The subject revolves around finding
and computing invariants associated with topological spaces. The first such
invariant is the fundamental group of a pointed topological space which we'll
study in detail along with the classification of covering spaces using fundamental
group actions.
Course Contents
• The Fundamental Group: Homotopy, Fundamental Group, Introduction to
Covering Spaces, The Fundamental Group of the circle S1, Retractions and
fixed points, Application to the Fundamental Theorem of Algebra, The
Borsuk-Ulam Theorem, Homotopy Equivalence and Deformation Retractions,
Fundamental group of a product of spaces, and Fundamental group the torus
T2=S1×S1, n-sphere Sn, and the real projective n-space RPn.
• Van Kampen’s Theorem: Free Products of Groups, The Van Kampen
Theorem, Fundamental Group of a Wedge of Circles, Definition and
construction of Cell Complexes, Application to Van Kampen Theorem to Cell
Complexes, Statement of the Classification Theorem for Surfaces, and
Fundamental groups of the closed orientable and non-orientable surfaces of
genus g.
• Covering Spaces: Universal Cover and its existence, Unique Lifting Property,
Galois Correspondence of covering spaces and their Fundamental Groups,
Representing Covering Spaces by Permutations – Deck Transformations,
Group Actions, Covering Space Actions, Normal or Regular Covering Spaces,
and Application of Covering Spaces to Cayley Complexes.
Suggested Books
1. J. R. Munkres, Topology (2nd Edition), Pearson Publishing Inc, 2000
2. Hatcher, Algebraic Topology, Cambridge University Press, 2002
3. M. A. Amstrong, Basic Topology, Springer International Edition, 2004
4. W. S. Massey, Algebraic Topology: An Introduction , Springer, 1977
5. J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988
 6. M. J. Greenberg and J. R. Harper, Algebraic Topology: A First Course,
Perseus Books Publishing, 1981
7. E. H. Spanier, Algebraic Topology, Springer, 1994

MTH 508/608: Introduction to Differentiable Manifolds and Lie Groups (4)

Pre-requisites: MTH 303 Real Analysis I, MTH 304 General Topology, MTH 306
Ordinary Differential Equations, MTH 403 Real Analysis II
Learning Objectives
This course aims to extend calculus from domains of Euclidean space to more
general objects called differentiable manifolds, which are of fundamental
importance in the study of higher geometry. These are higher dimensional
analogues of curves and surfaces in space with which the students are familiar.
This is an important course for students who want to pursue research in
differential geometry, topology and related areas.
Course Contents
• Differentiable manifolds: definition and examples, differentiable functions,
existence of partitions of unity, tangent vectors and tangent space at a point,
tangent bundle, differential of a smooth map, inverse function theorem,
implicit function theorem, immersions, submanifolds, submersions, Sard’s
theorem, Whitney embedding theorem
• Vector fields: vector fields, statement of the existence theorem for ordinary
differential equations, one parameter and local one-parameter groups acting
on a manifold, the Lie derivative and the Lie algebra of vector fields,
distributions and the Frobenius theorem
• Lie groups: definition and examples, action of a Lie group on a manifold,
definition of Lie algebra, the exponential map, Lie subgroups and closed
subgroups, homogeneous manifolds: definition and examples
• Tensor fields and differential forms: cotangent vectors and the
cotangent space at a point, cotangent bundle, covector fields or 1-forms on a
manifold, tensors on a vector space, tensor product, symmetric and alternating
tensors, the exterior algebra, tensor fields and differential forms on a
manifold, the exterior algebra on a manifold
• Integration: orientation of a manifold, a quick review of Riemann integration
in Euclidean spaces, differentiable simplex in a manifold, singular chains,
integration of forms over singular chains in a manifold, manifolds with
boundary, integration of n-forms over regular domains in an oriented manifold
of dimension n, Stokes theorem, definition of de Rhamcohomology of a
manifold, statement of de Rham theorem, Poincare lemma
Suggested Books
Texts
1. J. Lee, Introduction to smooth manifolds, Springer, 2002
2. W. Boothby, An Introduction to differentiable manifolds and Riemannian
geometry, Academic Press, 2002
3. F. Warner, Foundations of differentiable manifolds and Lie groups,
Springer, GTM 94, 1983
4. M. Spivak, A comprehensive introduction todifferential geometry, Vol. 1,
Publish or Perish, 1999
References
1. G. de Rham, Differentiable manifolds: forms, currents and harmonic
forms, Springer, 1984
2. V. Guillemin and A. Pollack., Differential topology, AMS Chelsea, 2010
3. J. Milnor, Topology from the differentiable viewpoint, Princeton
University Press, 1997
4. J. Munkres, Analysis on manifolds, Westview Press, 1997
5. C. Chevalley, Theory of Lie groups, Princeton University Press, 1999
6. R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and
applications, Springer, 1988

MTH 509/609: Sturm-Liouville Theory (4)

Pre-requisites: MTH 306 Ordinary Differential Equations, MTH 404 Measure
and Integration
Course Contents
• Fourier Series: Fourier series of a periodic function, question of point-wise
convergence of such a series, behavior of the Fourier series under the
 operation of differentiation and integration , sufficient conditions for uniform
and absolute convergence of a Fourier series, Fourier series on intervals,
examples of boundary value problems for the one dimensional heat and wave
equations illustrating the use of Fourier series in solving them by separating
variables, a brief discussion on Cesarosummability and Gibbs phenomenon
• Orthogonal Expansions: A quick review of L2 spaces on an interval,
convergence, completeness, orthonormal systems, Bessel’s inequality,
Parseval’s identity, dominated convergence theorem
• Sturm-Liouville Systems: linear differential operators, formal adjoint of a
linear operator, Lagrange’s identity, self-adjoint operators, regular and
singular Sturm-Liouville systems, Sturm-Liouville series, Prufer substitution,
Sturm comparison and oscillation theorems, eigenfunctions, Liouville normal
form, distribution of eigenvalues, normalized eigenfunctions, Green’s
functions, completeness of eigenfunctions
• Illustrative boundary value problems: A technique to solve inhomogeneous
equations using Sturm-Liouville expansions, one dimensional heat and wave
equations with inhomogeneous boundary conditions, one dimensional
inhomogeneous heat and wave equations, mixed boundary conditions,
Dirichlet problem in a rectangle and a polar coordinate rectangle
• Maximum Principle and applications: maximum principle for linear, second-
order, ordinary differential equations, generalized maximum principle for such
equations, applications to initial and boundary value problems, the eigenvalue
problem, an extension of the principle to non-linear equations
• Orthogonal polynomials and their properties: Legendre polynomials,
Legendre equation, Legendre functions and spherical harmonics, Hermite
polynomials, Hermite functions, Hermite equation, Laguerre polynomials,
Laguerre equation, zeros of orthogonal polynomials on an interval, and a
recurrence relation satisfied by them
• Bessel Functions: Bessel’s equation, identities, asymptotics and zeros of
Bessel functions
Suggested Books:
Texts:
1. Birkhoff, G &Rota G., Ordinary Differential Equations, John Wiley &
Sons
2. Folland, G., Fourier Analysis & Its Applications, AMS
3. Protter, M. & Weinberger, H., Maximum Principles in Differential
Equations, Springer
References:
1. Brown, J. & Churchill, R., Fourier Series and Boundary Value Problems,
McGraw-Hill
2. Jackson, D., Fourier Series and Orthogonal Polynomials, Dover

MTH 510/615: Operator Theory and Operator Algebras (4)

Pre-requisites: MTH 503 Functional Analysis
Learning Objectives
The goal of this course is to prove a far-reaching generalization of the spectral
theorem for self-adjoint matrices. This is the classification of normal operators on
a Hilbert space by their spectrum and multiplicity. Operator algebras provide the
natural framework for the proof, and are a subject of vigorous research in modern
functional analysis.
Course Contents
• Banach Algebras, Ideals, Quotients, homomorphisms, Unitization
• Invertible Elements, Spectrum, Gelfand-Mazur Theorem, Spectral Radius
Formula
• Commutative Banach Algebras, The Gelfand Transform, Applications to
Fourier Transforms, Weiner's Theorem, Stone-Weierstrass Theorem
• Compact and Fredholm Operators, Atkinson's Theorem, Index Theory
• C* algebras, uniqueness of the norm, Commutative C* algebras, Gelfand-
Naimark theorem, Spectral Mapping theorem
• Functional Calculus, Positive Operators, Polar Decomposition
• Weak and Strong Operator Topologies, Von Neumann Algebras, Double
Commutant Theorem
• Spectral measure, Spectral Theorem for Normal Operators, Borel Functional
Calculus
• Multiplicity Theory, Abelian Von Neumann Algebras, Classification of
normal operators upto unitary equivalence
Suggesting Books
1. G. J. Murphy, C* Algebras and Operator Theory (Academic Press Inc,
1990)
2. J. B. Conway, A Course in Functional Analysis (2nd Ed) (Springer, 1990)
3. R. G. Douglas, Banach Algebra Techniques in Operator Theory (2nd Ed)
(Springer, 1998)
4. K. R. Davidson, C* Algebras by Example (Fields Institute Monograph,
AMS 1996)
5. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of
Operator Algebras - Vol. I (Academic Press Inc, 1983)
6. W. A. Arveson, A Short Course in Spectral Theory (Springer 2002)

MTH 512/612: Non-commutative Algebra (4)

Pre-requisites: MTH 301, MTH 302, MTH 401
Course Contents
• Matrix Rings and PLIDs, Tensor Products of Matrix Algebras, Ring
constructions using Regular Representation
• Basic notions for Noncommutative Rings, Structure of
Hom(M,N), Semisimple Modules & Rings, the Wedderburn Structure
Theorem, Simple Rings, Rings with Involution
• The Jacobson Radical and its properties, Primitive Rings and
Ideals, Hopkins-Levitzki Theorem, Nakayama’s Lemma , Radical of a
Module, Local Rings, Chevalley-Jacobson Theorem, Kolchin’s Theorem,
Clifford Algebras.
• Prime and Semiprime rings, Rings of Fractions and Goldie’s Theorems, Rings
with ACC (ideals), Tensor Algebras, Algebras over large Fields,
Deformations and Quantum Algebras.
• Hereditary Rings and their Modules, Division rings.
• Central Simple Algebras, Cyclic Algebras, Symbol Algebras, Crossed
Products, the Brauer Group, the functor Br, the Skolem-Noether Theorem, the
centralizer Theorem, calculation of Brauer group of commutative rings.
Suggesting Books:
1. L. Rowen, Graduate algebra: noncommutative view, Graduate Studies in
Mathematics, 91.
2. B. Farb, R. Dennis, Noncommutative algebra, GTM, Springer-Verlag.
3. T. Y. Lam, A first course in noncommutative rings, GTM, Springer.
4. J. Golan and T. Head, Modules and the structure of rings: A primer, Pure
and applied mathematics

MTH 513/613: Introduction to Riemannian Geometry (4)

Pre-requisites: MTH 405 and MTH 508
Learning Objectives
A Riemannian manifold is a smooth manifold equipped with additional geometric
structure called a Riemannian metric and this structure provides a framework to
measure geometric quantities such as length and angles on the manifold.
Associated with a Riemannian metric are the fundamental concepts of a
Riemannian connection, geodesics and curvature. First, the basic properties and
results associated to these are studied. The course then explores the relationship
between geodesics and curvature. After studying such questions (which are local
in nature), the focus turns to global questions and the course culminates in a study
of certain important results concerning how curvature affects the topology of the
manifold.
Course Contents
• Review of differentiable manifolds: vector bundles, tensors, vector fields,
differential forms, Lie groups
• Riemannian metrics: Definition, examples, existence theorem; model spaces
of Riemannian geometry
• Connections: connections on a vector bundle, linear connections, covariant
derivative, parallel transport, geodesics
• Riemannian connections and geodesics: torsion tensor, Fundamental Theorem
of Riemannian Geometry, geodesics of the model spaces, exponential
map, convex neighborhoods, Riemannian distance function, first variation
formula, Gauss' lemma, geodesics as locally minimizing curves;
completeness, statement of Hopf-Rinow Theorem
• Curvature: Riemann Curvature Tensor, Bianchi identity, scalar, sectional and
Ricci curvatures
• Jacobi Fields: Jacobi equation, conjugate points, second variation formula,
spaces of constant curvature (if time permits)
• Curvature and topology: Gauss-Bonnet Theorem, Bonnet-Myers Theorem,
Cartan-Hadamard Theorem
Suggesting Books
Texts:
1. J. M. Lee. Riemannian Manifolds, An introduction to Curvature. Graduate
Texts in Mathematics. Springer (1997).
2. M. P. doCarmo. Riemannian Geometry. Birkhauser (1991).
3. S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Springer (2004).

References:
1. I. Chavel. Riemannian geometry, a modern introduction. Cambridge
University Press (2006)
2. S. Kobayashi, K. Nomizu. Foundations of differential geometry, vol. -I,
Wiley Interscience Publication (1996).

MTH 516/616: Topology II (4)

Pre-requisites (Desirable): MTH 507 or MTH 605 and MTH 302 or MTH 601
Learning Objectives
This is an advanced course in algebraic topology that builds on the concepts
introduced in MTH 507 and MTH 605. Homology, which is the central topic of
this course, finds applications in several areas of mathematical research, including
low-dimensional topology and operator algebras. It is primarily meant for
students who wish to pursue research in topology and related areas.
Course Contents
• Simplicial Homology: Simplicial Complexes, Barycentric Subdivision, and
Simplicial Homology with examples
• Singular and Cellular Homology: Definition with examples, Homotopy
Invariance, Exact Sequence of Relative Homology, Excision, Mayer-Vietoris
Sequence, Degree of Maps, and Cellular Homology, Jordan-Brouwer
Separation Theorem, Invariance of domain and dimension, Borsuk-Ulam
Theorem, Lefschetz-Hopf Fixed Point Theorem, Axioms for homology,
Fundamental group and homology, and Simplicial Approximation Theorem
• Cohomology: Universal Coefficient Theorem, Künneth Formula, Cup Product
and the Cohomology Ring, Cap Product, Orientations on Manifolds, and
Poincaré Duality
• Higher Homotopy Groups: Definition with examples, Aspherical Spaces,
Relative Homotopy Groups, Long Exact Sequence of a triple, n-connected
spaces, and Whitehead's Theorem
Suggesting Books:
1. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
2. E. H. Spanier, Algebraic Topology, Springer, 1994.
3. J. R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.
4. J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988.
5. M. J. Greenberg & J. R. Harper, Algebraic Topology: A First Course,
Perseus Books Publishing, 1981.
6. W. S. Massey, A Basic Course in Algebraic Topology, Springer
International Edition, 2007.
7. G. Bredon, Topology and Geometry, Springer International Edition.

MTH 517/617: Introduction to Algebraic Geometry (4)

Pre-requisites (Desirable): MTH 301, MTH 302, and MTH 401
Learning Objectives
This course aims to provide an introduction to some of the basic objects and
techniques and objects of algebraic geometry with minimal prerequisites. The
main emphasis is on geometrical ideas and so most of the treatment will be over
algebraically closed fields. Results from commutative algebra will introduced and
proved as required and so no prior experience with commutative algebra will be
assumed. After introducing the basic objects and techniques, they will be
illustrated by application to the theory of algebraic curves.
Course Contents
1. Closed subsets of affine space, coordinate rings, correspondence between
ideals and closed subsets, affine varieties, regular maps, rational functions,
Hilbert's nullstellensatz
2. Projective and quasi-projective varieties, regular and rational functions on
projective varieties, products and maps of quasi-projective varieties
3. Dimension of varieties, examples and applications
4. Local ring of a point, tangent and cotangent space, local parameters, non-
singular points and non-singular varieties
5. Birational maps, blowups, disingularization of curves
6. Intersection numbers for plane curves, divisors on curves, Bezout's theorem,
Riemann-Roch theorem for curves, Residue theorem, Riemann-Hurwitz
formula
Suggesting Books:
1. W. Fulton, Algebraic curves: An introduction to algebraic geometry, 2008
ed. (available online).
2. R. Shafarevich, Basic Algebraic Geometry, Vol. 1, Third Edition,
Springer, Heidelberg, 2013.
3. S. Abhyankar, Algebraic geometry for scientists and engineers,
Mathematical Surveys and Monographs 35, American Mathematical
Society, 1990.
4. K. Smith et al, An invitation to algebraic geometry, Springer, 2004.

MTH 518/618: Commutative Algebra (4)

Pre-requisites: MTH 401 and its pre-requisites
Learning Objectives
The aim of this course is to introduce commutative algebra. This theory has
developed not just as a standalone area of algebra, but also as a tool to study other
important branches of Mathematics including Algebraic Geometry and Algebraic
Number Theory.
Course Contents
• Quotient Rings, Prime and Maximal ideals, units, Nilradical, Jacobson
Radical, Operations on ideals, Extensions and contractions
• Tensor product of Algebras (only existence theorem), Rings and Modules of
fractions, Local properties, Structure passing between R and S-1R (resp. M and
S-1M)
• Primary decompositions, Uniqueness theorems, Chain conditions, Noetherian
and Artinian Rings, Lasker-Noether theorem, Hilbert basis theorem,
Nakayama's lemma, Krull intersection theorem
• Integral dependence, Going up theorem, Integrally closed integral domains,
Going down theorem
• Valuation rings, Discrete valuation rings, Dedekind domains, Fractional ideals
• Valuations, Completions, Extensions of absolute values, residue field, Local
fields, Ostrowski's theorem
• Hilbert's Nullstellensatz
Suggesting Books:
1. Introduction to Commutative Algebra, Atiyah, M and Macdonald, I.G.,
Levant Books, Kolkata
2. Graduate Algebra: Commutative View, Rowen, L.H., Graduate Studies in
Mathematics, AMS
3. Commutative Algebra with a view towards Algebraic Geometry,
Eisenbud, D., Springer

MTH 519/619: Introduction to Modular Forms (4)

Pre-requisites (Desirable): MTH 407: Complex Analysis I
Learning Objectives
The aim of this course is to introduce the theory of modular forms with minimal
prerequisites. The course is intended for the students who have done the standard
courses in Linear Algebra and Complex Analysis. The results and techniques
from these courses will be used to understand the space of modular forms and
hence the students will solidify their understandings of some basic tools learned
throughout mathematics. Numerous examples of modular forms will be given
which are useful in solving some classical problems in number theory. The
purpose is to make the modular form theory accessible without going into the
advanced algebraically oriented treatments of the subject. At the same time this
course introduces the topics that are at the forefront of the current research.
Course Contents
• The full modular group SL2(Z), Congruence subgroups, The upper half-plane
H, Action of groups on H, Fundamental domains, The invariant metric on H
• Modular forms of integral weight of level one, Eisenstein series, The
Ramanujan τ-function, Dedekind η-function, Poincare series, The valence
formula and dimension formula, Modular forms of integral weight of higher
level
• The Petersson inner product, Hecke operators, Oldforms and newforms,
Dirichlet series associated to modular forms: Convergence, Analytic
continuation, Functional equation
• (if time permits) Modular forms of half-integral weight: Definition and
examples, Hecke operators, Shimura-Shintani correspondences between
modular forms of integral weight and half-integral weight.
Suggesting Books:
1. M. Ram Murty, M. Dewar, H. Graves, Problems in the theory of modular
forms, Institute of Mathematical Sciences - Lecture Notes 1, Hindustan
Book Agency, 2015.
2. N. Koblitz, Introduction to elliptic curves and modular forms, Graduate
Texts in Mathematics 97, Springer, 1993.
3. J. P. Serre, A course in arithmetic, Graduate Texts in Mathematics 7,
Springer, 1973.
4. T. M. Apostol, Modular functions and Dirichlet series in number theory,
GTM 41, Springer, 1990.
5. H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in
Mathematics 17, AMS, 1997.
 6. F. Diamond and J. Shurman, A first course in modular forms, Graduate
Texts in Mathematics 228, Springer, 2005.
7. T. Miyake, Modular forms, Springer Monographs in Mathematics,
Springer, 2006.
8. G. Shimura, Modular forms: basics and beyond, Springer Monographs in
Mathematics, Springer, 2012.

MTH 520/622: Introduction to Hyperbolic Geometry (4)

Pre-requisites (Desirable): MTH 304, MTH 407
Learning Objectives
Hyperbolic geometry is arguably the most important area in modern geometry and
topology. This course is intended to expose the student to the foundational
concepts in hyperbolic geometry, and is specially tailored to prepare the student
for advance topics in geometric topology.
Course Contents
• The general Möbius group: The extended complex plane (or the Riemman
sphere) C; The general Möbius group Mob(Ĉ); Identifying Mob+(Ĉ)with the
matrix group PGL(2; C); Classification of elements of elements of Mob+(Ĉ);
Reflections and the general Möbius group Mob(Ĉ); Conformality of elements
in Mob(Ĉ).
• The upper-half plane model H2: The upper half plane H2; The
subgroup Mod(H2); Transitivity properties of Mob+(H2); Geometry of the
action of Mob+(H2); The metric in H2; Element of arc-length inH2; Path
metric in H2; The Poincaré metric dH on H2; Geodesics in H2; Identifying the
group Mob+(H2) of isometries of (H2, dH) with PSL(2; R); Ultraparallel lines
in H2.
• The Poincaré disk model D: The Poincaré disk D; Transitioning from H2 to D
via Mob+(H2); Element of arc-length and the metric dD in D; The
Group Mob(D) of isometries of (D, dD); Centre, radii, and length of
hyperbolic circles in D; Hyperbolic structures on holomorphic disks.
• Properties of H2: Curvature of H2; Convex subsets of H2; Hyperbolic
polygons; Area of a subset of H2; Gauss-Bonnet formula - area of a
hyperbolic triangle; Applications of Gauss-Bonnet Formula: Area of
 reasonable hyperbolic polygons, existence of certain hyperbolic n-gons,
hyperbolic dilations; Putting a hyperbolic structure on a surface using
hyperbolic polygons; Hyperbolic trigonometry: triogometric identities, law of
sines and cosines, Pythagorean theorem.
• Non-planar models (if time permits): Hyperboloid model for the hyperbolic
plane; Higher dimensional hyperbolic spaces.
Suggesting Books
1. James W. Anderson, Hyperbolic Geometry (2nd Edition), Springer, 2005.
2. Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic
Geometry, Springer, 1995.
3. Harold E. Wolfe, Introduction to Non-Euclidean Geometry, Dover, 2012
4. Alan F. Beardon, The geometry of discrete groups (Chapter 7), Springer,
1983.
5. Svetlana Katok, Fuchsian Groups (Chapter 1), Chicago Lectures in
Mathematics, 1992.
6. John Stillwell, Geometry of surfaces (Chapter 4), Springer, 1992.

MTH 521/621: Introduction to Wavelets (4)

Pre-requisites (Desirable): MTH 311, MTH 404
Learning Objectives
This is an introductory course on wavelet analysis. In this course we will
introduce the basic notion of wavelets in different settings, namely for finite
groups, discrete infinite groups and real line. This will provide the students an
opportunity to know perspective applications of linear algebra and real analysis in
mathematics and beyond.
Course Contents
• Review of Linear Algebra: Complex Series, Euler’s Formula, Roots of Unity,
Linear Transformations and Matrices, Change of Basis, diagonalization of
Linear Transformations and Matrices, Inner Product, Orthogonal Bases,
Unitary Matrices.
• The Discrete Fourier Transform: Definition and Basic Properties of Discrete
Fourier Transform, Translation-Invariant Linear Transformations, The Fast
Fourier Transform.
• Wavelets on Finite Group ZN: Convolution on ZN, Fourier Transform on ZN,
Definition of Wavelets and Basic Properties, Construction of Wavelets on ZN.
• Wavelets on Infinite Discrete Group Z: Definition and Basic Properties of
Hilbert spaces, Complete orthonormal Sets in Hilbert Spaces, The spaces l2(Z)
and L2([-π, π)), Basic Fourier Series, The Fourier Transform and Convolution
on l2(Z) Wavelets on Z.
• Wavelets on R: Convolution and Approximate Identities, Fourier Transform
on R, Bases for The Space L2(R), Belian-Low Theorem, Wavelets on R,
Multiresolution Analysis, Construction of Wavelets from multiresolution
Analysis, Construction of Compactly supported Wavelets, Haar Wavelets,
Band-Limited Wavelets, Applications.
Suggesting Books
1. Michael W. Frazier: An Introduction to Wavelets Through Linear
Algebra, Undergraduate Texts in Mathematics. Springer-Verlag, New
York, 1999.
2. Eugenio Hernandez, Guido Weiss: A First Course on Wavelets, Studies in
Advanced Mathematics. CRC Press, Boca Raton, FL, 1996.
3. Ingrid, Daubechies: Ten Lectures on Wavelets, CBMS-NSF Regional
Conference Series in Applied Mathematics, 61. Society for Industrial and
Applied Mathematics (SIAM), Philadelphia, PA, 1992

MTH 601: Algebra I (4)

Course Contents
Group Theory:
• Review of basics: Groups, subgroups, cyclic groups, quotient groups,
Lagrange’s theorem and some applications, isomorphism theorems,
composition series, Jordan-Holder theorem.
• Group actions: Definition and examples, Cauchy’s theorem, class equation,
Sylow’s theorems and applications.
• Direct product: Definition and examples, structure theorem for finitely
generated abelian groups, examples.
• Solubility: Derived and lower central series, soluble groups, examples.
• Free groups, group presentation, nilpotent groups (if time permits)
Rings and Modules:
• Review of basics: Rings, subrings, ideals, examples, ring homomorphism,
quotient rings, isomorphism theorems, field, integral domain, prime and
maximal ideals, characterization of prime and maximal ideals, direct product
of rings, chinese remainder theorem.
• Localization: Definition and examples, universal property of localization,
local ring, localization at a prime ideal.
• Integral domains: Euclidean domain, Principal ideal domain, Unique
factorization domain, primes and irreducible elements, examples, Gauss’s
lemma, Eisenstein’s criterion.
• Polynomial rings: polynomial rings in one and several variables, universal
property, unique factorization property.
• Basics on modules: Modules, submodules, homomorphism of modules,
isomorphism theorems, ring of endomorphisms of a module.
• Structure theorems: Direct product of modules, direct sum of modules,
universal property of direct product and direct sum of modules, short exact
sequence, short five lemma, structure theorem for finitely generated modules
over a principal ideal domain (with proof), review the fundamental theorem of
finitely generated abelian groups.
• Canonical forms: Rational and Jordan canonical forms.
Fields and Galois Theory:
• Review of basics: Fields, subfield, characteristic of a field, field
homomorphism.
• Field extensions: Finite and algebraic extensions, splitting fields, normal
extensions, algebraic closure, separable extensions, inseparable extensions,
cyclotomic fields, finite fields.
• Galois Theory: Primitive element theorem, fundamental theorem of Galois
theory, applications, simple extensions.
Suggesting Books:
1. Abstract Algebra, Dummit and Foote, Wiley Publications, 2nd edition.
2. Algebra, Hungerford, Springer Publications.
3. Algebra (3rd Edition), Serge Lang, Addison Welsey.
4. Basic Algebra, Jacobson, parts-I and II, Dover Publications Inc.; 2nd
edition.
5. Algebra, Birkhoff and McLane, Chelsea Publishing Co.
6. A course in the theory of groups, D.J.S. Robinson, Springer; 2nd edition.

MTH 602: Algebra II (4)

Course Contents
• Extension of rings, integral extensions, going up and going down theorems,
integral closure, integral galois extensions
• Transcendental extension, transcendence basis, Noether normalization
theorem, separable and regular extensions
• Algebraic varieties, Hilbert's Nullstellensatz, Spec of a ring
• Noetherian (and Artinian) rings and Modules
• Matrices and linear maps, determinants, duality, bilinear and quadratic forms
• Tensor product, basic properties, bimodules, Flat modules, extension of
scalars, Algebras, Graded algebras, Tensor, symmetric and exterior algebras
Suggesting Books:
• S. Lang, Algebra, 3rd Edition, Addison Wesley.
• Jacobson, Basic Algebra I and II.
• Birkhoff and McLane, Algebra.
• Dummit and Foote, Abstract Algebra, 2nd Edition, Wiley.
MTH 603: Real Analysis (4)
Course Contents
• Several variable calculus: A quick overview, the contraction mapping
theorem, the inverse function theorem, the implicit function theorem.
• Riemann integration in Rn, n≥1.
• Lebesgue measure and integration: Measures, measurable functions,
integration of nonnegative and complex functions, modes of convergence,
convergence theorems, product measure, Fubini's theorem, convolution,
integration in polar coordinates.
• Signed measures and differentiation, complex measures, total variation,
absolute continuity, Fundamental theorem of calculus for Lebesgue integral,
the Radon-Nikodym theorem and consequences.
• Lp spaces, the Hölder and Minkowski inequalities, Jensen's inequality,
completeness, the Riesz representation theorem, dual of Lp spaces.
Suggested Books:
1. G.B. Folland, Real analysis: Modern techniques and their applications, 2nd
Edition, Wiley.
2. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, Tata
McGraw-Hill.
3. W. Rudin, Real and Complex Analysis, 3rd Edition, Tata McGraw-Hill.
4. E.M. Stein and R. Shakarchi, Functional Analysis: Introduction to further
topics in analysis, Princeton lectures in analysis.
5. T. Tao, Analysis I and II, 2nd Edition, TRIM Series 37, 38, Hindustan
Book Agency.

MTH 605: Topology I (4)

Course Contents
• General Topology: Connectedness, Compactness, Local Compactness,
Paracompactness, Quotient Spaces, Topological Groups, and Baire Category
Theorem.
• Homotopy Theory: Homotopy, Homotopy Equivalence and Deformation
Retractions, Fundamental Group, Van Kampen Theorem, Deck
Transformations, Group Actions, and Classification of covering spaces. Basic
definitions of higher homotopy groups and long exact sequence of a bration.
• Cellular and Simplicial Complexes: Operations on Cell Complexes and
Homotopy Extension Property. Simplicial Complexes - Barycentric
Subdivision and Simplicial Approximation Theorem.
Suggesting Books
1. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
2. J. R. Munkres, Topology, Second Edition, Prentice Hall, 2011.
3. G. Bredon, Topology and Geometry, Springer International Edition, 2006.
4. W. S. Massey, A Basic Course in Algebraic Topology, Springer
International Edition, 2007.
5. J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988.
6. J. R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.

MTH 606: Ordinary Differential Equations (4)

Course Contents
• First-order equations
o Direction fields, approximate solutions , the fundamental
inequality, uniqueness and existence theorems, solutions of equations
containing parameters
o Comparison theorems
• Systems of first-order equations
o Linear systems with constant coefficients: exponentials of linear operators,
the fundamental theorem for linear systems, linear systems in the plane,
canonical forms of linear operators on a complex vector space (S+N
decomposition, nilpotent canonical forms, Jordan and real canonical
forms), stability theory (saddle, spiral, and nodal points), phase portraits
o Linear equations of higher order: fundamental systems, Wronskian,
reduction of order, non-homogeneous linear systems, Green’s function
 o Non-linear systems: the fundamental existence-uniqueness theorem,
dependence on initial conditions and parameters, the maximal interval of
existence, the flow defined by a differential equation, linearization, the
Stable Manifold theorem, the Hartman-Grobman theorem, stability theory
of equilibria (saddles, nodes, foci and centres), Liapunov functions, La-
Salle’s invariance principle, gradient systems
• Poincare-Bendixson theory: Limit sets, local sections, theorem of Poincare-
Bendixson, Poincare's index, orbital stability of limit cycles, index of simple
singularities
Suggesting Books
1. G. Birkhoff & G. C. Rota, Ordinary differential equations, Paperback edition,
John Wiley & Sons, 1989
2. W. Hurewicz, Lectures on ordinary differential equations, Dover, New York,
1997
3. Morris Hirsch and Stephen Smale, Differential Equations, Dynamical
Systems, and Linear Algebra (Pure and Applied Mathematics (Academic
Press), 1974
4. P. Hartman, Ordinary Differential Equations, New York, Wiley, 1964