Course Curriculum Booklet

1
Department of Mathematics, IIT Bombay

July 3, 2023
Course Curriculum Booklet
Contents
1 B.S. program in Mathematics (for those who joined in July 2022 or later) 3
1.1 Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 List of Electives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Integrated M.Sc. program in Mathematics 5

2.1 Minimum Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Curriculum for students switching after first year . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Curriculum for students switching after second year . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Eligibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 List of Electives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Two year M.Sc. program in Mathematics 11

3.1 Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 List of Electives for Mathematics programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Two year M.Sc. program in Statistics 15

4.1 Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 List of Electives for Statistics programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 LIST OF ALL MATHEMATICS COURSES 17
6 LIST OF ALL STATISTICS COURSES 125
1
The latest version of this file can be found at https://www.math.iitb.ac.in/−→ Academic Programs−→Curriculum Booklet.
Created and maintained by Shri Ashutosh R. Mulik and Professor Ronnie Sebastian. Please write to ronnie@iitb.ac.in if you find
any errors.
1
2 CONTENTS
Chapter 1
B.S. program in Mathematics (for those who

joined in July 2022 or later)
1.1 Curriculum
First Semester Second Semester

Course Name L T P C Course Name L T P C
CH105 Organic and Inorganic 4 BB101 Biology 6
Chemistry CH117 Chemistry Lab 3
CH107 Physical Chemistry 4 MA106 Linear Algebra 3 1 0 4
CS101 Computer Program- 6 MA108 Differential Equations 3 1 0 4
ming and Utilization MA114 Introduction to Math- 3 0 0 6
MA109 Calculus 1 3 1 0 4 ematical Concepts
MA111 Calculus 2 3 1 0 4 ME113 Workshop Practice 4
MA113 Mathematics and Its 2 1 0 6 NOCS02 National Cadet Corps P/
History National Sports Or- NP
ME119 Engineering Graphics 5 ganization National
and Drawing Sports Organization
NOCS01 National Cadet Corps P/ National Service
National Sports Or- NP Scheme
ganization National PH103 Electricity and Mag- 6
Sports Organization netism
National Service
Scheme
PH107 Quantum Physics and 6
application
PH117 Physics Lab 6
3
4CHAPTER 1. B.S. PROGRAM IN MATHEMATICS (FOR THOSE WHO JOINED IN JULY 2022 OR LATER)
Third Semester Fourth Semester

MA401 Linear Algebra 3 1 0 8 ES200 Environmental Stud- 3
MA403 Real Analysis 3 1 0 8 ies
MA419 Basic Algebra 3 1 0 8 HS200 Environmental Stud- 3
SI427 Probability 1 3 1 0 8 ies
HSS101 Economics 6
MA406 General Topology 3 1 0 8
MA410 Multivariable Calcu- 2 1 0 6
lus
MA412 Complex Analysis 3 1 0 8
Fifth Semester Sixth Semester

HS30X Introduction to Lit- 6 MA214 Numerical Analysis 3 1 0 8
erature/ Philosophy/ MA408 Measure Theory 3 1 0 8
Sociology MA414 Algebra 1 2 1 0 6
MA417 Ordinary Differential 3 1 0 8 Dept Elective 6
Equations Inst Elective 6
SI419 Combinatorics 3 1 0 8
Dept Elective 6
Inst Elective 6
Seventh Semester Eighth Semester

MA503 Functional Analysis 3 1 0 8 Dept Elective 6
MA515 Partial Differential 3 1 0 8 Dept Elective 6
Equations Dept Elective 6
Dept Elective 6 Dept Elective 6
Dept Elective 6 Inst Elective 6
Inst Elective 6
1.2 List of Electives

(click on the link)
Chapter 2
Integrated M.Sc. program in Mathematics
2.1 Minimum Requirements

The minimum requirements for students to obtain a 5-year Integrated M.Sc. degree are as follows:
(1) Number of core courses: 2 HSS courses + HSS Environmental Science + CESE Environmental Science +
10 MA courses (excludes MA 1xx courses) = 14.
(2) Compulsory Project in the 5th year.
(3) Number of Credits in core courses (counted as above): 92
(4) Number of credits for the project: 30
(5) Minimum number of Department electives: 9
(6) Minimum no of credits in department electives (including the Advanced Electives): 54
(7) Minimum no of Institute electives: 2
(8) Minimum No of credits in institute electives: 12
(9) Minimum No of credits: 330
5
6 CHAPTER 2. INTEGRATED M.SC. PROGRAM IN MATHEMATICS
2.2 Curriculum for students switching after first year

Code Course Title C Code Course Title C
MA401 Linear Algebra 8 MA214 Numerical Analysis 8
MA403 Real Analysis 8 MA406/ General Topology/Complex 8
MA419 Basic Algebra 8 MA412 Analysis
SI419 Combinatorics 6 MA410 Multivariable Calculus 6
MA414 Algebra I 8
Fifth Semester Sixth Semester

HS301 6 ES200 Environmental Science 3
MA417 Ordinary Differential Equa- 8 HS200 Environmental Science 3
tions MA406/ General Topology/Complex 8
SI427 Probability I 8 MA412 Analysis
Dept Elective 6 MA408 Measure Theory 8
Inst Elective 6 SI404 Applied Stochastic Pro- 8
cesses
Inst Elective 6
Seventh Semester Eighh Semester

MA503 Functional Analysis 8 MA450 Independent study 6
MA515 Partial Differential Equa- 8 Dept Elective 6
tions Dept Elective 6
Inst Elective 6 Inst Elective 6
Ninth Semester Tenth Semester

Advance Elective 6 Advance Elective 6
Dept Elective 6 Project 18
Dept Elective 6
Project 12
Total credits for eight semesters = 264.

2.2. CURRICULUM FOR STUDENTS SWITCHING AFTER FIRST YEAR 7
Independent Study: The student should pursue a topic of his or her choice for one semester under the supervision
of a faculty member. The course is the analogue of the “Seminar” courses that undergraduate students were
required to take in previous years. The Independent Study should end with a presentation to the supervising
faculty member and the preparation of a brief report of about ten pages.
Department Electives may include any Department Elective offered or Ph.D. course offered in the relevant
semester. The existing departmental rules for prerequisites for these courses will apply.
The Advanced Elective listed in Semesters 9 and 10 must be one of the core Ph.D. courses (that is, not a
“Topics” course). These are:
For Semester 9 (Advanced Elective): MA 813 (Algebra I), MA 819 (Measure Theory), MA 815 (Differen-
tial Topology), MA 817 (Partial Differential Equations), MA 833 (Weak Convergence and Martingale Theory),
MA 821 (Theory of Estimation)
For Semester 10 (Advanced Elective): MA 812 (Algebra II), MA 814 (Complex Analysis), MA 816 (Alge-
braic Topology), MA818 (Partial Differential Equations II), MA 820 (Stochastic Processes), MA 822 (Testing
of Hypothesis), MA 824 (Functional Analysis
2.3 Curriculum for students switching after second year

2.3.1 Eligibility
The department will use its discretion to admit students who apply for a branch change after their second
year. Students will not be admitted after their second year unless they have already completed MA 403 (Real
Analysis) and at least one of MA 406 (General Topology), MA 410 (Multivariable calculus), MA 412 (Complex
Analysis), MA 419 (Basic Algebra) or MA 414 (Algebra I). Other criteria, such as performance in these courses,
may also be used to determine eligibility.
Fifth Semester Sixth Semester*

HS301 6 ES200 Environmental Science 3
MA401 Linear Algebra 8 HS200 Environmental Science 3
MA417 Ordinary Differential Equa- 8 MA406 General Topology 8
tions MA408 Measure Theory 8
MA419 Basic Algebra 8 MA410 Multivariable Calculus 6
SI419 Combinatorics 6 MA412 Complex Analysis 8
MA414 Algebra I 8
SI416 Optimization 8
*In Semester 6, it is expected that the student will take 4 out of the 5 MA courses.
Seventh Semester Eighh Semester

MA503 Functional Analysis 8 MA450 Independent study 6
MA515 Partial Differential Equa- 8 SI404 Applied Stochastic Pro- 8
tions cesses
SI427 Probability I 8 Dept Elective 6
Inst Elective 6 Dept Elective 6
Inst Elective 6
Ninth Semester Tenth Semester

Advance Elective 6 Advance Elective 6
Dept Elective 6 Project 18
Project 12
Total credits for six semesters = 220.

2.4. LIST OF ELECTIVES 9
Independent Study: The student should pursue a topic of his or her choice for one semester under the supervision
of a faculty member. The course is the analogue of the “Seminar” courses that undergraduate students were
required to take in previous years. The Independent Study should end with a presentation to the supervising
faculty member and the preparation of a brief report of about ten pages.
Department Electives may include any Department Elective offered or Ph.D. course offered in the relevant
semester. The existing departmental rules for prerequisites for these courses will apply.
The Advanced Elective listed in Semesters 9 and 10 must be one of the core Ph.D. courses (that is, not a
“Topics” course). These will be courses with numbers MA8xx.
For Semester 9 (Advanced Elective): MA 813 (Algebra I), MA 819 (Measure Theory), MA 815 (Differen-
tial Topology), MA 817 (Partial Differential Equations), MA 833 (Weak Convergence and Martingale Theory),
MA 821 (Theory of Estimation)
For Semester 10 (Advanced Elective): MA 812 (Algebra II), MA 814 (Complex Analysis), MA 816 (Alge-
braic Topology), MA818 (Partial Differential Equations II), MA 820 (Stochastic Processes), MA 822 (Testing
of Hypothesis), MA 824 (Functional Analysis
2.4 List of Electives

(click on the link)
Chapter 3
Two year M.Sc. program in Mathematics
3.1 Curriculum

CourseName L T P C CourseName L T P C
SI427 Probability 1 3 1 0 8 MA406 General Topology 3 1 0 8
MA401 Linear Algebra 3 1 0 8 MA408 Measure Theory 3 1 0 8
MA403 Real Analysis 3 1 0 8 MA410 Multivariable Calcu- 2 1 0 6
MA417 Ordinary Differential 3 1 0 8 lus
Equations MA412 Complex Analysis 3 1 0 8
MA419 Basic Algebra 3 1 0 8 MA414 Algebra 1 3 1 0 8

MA503 Functional Analysis 3 1 0 8 MA598 Project/Dept Elec- 6
MA515 Partial Differential 3 1 0 8 tive/Inst Elective
Equations Dept Elective 6
MA593 Project (Optional) 4 Dept Elective 6
Dept Elective 6 Environmental Stud- 6
ies/Dept Elective/Inst
Elective
11
12 CHAPTER 3. TWO YEAR M.SC. PROGRAM IN MATHEMATICS
3.2 List of Electives for Mathematics programs
Odd Semester Electives Even Semester Electives

Course Name Course Name
MA521 Theory of Analytic Functions MA504 Operators on Hilbert Spaces
MA523 Basic Number Theory MA510 Introduction to Algebraic Geometry
MA525 Dynamical Systems MA518 Spectral Approximation
MA538 Representation Theory of Finite MA524 Algebraic Number Theory
Groups MA526 Commutative Algebra
MA539 Spline Theory and Variational MA528 Hyperplane Arrangements
Methods MA530 Nonlinear Analysis
MA556 Differential Geometry MA532 Analytic Number Theory
MA581 Elements of Differential Topology MA534 Modern Theory of PDE
MA5101 Algebra 2 MA540 Numerical Methods for Partial Differential
MA5103 Algebraic Combinatorics 2 1 0 6 Equations
MA5105 Coding Theory MA562 The Mathematical Theory of Finite Ele-
MA5107 Continuum Mechanics ments
MA5109 Graph Theory MA5102 Basic Algebraic Topology
MA5111 Theory of Finite Semigroups MA5104 Hyperbolic Conservation Laws
MA5113 Category Theory 1 MA5106 Introduction to Fourier Analysis
MA5115 Hopf Algebras MA5108 Lie Groups and Lie Algebras
SI507 Numerical Analysis MA5110 Non-commutative Algebra
SI537 Probability 2 MA5112 Introduction to Mathematical Methods
MA5116 Species and Operads
MA5118 Category Theory 2
MA606 Coxeter Groups
SI416 Optimization
SI527 Introduction to Derivative Pricing
Apart from the above listed electives, a student may also opt for a Ph.D. level course as an
elective subject to the approvals from the course instructor and the faculty advisor. The list of
Ph.D. courses offered in the department are as follows:
3.2. LIST OF ELECTIVES FOR MATHEMATICS PROGRAMS 13

MA811 Algebra 1 MA812 Algebra 2
MA813 Measure Theory MA814 Complex Analysis
MA815 Differential Topology MA816 Algebraic Topology
MA817 Partial Differential Equations 1 MA818 Partial Differential Equations 2
MA833 Weak Convergance and Martin- MA820 Stochastic Processes
gale Theory MA823 Probability
MA839 Advanced Commutative Algebra MA824 Functional Analysis
MA841 Topics in Algebra 1 MA842 Topics in Algebra 2
MA843 Topics in Analysis 1 MA844 Topics in Analysis 2
MA845 Topics in Combinatorics 1 MA846 Topics in Combinatorics 2
MA847 Topics in Geometry 1 MA848 Topics in Geometry 2
MA849 Topics in Topology 1 MA850 Topics in Topology 2
MA851 Topics in Number Theory 1 MA852 Topics in Number Theory 2
MA853 Topics in Differential Equations 1 MA854 Topics in Differential Equations 2
MA855 Topics in Numerical Analysis 1 MA856 Topics in Numerical Analysis II
MA857 Topics in Probability 1 MA858 Topics in Probability II
MA859 Topics in Statistics I MA860 Topics in Statistics II
MA861 Combinatorics 1 MA862 Combinatorics 2
MA863 Theoretical Statistics 1 MA865 Topics in Category Theory 2
MA864 Topics in Category Theory 1 MA867 Statistical Modelling - 1
14 CHAPTER 3. TWO YEAR M.SC. PROGRAM IN MATHEMATICS
Chapter 4
Two year M.Sc. program in Statistics
4.1 Curriculum

SI419 Combinatorics 3 1 0 8 SI404 Applied Stochastic 3 1 0 8
SI423 Linear Algebra and its 3 1 0 8 Processes
Applications SI416 Optimization 2 1 0 6
SI427 Probability 1 3 1 0 8 SI422 Regression Analysis 3 1 0 8
SI429 Real analysis 3 1 0 8 SI424 Statistical Inference 1 3 1 0 8
SI431 Introduction to Data 2 0 2 6 SI426 Algorithms 3 1 0 8
Analysis using R

SI503 Categorical Data 3 1 0 8 SI509 Time Series Analysis 3 1 0 8
Analysis SI526 Experimental Designs 2 1 0 6
SI505 Multivariate Analysis 3 1 0 8 SI598 Project/Dept Elec- 6
SI593 Project 1 (Optional) 4 tive/Inst Elective
Environmental Stud- 6 Dept Elective 2 1 0 6
ies/Dept Elective/Inst Dept Elective 2 1 0 6
Elective Dept Elective 2 1 0 6
Dept Elective 3 1 0 8
Dept Elective 3 1 0 8
15
16 CHAPTER 4. TWO YEAR M.SC. PROGRAM IN STATISTICS
4.2 List of Electives for Statistics programs

SI507 Numerical Analysis SI514 Statistical Modelling
SI513 Theory of Sampling SI527 Introduction to Derivative Pricing
SI515 Statistical Techniques in Data SI534 Nonparametric Statistics
Mining SI536 Analysis of Multi-Type and Big Data
SI537 Probability 2 SI544 Martingale theory
SI539 Random Graphs SI546 Statistical Inference II
SI541 Statistical Epidemiology SI548 Computational Statistics
SI543 Asymptotic Statistics SI550 Weak Convergence and Empirical Processes
Apart from the above listed electives, a student may also opt for a Ph.D. level course as an elective
subject to the approvals from the course instructor and the faculty advisor. The list of Ph.D. courses
offered in the department are as follows:

MA811 Algebra 1 MA812 Algebra 2
MA813 Measure Theory MA814 Complex Analysis
MA815 Differential Topology MA816 Algebraic Topology
MA817 Partial Differential Equations 1 MA818 Partial Differential Equations 2
MA833 Weak Convergance and Martin- MA820 Stochastic Processes
gale Theory MA823 Probability
MA839 Advanced Commutative Algebra MA824 Functional Analysis
MA841 Topics in Algebra 1 MA842 Topics in Algebra 2
MA843 Topics in Analysis 1 MA844 Topics in Analysis 2
MA845 Topics in Combinatorics 1 MA846 Topics in Combinatorics 2
MA847 Topics in Geometry 1 MA848 Topics in Geometry 2
MA849 Topics in Topology 1 MA850 Topics in Topology 2
MA851 Topics in Number Theory 1 MA852 Topics in Number Theory 2
MA853 Topics in Differential Equations 1 MA854 Topics in Differential Equations 2
MA855 Topics in Numerical Analysis 1 MA856 Topics in Numerical Analysis II
MA857 Topics in Probability 1 MA858 Topics in Probability II
MA859 Topics in Statistics I MA860 Topics in Statistics II
MA861 Combinatorics 1 MA862 Combinatorics 2
MA863 Theoretical Statistics 1 MA865 Topics in Category Theory 2
MA864 Topics in Category Theory 1 MA867 Statistical Modelling - 1
Chapter 5
LIST OF ALL MATHEMATICS COURSES
MA001 . . . . . . . . . . . . . . . . . . . . . 19 MA528 . . . . . . . . . . . . . . . . . . . . . 52
MA002 . . . . . . . . . . . . . . . . . . . . . 20 MA530 . . . . . . . . . . . . . . . . . . . . . 53
MA106 . . . . . . . . . . . . . . . . . . . . . 21 MA532 . . . . . . . . . . . . . . . . . . . . . 54
MA108 . . . . . . . . . . . . . . . . . . . . . 22 MA533 . . . . . . . . . . . . . . . . . . . . . 55
MA109 . . . . . . . . . . . . . . . . . . . . . 23 MA534 . . . . . . . . . . . . . . . . . . . . . 56
MA111 . . . . . . . . . . . . . . . . . . . . . 24 MA538 . . . . . . . . . . . . . . . . . . . . . 57
MA113 . . . . . . . . . . . . . . . . . . . . . 25 MA539 . . . . . . . . . . . . . . . . . . . . . 58
MA114 . . . . . . . . . . . . . . . . . . . . . 27 MA540 . . . . . . . . . . . . . . . . . . . . . 59
MA205 . . . . . . . . . . . . . . . . . . . . . 28 MA556 . . . . . . . . . . . . . . . . . . . . . 60
MA207 . . . . . . . . . . . . . . . . . . . . . 29 MA562 . . . . . . . . . . . . . . . . . . . . . 61
MA214 . . . . . . . . . . . . . . . . . . . . . 30 MA581 . . . . . . . . . . . . . . . . . . . . . 62
MA401 . . . . . . . . . . . . . . . . . . . . . 31 MA593 . . . . . . . . . . . . . . . . . . . . . 63
MA403 . . . . . . . . . . . . . . . . . . . . . 32 MA598 . . . . . . . . . . . . . . . . . . . . . 64
MA406 . . . . . . . . . . . . . . . . . . . . . 33 MA5101 . . . . . . . . . . . . . . . . . . . . 65
MA408 . . . . . . . . . . . . . . . . . . . . . 34 MA5102 . . . . . . . . . . . . . . . . . . . . 66
MA410 . . . . . . . . . . . . . . . . . . . . . 35 MA5103 . . . . . . . . . . . . . . . . . . . . 67
MA412 . . . . . . . . . . . . . . . . . . . . . 36 MA5104 . . . . . . . . . . . . . . . . . . . . 68
MA414 . . . . . . . . . . . . . . . . . . . . . 37 MA5105 . . . . . . . . . . . . . . . . . . . . 69
MA417 . . . . . . . . . . . . . . . . . . . . . 38 MA5106 . . . . . . . . . . . . . . . . . . . . 70
MA419 . . . . . . . . . . . . . . . . . . . . . 40 MA5107 . . . . . . . . . . . . . . . . . . . . 71
MA450 . . . . . . . . . . . . . . . . . . . . . 41 MA5108 . . . . . . . . . . . . . . . . . . . . 72
MA503 . . . . . . . . . . . . . . . . . . . . . 42 MA5109 . . . . . . . . . . . . . . . . . . . . 73
MA504 . . . . . . . . . . . . . . . . . . . . . 43 MA5110 . . . . . . . . . . . . . . . . . . . . 74
MA510 . . . . . . . . . . . . . . . . . . . . . 44 MA5111 . . . . . . . . . . . . . . . . . . . . 75
MA515 . . . . . . . . . . . . . . . . . . . . . 45 MA5112 . . . . . . . . . . . . . . . . . . . . 76
MA518 . . . . . . . . . . . . . . . . . . . . . 46 MA5113 . . . . . . . . . . . . . . . . . . . . 77
MA521 . . . . . . . . . . . . . . . . . . . . . 47 MA5115 . . . . . . . . . . . . . . . . . . . . 78
MA523 . . . . . . . . . . . . . . . . . . . . . 48 MA5116 . . . . . . . . . . . . . . . . . . . . 79
MA524 . . . . . . . . . . . . . . . . . . . . . 49 MA5118 . . . . . . . . . . . . . . . . . . . . 80
MA525 . . . . . . . . . . . . . . . . . . . . . 50 MA606 . . . . . . . . . . . . . . . . . . . . . 81
MA526 . . . . . . . . . . . . . . . . . . . . . 51 MA811 . . . . . . . . . . . . . . . . . . . . . 82
17
18 CHAPTER 5. LIST OF ALL MATHEMATICS COURSES
MA812 . . . . . . . . . . . . . . . . . . . . . 83 MA849 . . . . . . . . . . . . . . . . . . . . . 104

MA813 . . . . . . . . . . . . . . . . . . . . . 84 MA850 . . . . . . . . . . . . . . . . . . . . . 105
MA814 . . . . . . . . . . . . . . . . . . . . . 85 MA851 . . . . . . . . . . . . . . . . . . . . . 106
MA815 . . . . . . . . . . . . . . . . . . . . . 86 MA852 . . . . . . . . . . . . . . . . . . . . . 107
MA816 . . . . . . . . . . . . . . . . . . . . . 87 MA853 . . . . . . . . . . . . . . . . . . . . . 108
MA817 . . . . . . . . . . . . . . . . . . . . . 88 MA854 . . . . . . . . . . . . . . . . . . . . . 110
MA818 . . . . . . . . . . . . . . . . . . . . . 89 MA855 . . . . . . . . . . . . . . . . . . . . . 111
MA820 . . . . . . . . . . . . . . . . . . . . . 90 MA856 . . . . . . . . . . . . . . . . . . . . . 112
MA823 . . . . . . . . . . . . . . . . . . . . . 91 MA858 . . . . . . . . . . . . . . . . . . . . . 113
MA824 . . . . . . . . . . . . . . . . . . . . . 92 MA859 . . . . . . . . . . . . . . . . . . . . . 114
MA833 . . . . . . . . . . . . . . . . . . . . . 93 MA860 . . . . . . . . . . . . . . . . . . . . . 115
MA839 . . . . . . . . . . . . . . . . . . . . . 94 MA861 . . . . . . . . . . . . . . . . . . . . . 116
MA841 . . . . . . . . . . . . . . . . . . . . . 95 MA862 . . . . . . . . . . . . . . . . . . . . . 117
MA842 . . . . . . . . . . . . . . . . . . . . . 96 MA863 . . . . . . . . . . . . . . . . . . . . . 118
MA843 . . . . . . . . . . . . . . . . . . . . . 97 MA864 . . . . . . . . . . . . . . . . . . . . . 119
MA844 . . . . . . . . . . . . . . . . . . . . . 98 MA865 . . . . . . . . . . . . . . . . . . . . . 120
MA845 . . . . . . . . . . . . . . . . . . . . . 100 MA867 . . . . . . . . . . . . . . . . . . . . . 121
MA846 . . . . . . . . . . . . . . . . . . . . . 101 MA899 . . . . . . . . . . . . . . . . . . . . . 122
MA847 . . . . . . . . . . . . . . . . . . . . . 102 MAS801 . . . . . . . . . . . . . . . . . . . . 124
MA848 . . . . . . . . . . . . . . . . . . . . . 103 MAS802 . . . . . . . . . . . . . . . . . . . . 124
19
Course Code MA 001

Course Name Preparatory Mathematics 1
Total Credits 0
Type T
Lecture 3
Tutorial 2
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
Text Reference 1. nil
Complex numbers as ordered pairs. Argand’s diagram. Triangle inequality.

De Moivre’s Theorem.
Algebra: Quadratic equations and expressions. Permutations and combina-
tions. Binomial theorem for a positive integral index.
Coordinate Geometry: Locus, Straight lines. Equations of circle, parabola,
Description
ellipse and hyperbola in standard forms. Parametric representation.
Vectors: Addition of vectors. Multiplication by a scalar. Scalar product, cross
product and scalar triple product with geometrical applications.
Matrices and Determinants: Algebra of matrices. Determinants and their
properties. Inverse of a matrix. Cramer’s rule.
Course Code MA 002

Course Name Preparatory Mathematics 2
Total Credits 0
Type T
Lecture 3
Tutorial 2
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
Text Reference 1.
Function, Inverse function, Elementary functions and their graphs, Limit, Con-
tinuity, Derivative and its geometrical significance. Differentiability. Deriva-
tives of sum, difference, product and quotient of functions. Derivatives of poly-
nomial, rational, trigonometric, logarithmic, exponential, hyperbolic, inverse
Description trigonometric and inverse hyperbolic functions. Differentiation of compos-
ite and implicit functions. Tangents and Normals, Increasing and decreasing
functions. Maxima and Minima. Integrations as the inverse process of differ-
entiation, Integration by parts and by substitution. Definite integrals and its
application to the determination of areas.
21
Course Code MA 106

Course Name Linear Algebra
Total Credits 4
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester Y
Prerequisite Nil
1. H. Anton, Elementary Linear Algebra with Applications (8th Edition),

John Wiley, 1995.
2. G. Strang, Linear Algebra and its Applications (4th Edition), Thomson,
2006.
Text Reference
3. S. Kumaresan, Linear algebra - A Geometric Approach, Prentice Hall of
India, 2000.
4. E. Kreyszig, Advanced Engineering Mathematics (8th Edition), John
Wiley, 1999.
Vectors in Rn , linear independence and dependence, linear span of a set of

vectors, vector subspaces of Rn , basis of a vector subspace. Systems of linear
equations, matrices and Gauss elimination, row space, null space, and col-
umn space, rank of a matrix. Determinants and rank of a matrix in terms
of determinants. Abstract vector spaces, linear transformations, matrix of
a linear transformation, change of basis and similarity, rank-nullity theorem.
Description
Inner product spaces, Gram-Schmidt process, orthonormal bases, projections
and least squares approximation. Eigenvalues and eigenvectors, characteristic
polynomials, eigenvalues of special matrices (orthogonal, unitary, hermitian,
symmetric, skew-symmetric, normal), algebraic and geometric multiplicity, di-
agonalization by similarity transformations, spectral theorem for real symmet-
ric matrices, application to quadratic forms.
Course Code MA 108

Course Name Differential Equations
Total Credits 4
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester Y
Prerequisite Nil
1. E. Kreyszig, Advanced Engineering Mathematics (8th Edition), John

Wiley, 1999.
Text Reference 2. W. E. Boyce and R. DiPrima, Elementary Differential Equations (8th

Edition), John Wiley, 2005.
3. T. M. Apostol, Calculus, Volume 2 (2nd Edition), Wiley Eastern, 1980.
Exact equations, integrating factors and Bernoulli equations. Orthogonal tra-

jectories. Lipschitz condition, Picard‘s theorem, examples of non-uniqueness.
Linear differential equations generalities. Linear dependence and Wronskians.
Description Dimensionality of space of solutions, Abel-Liouville formula. Linear ODE
with constant coefficients, characteristic equations. Cauchy-Euler equations.
Method of undetermined coefficients. Method of variation of parameters.
Laplace transform generalities. Shifting theorems.
23
Course Code MA 109

Course Name Calculus 1
Total Credits 4
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester Y
Prerequisite Nil
1. Hughes-Hallett et al., Calculus - Single and Multivariable (3rd Edition),

John-Wiley, 2003
2. James Stewart, Calculus (5th Edition), Thomson, 2003.

Text Reference 3. T. M. Apostol, Calculus, Volumes 1 & 2 (2nd Edition), Wiley Eastern,
1980.
4. G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (9th

Edition), ISE Reprint, Addison-Wesley, 1998.
Review of limits, continuity, differentiation. Mean value theorem, Taylor’s

theorem, maxima and minima. Riemann integrals, fundamental theorem of
calculus. Improper integrals, applications to area and volume.
Description
Convergence of sequences and series: power series. Partial derivatives, gra-
dient and directional derivatives, chain rule, maxima and minima, Lagrange
multipliers.
Course Code MA 111

Course Name Calculus 2
Total Credits 4
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester Y
Prerequisite Nil
1. Hughes-Hallett et al., Calculus - Single and Multivariable (3rd Edition),

John-Wiley, 2003.
2. James Stewart, Calculus (5th Edition), Thomson, 2003.

Text Reference 3. T. M. Apostol, Calculus, Volumes 1 & 2 (2nd Edition), Wiley Eastern,
1980.
4. G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (9th

Edition), ISE Reprint, Addison-Wesley, 1998.
Double and triple integration, Jacobians and change of variables formula.

Description Parametrization of curves and surfaces, vector fields, line and surface inte-
grals. Divergence and curl. Theorems of Green, Gauss, and Stokes.
25
Course Code MA 113

Course Name Mathematics and Its History
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. (JB) The Ascent of Man, by Jacob Bronowski; BBC Books
2. (BS) The Ascent of Science, by Brian L. Silver; Oxford University Press.
3. (EM) This is Biology, by Ernst Mayr; Harvard University Press.
4. (Stillwell) Mathematics and its History, by John Stillwell; Springer (Un-

dergraduate Texts in Mathematics).
5. (PCM) The Princeton Companion to Mathematics, edited by Timothy

Text Reference Gowers, June Barrow Green, and Imre Leader; Princeton University
Press.
6. (Burton) Elementary number theory, by D. M. Burton; 6th edition,

McGraw-Hill, 2007.
7. (Goldberg) Methods of real analysis, by R. R. Goldberg; Oxform & IBH

Pub. (Indian Edition), 1970.
8. (JJ) Elementary number theory, by G. A. Jones and J. M. Jones; Springer

Math Undergrad Series, 1998.
Description Continued on next page ...

Course Code MA 113 ( ... continued from previous page)

Course Name Mathematics and Its History
Part I 1. Copernican revolution, Galileo versus Church, Kepler and Newton
(JB – Chapter 6 and 7). 2. Enlightenment movement and Romantic move-
ment and further professionalisation (BS – Chapter 6, 7 and 11). 3. Industrial
revolution and engines (JB – Chapter 8) 4. Electromagnetism (BS – Chap-
ter 8) 5. Darwin and Mendel (JB – Chapter 9 and 12; BS – Chapter 23).
Part II 1. History of Algebra: Quadratic equations, solutions to cubics and
quartics, higher degree equations and insolvability. algebra and geometry of
complex numbers, fundamental theorem of algebra (Stillwell – Chapters 6 and
14). 2. History of Calculus and Geometry: The regular polyhedra, conic sec-
tions, coordinate geometry (Stillwell – Chapter 2 and 7). Early results on areas
and volumes, maxima, minima and tangents, infinite series, Leibniz’s calculus
(Stillwell – Chapter 9). The isoperimetric inequality (PCM – III.94 , IV.26
Description and V.19). 3. History of Number Theory and Combinatorics: Pythagorean
triples, prime numbers, Euclidean algorithm, chinese remainder theorem, Pell’s
equation (Stillwell – Chapters 3, and 5). Divisibility, Bezout‘s identity, prime
factorisation, fundamental theorem of arithmetic, division algorithms, GCD
and LCM (Burton – Chapter 2, JJ – Chapter 1 and 2). Pigeonhole princi-
ple, Konigsberg problem (Stillwell – Chapter 25). 4. Elementary Concepts:
Statements and quantifiers, sets, functions and methods of proofs (Goldberg
– Chapter 1, Burton – Chapter 1, Jones and Jones – Appendix A). Relations,
equivalence, partitions, modular arithmetic (JJ – Appendix B, Section 3.1, 5.1-
5.3, 6.1, 8.2-8.5). Uncountability of R (Goldberg – Chapter 1). Double and
triple integration, Jacobians and change of variables formula. Parametrization
of curves and surfaces, vector fields, line and surface integrals. Divergence and
curl. Theorems of Green, Gauss, and Stokes.
27
Course Code MA 114

Course Name An Introduction to Mathematical Concepts
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. T. M. Apostol, Mathematical Analysis, (2nd edition) Narosa Publicating

House, 1974.
2. D. M. Burton, Elementary number theory, 6th edition, McGraw-Hill,
2007.
3. J. B. Conway, Functions of one complex variable, 2nd edition, Springer,
1978.
4. J. P. D’Angelo and D. B. West, Mathematical thinking: Problem-solving
Text Reference and Proofs, 2nd edition,Prentice Hall, 1997.
5. R. R. Goldberg, Methods of real analysis, Oxform & IBH Pub. (Indian
Edition), 1970.
6. P. R. Halmos, Naive set theory, Springer 1960 (Reprint 2017).
7. G. A. Jones and J. M. Jones, Elementary number theory, Springer Math
Undergrad Series, 1998 (Indian edition available).
8. A. Kumar and S. Kumerasan, A Basic course in real analysis, CRC Press,
2014.
Elementary Concepts: Statements and Quantifiers, Sets, Functions and Meth-

ods of proofs (Goldberg, Ch 1) (Burton, Ch 1) (Jones and Jones Appendix A).
Basic Real Analysis: Least upper bound and applications, Archimedean prop-
erty, Density of Q, R \ Q, Greatest integer function, Nested Interval Theorem,
Uncountability of R (Goldberg, Ch 1). Sequence of Real numbers: (Goldberg,
Ch 2). Operations, Monotone sequences, Cauchy sequences. Convergence of
Series: Convergence and divergence, Test for absolute convergence (Goldberg,
Ch 3). Basic Algebra: Divisibility, Bezout’s Identity, Prime Factorisation,
Description
Fundamental Theorem of Arithmetic, Division Algorithms, GCD and LCM
(Burton, Ch. 2) (Jones and Jones Ch. 1 and 2). Relations, Equivalence,
Partitions, Modular Arithmetic, Euler and Mobius functions and inversion.
Groups and Subgroups (basic properties and examples) (Jones and Jones Ap-
pendix B, Sec 3.1, 5.1-5.3, 6.1, 8.2-8.5). Complex Plane: Polar representation
and roots of unity, lines and half planes in C, C as a vector space over R,
conjugation as a linear map over R, extended complex plane and its spherical
representation (Conway, Ch. 1).
Course Code MA 205

Course Name Complex Analysis
Total Credits 4
Type T
Lecture 3
Tutorial 0
Practical 1
Selfstudy 0
Half Semester Y
Prerequisite Nil
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)

Text Reference 3. M. J. Ablowitz and A. S. Fokas, Complex Variables- Introduction and
Applications, Cambridge University Press, 1998 (Indian Edition)
4. E. Kreyszig, Advanced engineering mathematics (8th Edition), John Wi-

ley (1999).
Definition and properties of analytic functions. Cauchy-Riemann equations,

harmonic functions. Power series and their properties. Elementary functions.
Description Cauchy’s theorem and its applications. Taylor series ans Laurent expansions.
Residues and the Cauchy residue formula. Evaluation of improper integrals.
Conformal mappings. Inversion of Laplace transforms.
29
Course Code MA 207

Course Name Differential Equations 2
Total Credits 4
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester Y
Prerequisite Nil

ley (1999).
2. W. E. Boyce and R. DiPrima, Elementary Differential Equations (8th

Text Reference Edition), John Wiley (2005)
3. R. V. Churchill and J. W. Brown, Fourier series and boundary value

problems (7th Edition), McGraw-Hill (2006).
Review of power series and series solutions of ODE’s. Legendre’s equation

and Legendre polynomials. Regular and irregular singular points, method of
Frobenius. Bessel’s equation and Bessel’s functions. Sturm-Liouville problems.
Description Fourier series. D’Alembert solution to the Wave equation. Classification of
linear second order PDE in two variables. Laplace, Wave, and Heat equations
using separation of variables. Vibration of a circular membrane. Heat equation
in the half space.
Course Code MA 214

Course Name Introduction to Numerical Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. D. Conte and Carl de Bo or, Elementary Numerical Analysis- An

Algorithmic Approach (3rd Edition), McGraw-Hill, 1980.
2. C. E. Froberg, Introduction to Numerical Analysis (2nd Edition),

Text Reference Addison-Wesley, 1981.

ley (1999).
Interpolation by polynomials, divided differences, error of the interpolating

polynomial, piecewise linear and cubic spline interpolation. Numerical in-
tegration, composite rules, error formulae. Solution of a system of linear
equations, implementation of Gaussian elimination and Gauss-Seidel meth-
ods, partial pivoting, row echelon form, LU factorization Cholesky’s method,
ill-conditioning, norms. Solution of a nonlinear equation, bisection and secant
Description methods. Newton’s method, rate of convergence, solution of a system of non-
linear equations, numerical solution of ordinary differential equations, Euler
and Runge-Kutta methods, multistep methods, predictor-corrector methods,
order of convergence, finite difference methods, numerical solutions of elliptic,
parabolic, and hyperbolic partial differential equations. Eigenvalue problem,
power method, QR method, Gershgorin’s theorem. Exposure to software pack-
ages like IMSL subroutines, MATLAB.
31
Course Code MA 401

Course Name Linear Algebra
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. K. Hoffman and R. Kunze, Linear Algebra, Pearson Education (India),

2003.
2. S. Lang, Linear Algebra, Undergraduate Texts in Mathematics, Springer-

Text Reference Verlag, New York, 1989.
3. P. Lax, Linear Algebra, John Wiley & Sons, 1997.
4. H.E. Rose, Linear Algebra, Birkhauser, 2002.
Vector spaces over fields, subspaces, bases and dimension. Systems of linear
equations, matrices, rank, Gaussian elimination. Linear transformations, rep-
resentation of linear transformations by matrices, rank-nullity theorem, du-
ality and transpose. Determinants, Laplace expansions, cofactors, adjoint,
Cramer‘s Rule. Eigenvalues and eigenvectors, characteristic polynomials, min-
imal polynomials, Cayley-Hamilton Theorem, triangulation, diagonalization,
Description
rational canonical form, Jordan canonical form. Inner product spaces, Gram-
Schmidt ortho-normalization, orthogonal projections, linear functionals and
adjoints, Hermitian, self-adjoint, unitary and normal operators, Spectral The-
orem for normal operators. Bilinear forms, symmetric and skew-symmetric
bilinear forms, real quadratic forms, Sylvester’s law of inertia, positive defi-
niteness.
Course Code MA 403

Course Name Real Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. T. Apostol, Mathematical Analysis, 2nd Edition, Narosa, 2002.
2. K. Ross, Elementary Analysis: The Theory of Calculus, Springer Int.

Text Reference Edition, 2004.
3. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw-

Hill, 1983.
Review of basic concepts of real numbers: Archimedean property, Complete-

ness. Metric spaces, compactness, connectedness, (with emphasis on Rn ). Con-
tinuity and uniform continuity. Monotonic functions, Functions of bounded
variation; Absolutely continuous functions. Derivatives of functions and Tay-
Description lor‘s theorem. Riemann integral and its properties, characterization of Rie-
mann integrable functions. Improper integrals, Gamma functions. Sequences
and series of functions, uniform convergence and its relation to continuity,
differentiation and integration. Fourier series, pointwise convergence, Fejer‘s
theorem, Weierstrass approximation theorem.
33
Course Code MA 406

Course Name General Topology
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 403 (Real Analysis)
1. M. A. Armstrong, Basic Topology, Springer (India), 2004.
2. K. D. Joshi, Introduction to General Topology, New Age International,

2000.
Text Reference 3. J. L. Kelley, General Topology, Van Nostrand, 1955.J. R. Munkres,
Topology, 2nd Edition, Pearson Education (India), 2001.
4. G. F. Simmons, Introduction to Topology and Modern Analysis,

McGraw-Hill, 1963.
Topological Spaces: open sets, closed sets, neighbourhoods, bases, sub bases,
limit points, closures, interiors, continuous functions, homeomorphisms. Ex-
amples of topological spaces: subspace topology, product topology, metric
topology, order topology.
Quotient Topology: Construction of cylinder, cone, Moebius band, torus, etc.
Connectedness and Compactness: Connected spaces, Connected subspaces of
the real line, Components and local connectedness, Compact spaces, Heine-
Borel Theorem, Local -compactness.
Description Separation Axioms: Hausdorff spaces, Regularity, Complete Regularity, Nor-
mality, Urysohn Lemma, Tychonoff embedding and Urysohn Metrization The-
orem, Tietze Extension Theorem. Tychnoff Theorem, One-point Compact-
ification. Complete metric spaces and function spaces, Characterization of
compact metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Cate-
gory Theorem.
Applications: space filling curve, nowhere differentiable continuous function.
Optional Topics: Topological Groups and orbit spaces, Paracompactness and
partition of unity, Stone-Cech Compactification, Nets and filters.
Course Code MA 408

Course Name Measure Theory
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. P.R. Halmos, Measure Theory, Graduate Text in Mathematics, Springer-

Verlag, 1979.
Text Reference 2. Inder K. Rana, An Introduction to Measure and Integration (2nd Edi-
tion), Narosa Publishing House, New Delhi, 2004.
3. H.L. Royden, Real Analysis, 3rd Edition, Macmillan, 1988.
Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theo-

rem. Measure spaces. Outline of extension of measures from algebras to the
generated sigma-algebras, Measurable sets, Lebesgue Measure and its proper-
ties. Measurable functions and their properties, Integration and Convergence
Description
theorems. Introduction to Lp -spaces, Riesz-Fischer theorem, Riesz Representa-
tion theorem for L2 -spaces. Absolute continuity of measures, Radon-Nikodym
theorem. Dual of Lp -spaces. Product measure spaces, Fubini’s theorem. Fun-
damental Theorem of Calculus for Lebesgue Integrals (an outline).
35
Course Code MA 410

Course Name Multivariable Calculus
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. W. Fleming, Functions of Several Variables, 2nd Edition, Springer-

Verlag, 1977.
2. J.R. Munkres, Analysis on Manifolds, Addison-Wesley,1991.

Text Reference 3. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw-
Hill, 1984.
4. M. Spivak, Calculus on Manifolds, A Modern Approach to Classical The-

orems of Advanced Calculus, W. A. Benjamin, Inc., 1965.
Functions on Euclidean spaces, continuity, differentiability, partial and direc-

tional derivatives, Chain Rule, Taylor‘s Theorem, Inverse Function Theorem,
Implicit Function Theorem, Regular and critical values, Applications. Rie-
mann Integral of real-valued functions on Euclidean spaces, measure zero sets,
Description Fubini‘s Theorem, Partition of unity, change of variables, Integration by parts.
Partition of unity, change of variables, Integration by parts. Integration on
chains, tensors, differential forms, Poincare Lemma, singular chains, Stokes‘
Theorem for integrals of differential forms on chains (general version), Funda-
mental theorem of calculus.
Course Code MA 412

Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. J.B. Conway, Functions of One Complex Variable, 2nd Edition, Narosa,

New Delhi, 1978.
2. T.W. Gamelin, Complex Analysis, Springer International Edition, 2001.

Text Reference
3. R. Remmert, Theory of Complex Functions, Springer Verlag, 1991.
4. A.R. Shastri, An Introduction to Complex Analysis, Macmilan India,

New Delhi, 1999.
Complex numbers and the point at infinity. Analytic functions. Cauchy-

Riemann conditions. Mappings by elementary functions. Riemann surfaces.
Conformal mappings. Contour integrals, Cauchy-Goursat Theorem. Uniform
Description
convergence of sequences and series. Taylor and Laurent series. Isolated singu-
larities and residues. Evaluation of real integrals. Zeroes and poles, Maximum
Modulus Principle, Argument Principle, Rouche’s theorem.
37
Course Code MA 414

Course Name Algebra 1
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 401 (Linear Algebra), MA 419 (Basic Algebra)
1. M. Artin, Algebra, Prentice Hall of India, 1994.
2. D.S. Dummit and R. M. Foote, Abstract Algebra, 2nd Edition, John

Wiley, 2002.
Text Reference 3. J.A. Gallian, Contemporary Abstract Algebra, 4th Edition, Narosa,
1999.
4. N. Jacobson, Basic Algebra I, 2nd Edition, Hindustan Publishing Co.,

1984, W.H. Freeman, 1985.
Fields, Characteristic and prime subfields, Field extensions, Finite, algebraic

and finitely generated field extensions, Classical ruler and compass construc-
tions, Splitting fields and normal extensions, algebraic closures. Finite fields,
Cyclotomic fields, Separable and inseparable extensions. Galois groups, Fun-
Description damental Theorem of Galois Theory, Composite extensions, Examples (in-
cluding cyclotomic extensions and extensions of finite fields). Norm, trace and
discriminant. Solvability by radicals, Galois’ Theorem on solvability. Cyclic
extensions, Abelian extensions, Polynomials with Galois groups Sn . Transcen-
dental extensions.
Course Code MA 417

Course Name Ordinary Differential Equations
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Hirsch, S. Smale and R. Deveney, Differential Equations, Dynamical

Systems and Introduction to Chaos, Academic Press, 2004
2. L. Perko, Differential Equations and Dynamical Systems, Texts in Ap-

plied Mathematics, Vol. 7, 2nd Edition, Springer Verlag, New York,
1998.
Text Reference
3. M. Rama Mohana Rao, Ordinary Differential Equations: Theory and
Applications. Affiliated East-West Press Pvt. Ltd., New Delhi, 1980.
4. D. A. Sanchez, Ordinary Differential Equations and Stability Theory:

An Introduction, Dover Publ. Inc., New York, 1968.
Review of solution methods for first order as well as second order equations,
Power Series methods with properties of Bessel functions and Legendre poly-
nomials.
Existence and Uniqueness of Initial Value Problems: Picard’s and Peano’s The-
orems, Gronwall’s inequality, continuation of solutions and maximal interval
of existence, continuous dependence.
Higher Order Linear Equations and linear Systems: fundamental solutions,
Description
Wronskian, variation of constants, matrix exponential solution, behaviour of
solutions.
Two Dimensional Autonomous Systems and Phase Space Analysis: critical
points, proper and improper nodes, spiral points and saddle points.
Asymptotic Behavior: stability (linearized stability and Lyapunov methods).
Boundary Value Problems for Second Order Equations: Green‘s function,
Sturm comparison theorems and oscillations, eigenvalue problems.
39
Course Code MA 419

Course Name Basic Algebra
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Artin, Algebra, Prentice Hall of India, 1994.
2. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd Edition, John

Wiley, 2002.
3. J. A. Gallian, Contemporary Abstract Algebra, 4th Edition, Narosa,

1999.
Text Reference
4. K. D. Joshi, Foundations of Discrete Mathematics, Wiley Eastern, 1989.
5. T. T. Moh, Algebra, World Scientific, 1992.
6. S. Lang, Algebra, 3rd Edition, Springer (India), 2004.
7. J. Stillwell, Elements of Algebra, Springer, 1994.
Description Continued on next page ...


Course Name Basic Algebra
Review of basics: Equivalence relations, partitions, division algorithm for in-
tegers, primes, unique factorization, congruences, Chinese Remainder Theo-
rem, Euler φ-function. Permutations, sign of a permutation, inversions, cycles
and transpositions. Rudiments of rings, fields, elementary properties, polyno-
mials in one, several variables, divisibility, irreducible polynomials, Division
algorithm, Remainder Theorem, Factor Theorem, Rational Zeros Theorem,
Relation between the roots and coefficients, Newton’s Theorem on symmetric
functions, Newton’s identities, Fundamental Theorem of Algebra. Rational
functions, partial fraction decomposition, unique factorization of polynomials
in several variables, Resultants and discriminants. Groups, subgroups, factor
groups, Lagrange‘s Theorem, homomorphisms, normal subgroups. Quotients
Description
of groups, Basic examples of groups: symmetric groups, matrix groups, group
of rigid motions of the plane and finite groups of motions. Cyclic groups,
generators and relations, Cayley‘s Theorem, group actions, Sylow Theorems.
Direct products, Structure Theorem for finite abelian groups. Simple groups
and solvable groups, nilpotent groups, simplicity of alternating groups, compo-
sition series, Jordan-Holder Theorem. Semidirect products. Free groups, free
abelian groups. Rings, Examples (including polynomial rings, formal power
series rings, matrix rings and group rings), ideals, prime and maximal ideals,
rings of fractions, Chinese Remainder Theorem for pairwise comaximal ide-
als. Euclidean Domains, Principal Ideal Domains and Unique Factorization
Domains. Polynomial rings over UFD’s.
41
Course Code MA 450

Course Name Independent Study
Total Credits 6
Type S
Lecture 0
Tutorial 0
Practical 6
Selfstudy 0
Half Semester N
Prerequisite Nil
Text Reference
Independent Study: The student should pursue a topic of his or her choice for
one semester under the supervision of a faculty member. The course is the
analogue of the Seminar courses that undergraduate students were required to
Description
take in previous years. The Independent Study should end with a presentation
to the supervising faculty member and the preparation of a brief report of about
ten pages.
Course Code MA 503

Course Name Functional Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 401 (Linear Algebra), MA 408 (Measure Theory)
1. J.B. Conway, A Course in Functional Analysis, 2nd Edition, Springer,

Berlin, 1990.
2. C. Goffman and G. Pedrick, A First Course in Functional Analysis,

Prentice-Hall, 1974.
3. E. Kreyzig, Introduction to Functional Analysis with Applications, John

Text Reference Wiley & Sons, New York, 1978.
4. B.V. Limaye, Functional Analysis, 2nd Edition, New Age International,

New Delhi, 1996.
5. A. Taylor and D. Lay, Introduction to Functional Analysis, Wiley, New

York, 1980.
Normed spaces. Continuity of linear maps. Hahn-Banach Extension and Sep-

aration Theorems. Banach spaces. Dual spaces and transposes. Uniform
Boundedness Principle and its applications. Closed Graph Theorem, Open
Description Mapping Theorem and their applications. Spectrum of a bounded opera-
tor. Examples of compact operators on normed spaces. Inner product spaces,
Hilbert spaces. Orthonormal basis. Projection theorem and Riesz Represen-
tation Theorem.
43
Course Code MA 504

Course Name Operators on Hilbert Spaces
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 503 (Functional Analysis)
1. B.V. Limaye, Functional Analysis, 2nd Edition, New Age International,

1996.
2. J.B. Conway, A Course in Functional Analysis, 2nd Edition, Springer,

1990.
Text Reference 3. C. Goffman and G. Pedrick, First Course in Functional Analysis, Prentice
Hall, 1974.
4. I. Gohberg and S. Goldberg, Basic Operator Theory, Birkhauser, 1981.
5. E. Kreyzig, Introduction to Functional Analysis with Applications, John

Wiley & Sons, 1978.
Adjoints of bounded operators on a Hilbert space, Normal, self-adjoint and

unitary operators, their spectra and numerical ranges. Compact operators on
Description
Hilbert spaces. Spectral theorem for compact self-adjoint operators. Applica-
tion to Sturm-Liouville Problems.
Course Code MA 510

Course Name Introduction to Algebraic Geometry
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 414 (Algebra 1)
1. S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers, Amer-

ican Mathematical Society, 1990.
2. W. Fulton, Algebraic Curves, Benjamin, 1969.
3. J. Harris, Algebraic Geometry: A First Course, Springer-Verlag, 1992.

Text Reference 4. M. Reid, Undergraduate Algebraic Geometry, Cambridge University
Press, Cambridge, 1990.
5. I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, Berlin,

1974.
6. R.J. Walker, Algebraic Curves, Springer- Verlag, Berlin, 1950.
Varieties: Affine and projective varieties, coordinate rings, morphisms and

rational maps, local ring of a point, function fields, dimension of a variety.
Curves: Singular points and tangent lines, multiplicities and local rings, in-
Description
tersection multiplicities, Bezout’s theorem for plane curves, Max Noether’s
theorem and some of its applications, group law on a nonsingular cubic, ratio-
nal parametrization, branches and valuations.
45
Course Code MA 515

Course Name Partial Differential Equations
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 410 (Multivariable Calculus), MA417 (Ordinary Differential Equations)
1. E. DiBenedetto, Partial Differential Equations, Birkhauser, 1995.
2. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathe-

matics, Vol. 19, American Mathematical Society, 1998.
Text Reference
3. F. John, Partial Differential Equations, 3rd Edition, Narosa, 1979.
4. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd

Edition, John Wiley and Sons, 1989.
Cauchy Problems for First Order Hyperbolic Equations: method of char-

acteristics, Monge cone. Classification of Second Order Partial Differential
Equations: normal forms and characteristics. Initial and Boundary Value
Problems: Lagrange-Green‘s identity and uniqueness by energy methods. Sta-
bility theory, energy conservation and dispersion. Laplace equation: mean
value property, weak and strong maximum principle, Green‘s function, Pois-
Description
son‘s formula, Dirichlet‘s principle, existence of solution using Perron’s method
(without proof). Heat equation: initial value problem, fundamental solution,
weak and strong maximum principle and uniqueness results. Wave equation:
uniqueness, D‘Alembert‘s method, method of spherical means and Duhamel’s
principle. Methods of separation of variables for heat, Laplace and wave equa-
tions.
Course Code MA 518

Course Name Spectral Approximation
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. M. Ahues, A. Largillier and B. V. Limaye, Spectral Computations for

Bounded Operators, Chapman and Hall/CRC, 2000.
2. F. Chatelin, Spectral Approximation of Linear Operators, Academic

Text Reference Press, 1983.
3. T. Kato, Perturbation Theory of Linear Operators, 2nd Edition, Springer-

Verlag, 1980.
Spectral decomposition. Spectral sets of finite type. Adjoint and product

spaces.
Convergence of operators: norm, collectively compact and ν convergence. Er-
Description ror estimates. Finite rank approximations based on projections and approxi-
mations for integral operators.
A posteriori error estimates. Matrix formulations for finite rank operators.
Iterative refinement of a simple eigenvalue. Numerical examples.
47
Course Code MA 521

Course Name Theory of Analytic Functions
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 403 (Real Analysis), MA412 (Complex Analysis)
1. L. Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979.
2. J.B. Conway, Functions of One Complex Variable, 2nd Edition, Narosa,

1978.
3. T.W. Gamelin, Complex Analysis, Springer International, 2001.

Text Reference
4. R. Narasimhan, Theory of Functions of One Complex Variable, Springer
(India), 2001.
5. W. Rudin, Real and Complex Analysis, 3rd Edition, Tata McGraw-Hill,

1987.
Maximum Modulus Theorem. Schwarz Lemma. Phragmen-Lindelof Theorem.

Riemann Mapping Theorem. Weierstrass Factorization Theorem. Runge‘s
Description
Theorem. Simple connectedness. Mittag-Leffler Theorem. Schwarz Reflection
Principle. Basic properties of harmonic functions. Picard Theorems.
Course Code MA 523

Course Name Basic Number Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 419 (Basic Algebra)
1. W.W. Adams and L.J. Goldstein, Introduction to the Theory of Num-

bers, 3rd Edition, Wiley Eastern, 1972.
2. A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge

Text Reference University Press, 1984.
3. I. Niven and H.S. Zuckerman, An Introduction to the Theory of Numbers,

4th Edition, Wiley, 1980.
Infinitude of primes, discussion of the Prime Number Theorem, infinitude of

primes in specific arithmetic progressions, Dirichlet‘s theorem (without proof).
Arithmetic functions, Mobius inversion formula. Structure of units modulo n,
Euler’s phi function. Congruences, theorems of Fermat and Euler, Wilson’s
theorem, linear congruences, quadratic residues, law of quadratic reciprocity.
Description Binary quadratics forms, equivalence, reduction, Fermat’s two square theorem,
Lagrange’s four square theorem. Continued fractions, rational approximations,
Liouville’s theorem, discussion of Roth’s theorem, transcendental numbers,
transcendence of e and π. Diophantine equations: Brahmagupta’s equation
(also known as Pell’s equation), The equation, Fermat’s method of descent,
discussion of the Mordell equation.
49
Course Code MA 524

Course Name Algebraic Number Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number

Theory, 2nd Edition, Springer-Verlag, Berlin, 1990.
Text Reference 2. S. Lang, Algebraic Number Theory, Addison- Wesley, 1970.
3. D. A. Marcus, Number Fields, Springer-Verlag, 1977.
Algebraic number fields. Localisation, discrete valuation rings. Integral ring

extensions, Dedekind domains, unique factorisation of ideals. Action of the
Galois group on prime ideals. Valuations and completions of number fields,
Description discussion of Ostrowski’s theorem, Hensel’s lemma, unramified, totally rami-
fied and tamely ramified extensions of p-adic fields. Discriminants and Ram-
ification. Cyclotomic fields, Gauss sums, quadratic reciprocity revisited. The
ideal class group, finiteness of the ideal class group, Dirichlet units theorem.
Course Code MA 525

Course Name Dynamical Systems
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 417 (Ordinary Differential Equations)
1. L. Perko, Differential Equations and Dynamical Systems, Springer Ver-

lag, 1991.
2. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems

and Linear Algebra, Academic Press, 174.
Text Reference
3. P. Hartman, Ordinary Differential Equations, 2nd edition, SIAM 2002.
4. C. Chicone, Ordinary Differential Equations with Applications, 2nd Edi-

tion, Springer, 2006.
Linear Systems: Review of stability for linear systems of two equations.

Local Theory for Nonlinear Planar Systems: Flow defined by a differen-
tial equation, Linearization and stable manifold theorem, Hartman-Grobman
theorem, Stability and Lyapunov functions, Saddles, nodes, foci, centers and
Description nonhyperbolic critical points. Gradient and Hamiltonian systems.
Global Theory for Nonlinear Planar Systems: Limit sets and attractors,
Poincare map, Poincare Benedixson theory and Poincare index theorem.
Bifurcation Theory for Nonlinear Systems: Structural stability and
Peixoto’s theorem, Bifurcations at nonhyperbolic equilibrium points.
51
Course Code MA 526

Course Name Commutative Algebra
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. D. Eisenbud, Commutative Algebra (with a view toward algebraic ge-

ometry), Graduate Texts in Mathematics 150, Springer-Verlag, 2003.
2. H. Matsumura, Commutative ring theory, Cambridge Studies in Ad-

Text Reference vanced Mathematics No. 8, Cambridge University Press, 1980.
3. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised edition, Cam-

bridge Studies in Advanced Mathematics No. 39, Cambridge University
Press, 1998.
Dimension theory of affine algebras: Principal ideal theorem, Noether nor-

malization lemma, dimension and transcendence degree, catenary property
of affine rings, dimension and degree of the Hilbert polynomial of a graded
ring, Nagata’s altitude formula, Hilbert’s Nullstellensatz, finiteness of inte-
gral closure. Associated primes of modules, degree of the Hilbert polyno-
mial of a graded module, Hilbert series and dimension, Dimension theorem,
Hilbert-Samuel multiplicity, associativity formula for multiplicity, Complete
local rings: Basics of completions, Artin-Rees lemma, associated graded rings
Description of filtrations, completions of modules, regular local rings Basic Homological
algebra: Categories and functors, derived functors, Hom and tensor prod-
ucts, long exact sequence of homology modules, free resolutions, Tor and Ext,
Koszul complexes. Cohen-Macaulay rings: Regular sequences, quasi-regular
sequences, Ext and depth, grade of a module, Ischebeck’s theorem, Basic
properties of Cohen-Macaulay rings, Macaulay’s unmixed theorem, Hilbert-
Samuel multiplicity and Cohen-Macaulay rings, rings of invariants of finite
groups.Optional Topics: Face rings of simplicial complexes, shellable simpli-
cial complexes and their face rings. Dedekind Domains and Valuation Theory.
Course Code MA 528

Course Name Hyperplane Arrangements
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. P. Orlik and H. Terao, Arrangements of hyperplanes, Springer, 1992.
2. C. De Concini and C. Procesi,.Topics in hyperplane arrangements, poly-

topes and boxsplines. Springer, 2011.
Text Reference
3. A. Dimca Hyperplane arrangements. An introduction. Springer, 2017.
4. M. Aguiar and S. Mahajan. Topics in hyperplane arrangements. AMS,

2017.
Faces, flats, cones, lunes. Distance functions Birkhoff monoid, Tits monoid
and Janus monoid Lie and Zie elements Eulerian idempotents, Dynkin idem-
Description
potents, JoyalKlyachko-Stanley theorem Orlik-Solomon algebra Incidence al-
gebras and operads.
53
Course Code MA 530

Course Name Nonlinear Analysis
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. M.C. Joshi and R.K. Bose, Some Topics in Nonlinear Functional Analy-
sis, Wiley Eastern Ltd., New Delhi, 1985.
Text Reference
2. E. Zeilder, Nonlinear Functional Analysis and Its Applications, Vol. I
(Fixed Point Theory), Springer Verlag, Berlin, 1985.
Fixed Point Theorems with Applications: Banach contraction mapping theo-

rem, Brouwer fixed point theorem, LeraySchauder fixed point theorem. Cal-
culus in Banach spaces: Gateaux as well as Frechet derivatives, chain rule,
Taylor’s expansions, Implicit function theorem with applications, subdiffer-
Description
ential. Monotone Operators: maximal monotone operators with properties,
surjectivity theorem with applications. Degree theory and condensing opera-
tors with applications.
Course Code MA 532

Course Name Analytic Number Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 414 (Algebra I), MA 412 (Complex Analysis)
1. S. Lang, Algebraic Number Theory, AddisonWesley, 1970.
2. J.P. Serre, A Course in Arithmetic, SpringerVerlag, 1973.

Text Reference
3. T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976.
The Wiener-Ikehara Tauberian theorem, the Prime Number Theorem. Dirich-

let’s theorem for primes in an Arithmetic Progression. Zero free regions for the
Riemann-zeta function and other L-functions. Euler products and the func-
tional equations for the Riemann zeta function and Dirichlet L-functions. Mod-
ular forms for the full modular group, Eisenstein series, cusp forms, structure
of the ring of modular forms. Hecke operators and Euler product for modular
forms. The L-function of a modular form, functional equations. Modular forms
Description
and the sums of four squares. Optional topics: Discussion of L-functions of
number fields and the Chebotarev Density Theorem. Phragmen-Lindelof Prin-
ciple, Mellin inversion formula, Hamburger’s theorem. Discussion of Modular
forms for congruence subgroups. Discussion of Artin’s holomorphy conjecture
and higher reciprocity laws. Discussion of elliptic curves and the Shimura-
Taniyama conjecture (Wiles’ Theorem)
55
Course Code MA 533

Course Name Advanced Probability Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 6
Half Semester N
Prerequisite Nil
1. P. Billingsley, Probability and Measure, 3rd Edition, John Wiley and

Sons, New York, 1995.
2. J. Rosenthal, A First Look at Rigorous Probability, World Scientific,

Singapore, 2000.
Text Reference
3. A.N. Shiryayev, Probability, 2 nd Edition, Springer, New York, 1995.
4. K.L. Chung, A Course in Probability Theory, Academic Press, New York,

1974.
Probability measure, probability space, construction of Lebesgue measure, ex-

tension theorems, limit of events, Borel-Cantelli lemma. Random variables,
Random vectors, distributions, multidimensional distributions, independence.
Expectation, change of variable theorem, convergence theorems. Sequence of
Description
random variables, modes of convergence. Moment generating function and
characteristics functions, inversion and uniqueness theorems, continuity theo-
rems, Weak and strong laws of large number, central limit theorem. Radon
Nikodym theorem, definition and properties of conditional expectation,
Course Code MA 534

Course Name Modern Theory of PDE
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 503 (Functional Analysis), MA 515 (Partial Differential Equations)
1. S. Kesavan, Topics in Functional Analysis Wiley Eastern Ltd., New

Delhi, 1989.
2. M. Renardy and R.C. Rogers, An Introduction to Partial Differential

Text Reference Equations,2nd Edition, Springer Verlag International Edition, New York,
2004.
3. L.C. Evans, Partial Differential Equations, American Mathematical So-

ciety, Providence, 1998.
Theory of distributions: supports, test functions, regular and singular distri-

butions, generalised derivatives. Sobolev Spaces: definition and basic prop-
erties, approximation by smooth functions, dual spaces, trace and imbedding
results (without proof). Elliptic Boundary Value Problems: abstract varia-
Description
tional problems, Lax-Milgram Lemma, weak solutions and wellposedness with
examples, regularity result, maximum principles, eigenvalue problems. Semi-
group Theory and Applications: exponential map, C0 -semigroups, Hille-Yosida
and Lummer-Phillips theorems, applications to heat and wave equations.
57
Course Code MA 538

Course Name Representation Theory of Finite Groups
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. M. Burrow, Representation Theory of Finite Groups, Academic Press,

1965.
2. N. Jacobson, Basic Algebra II, Hindustan Publishing Corporation, 1983.

Text Reference
4. J.-P. Serre, Linear Representation of Groups, Springer-Verlag, 1977.
Representations, Subrepresentations, Tensor products, Symmetric and Alter-

nating Squares.Characters, Schur’s lemma, Orthogonality relations, Decompo-
sition of regular representation, Number of irreducible representations, canon-
ical decomposition and explicit decompositions. Subgroups, Product groups,
Description Abelian groups. Induced representations.Examples: Cyclic groups, alternat-
ing and symmetric groups. Integrality properties of characters, Burnside’s pq
theorem. The character of induced representation, Frobenius Reciprocity The-
orem, Meckey’s irreducibility criterion, Examples of induced representations,
Representations of supersolvable groups.
Course Code MA 539

Course Name Spline Theory and Variational Methods
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. C. De Boor, A Practical Guide to Splines, Springer-Verlag, Berlin, 1978.
2. H.N. Mhaskar and D.V. Pai, Fundamentals of Approximation Theory,

Text Reference Narosa Publishing House, New Delhi, 2000.
3. P.M. Prenter, Splines and Variational Methods, Wiley-Interscience, 1989.
Even Degree and Odd Degree Spline Interpolation, end conditions, error anal-
ysis and order of convergence. Hermite interpolation, periodic spline interpo-
Description lation. B-Splines, recurrence relation for B-splines, curve fitting using splines,
optimal quadrature. Tensor product splines, surface fitting, orthogonal spline
collocation methods.
59
Course Code MA 540

Course Name Numerical Methods for Partial Differential Equations
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 515 (Partial Differential Equations), SI 517 (Numerical Analysis)
1. K. W. Morton and D. Mayers, Numerical Solution for Partial Differential

Equations, 2nd edition, Cambridge, 2005.
2. G. D. Smith, Numerical Solutions of Partial Differential Equations, 3rd

Edition, Calrendorn Press, 1985.
3. J. C. Strikwerda, Finite difference Schemes and Partial Differential Equa-

tions, Wadsworth and Brooks/ Cole, 1989.
Text Reference 4. J. W. Thomas, Numerical Partial Differential Equations : Finite Differ-

ence Methods, Texts in Applied Mathematics, Vol. 22, Springer Verlag,
1999.
5. J. W. Thomas, Numerical Partial Differential Equations: Conservation

Laws and Elliptic Equations, Texts in Applied Mathematics, Vol. 33,
Springer Verlag, 1999.
6. R. Mitchell and S. D. F. Griffiths, The Finite Difference Methods in

Partial Differential Equations, Wiley and Sons, NY, 1980.
Finite differences: Grids, Finite-difference approximations to derivatives. Lin-

ear Transport Equation: Upwind, Lax-Wendroff and Lax-Friedrich schemes,
von-Neumann stability analysis, CFL condition, Lax-Richtmyer equivalence
theorem, Modified equations, Dissipation and dispersion. Heat Equation: Ini-
tial and boundary value problems (Dirichlet and Neumann), Explicit and im-
plicit methods (Backward Euler and Crank-Nicolson schemes) with consistency
and stability, Discrete maximum principle, ADI methods for two dimensional
Description heat equation (including LOD algorithm). Poisson‘s Equation: Finite differ-
ence scheme for initial and boundary value problems, Discrete maximum prin-
ciple, Iterative methods for linear systems (Jacobi, Gauss-Seidel, SOR meth-
ods and Conjugate Gradient method), Peaceman-Rachford algorithm (ADI)
for linear systems. Wave Equation: Explicit schemes and their stability anal-
ysis, Implementation of boundary conditions.Lab Component: Exposure to
MATLAB and computational experiments based on the algorithms discussed
in the course.
Course Code MA 556

Course Name Differential Geometry
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 410 (Multivariable Calculus)
1. M. doCarmo, Differential Geometry of Curves and Surfaces, Prentice

Hall, 1976.
2. B. O‘Neill, Elementary Differential Geometry, Academic Press, 1966.

Text Reference
3. J.J. Stoker, Differential Geometry, Wiley-Interscience, 1969.
4. J. A. Thorpe, Elementary Topics in Differential Geometry, Springer (In-

dia), 2004.
Graphs and level sets of functions on Euclidean spaces, vector fields, integral
curves of vector fields, tangent spaces. Surfaces in Euclidean spaces, vec-
tor fields on surfaces, orientation, Gauss map. Geodesics, parallel transport,
Description
Weingarten map. Curvature of plane curves, arc length and line integrals,
Curvature of surfaces. Parametrized surfaces, local equivalence of surfaces.
Gauss-Bonnet Theorem, Poincare-Hopf Index Theorem.
61
Course Code MA 562

Course Name The Mathematical Theory of Finite Elements
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 515 (Partial Differential Equations), MA 503 (Functional Analysis)
1. K. E. Brenner and R. Scott, The Mathematical Theory of Finite Element

Methods, Springer- Verlag, 1994.
2. P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North
Holland, 1978.
Text Reference 3. C. Johnson, Numerical solutions of Partial Differential Equations by Fi-
nite Element Methods, Cambridge University Press, 1987.
4. C. Mercier, Lectures on Topics in Finite Element Solution of Elliptic

Problems, TIFR Lectures on Mathematics and Physics Vol. 63, Narosa,
1979.
Sobolev Spaces: basic elements, Poincare inequality. Abstract variational

formulation and elliptic boundary value problem. Galerkin formulation and
Cea’s Lemma. Construction of finite element spaces. Polynomial approxima-
tions and interpolation errors. Convergence analysis: Aubin-Nitsche duality
Description argument; non-conforming elements and numerical integration; computation
of finite element solutions. Parabolic initial and boundary value problems:
semidiscrete and completely discrete schemes with convergence analysis. Lab
component: Implementation of algorithms and computational experiments us-
ing MATLAB
Course Code MA 581

Course Name Elements of Differential Topology
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 410 (Multivariable Calculus)
1. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geome-

try Methods and Applications Part II: The Geometry and Topology of
Manifolds, Springer-Verlag, 1985.
2. V. Guillemin and A Pollack, Differential Topology Prentice-Hall Inc.,

Text Reference Englewood Cliffs, New Jersey, 1974.
3. J. Milnor, Topology from the Differential View-point, University Press

of Virginia, Charlottsville 1990.
4. A. R. Shastri, Elements of Differential Topology, CRC Press, 2011.
Differentiable Manifolds in Rn : Review of inverse and implicit function the-

orems; tangent spaces and tangent maps; immersions; submersions and em-
beddings. Regular Values: Regular and critical values; regular inverse image
theorem; Sard’s theorem; Morse lemma. Transversality: Orientations of man-
ifolds; oriented and mod 2 intersection numbers; degree of maps. Application
Description to the Fundamental Theorem of Algebra.
*Lefschetz theory of vector fields and flows: Poincare-Hopf index theorem;
Gauss-Bonnet theorem.
*Abstract manifolds: Examples such as real and complex projective spaces
and Grassmannian varieties; Whitney embedding theorems.
(*indicates expository treatment intended for these parts of the syllabus.)
63
Course Code MA 593

Course Name Project (Optional)
Total Credits 4
Type
Lecture
Tutorial
Practical
Selfstudy
Half Semester
Prerequisite
Text Reference 1.
Description
Course Code MA 598

Course Name Project 2 (Optional)
Total Credits 6
Type
Lecture
Tutorial
Practical
Selfstudy
Half Semester
Prerequisite
Text Reference 1.
Description
65
Course Code MA 5101

Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Alge-

bra, Addison Wesley, 1969.
2. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd Edition, John

Wiley, 2002.
Text Reference 3. N. Jacobson, Basic Algebra I and II, 2nd Edition, W. H. Freeman, 1985
and 1989.
5. O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Springer, 1975.
Modules, submodules, quotient modules and module homomorphisms. Gener-

ation of modules, direct sums and free modules. Tensor products of modules.
Exact sequences. Hom and Tensor duality. Finitely generated modules over
principal ideal domains, invariant factors, elementary divisors, rational canon-
Description ical forms. Applications to finitely generated abelian groups and linear trans-
formations. Noetherian rings and modules, Hilbert basis theorem, Primary
decomposition of ideals in noetherian rings. Integral extensions, Going-up and
Going-down theorems, Extension and contraction of prime ideals, Noether’s
Normalization Lemma, Hilbert’s Nullstellensatz.
Course Code MA 5102

Course Name Basic Algebraic Topology
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 406 (General Topology)
1. M. J. Greenberg and J. R. Harper, Algebraic Topology, Benjamin, 1981.
2. W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.
3. A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge,

2002.
4. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag,

Berlin, 1991.
Text Reference 5. J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
6. J. J. Rotman, An Introduction to Algebraic Topology, Springer (India),

2004.
7. H. Seifert and W. Threlfall, A Textbook of Topology, translated by M.

A. Goldman, Academic Press, 1980.
8. J. W. Vick, Homology Theory: An Introduction to Algebraic Topology,

2nd Edition, Springer-Verlag, 1994.
Paths and homotopy, homotopy equivalence, contractibility, deformation re-

tracts. Basic constructions: cones, mapping cones, mapping cylinders, suspen-
sion. Fundamental groups. Examples (including the fundamental group of the
circle) and applications (including Fundamental Theorem of Algebra, Brouwer
Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two).
Van Kampen‘s Theorem, Covering spaces, lifting properties, deck transforma-
Description tions. Universal coverings (existence theorem optional). Singular Homology.
Mayer-Vietoris Sequences. Long exact sequence of pairs and triples. Homotopy
invariance and excision theorem (without proof). Applications of homology:
Jordan-Brouwer separation theorem, invariance of dimension, Hopf’s Theo-
rem for commutative division algebras with identity, Borsuk-Ulam Theorem,
Lefschetz Fixed Point Theorem.
67
Course Code MA 5103

Course Name Algebraic Combinatorics
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 401 (Linear Algebra)
1. N. Alon, Combinatorial Nullstellensatz, Combinatorics, Probability, and

Computing, Vol. 8 (1999), pp. 7-29.
2. R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and

More, Springer, 2013.
3. C. Godsil and G. F. Royle, Algebraic Graph Theory, Springer, 2001.

Text Reference
4. F. Chung, Spectral Graph Theory, CBMS Regional Conference Series in
Math., No. 92, American Mathematical Society, 1991.
5. L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, De-

partment of Computer Science, University of Chicago, Preliminary ver-
sion, 1992.
A selection of topics from the following:

Algebraic Graph theory: adjacency and Laplacian matrices of a graph, Matrix-
Tree theorem, Cycle space and Bond space.
Algebraic Sperner theory: Sperner property of posets, algebraic characteriza-
tion of strong Sperner property, unimodality of q-binomial coefficients.
Young Tableaux: Up-Down operators on the Young lattice and counting
Description tableaux, RSK correspondence.
Enumeration under group action: Burnside’s lemma, Polya theory.
Spectral Graph theory: Isoperimetric problems, Flows and Cheeger constants,
Quasirandomness, expanders, and eigenvalues, random walks on graphs. The
Combinatorial Nullstellensatz and some of its applications. Linear Algebra
methods in Combinatorics. Association Schemes. Electrical Networks and
resistances. Connections to Graph sparsification.
Course Code MA 5104

Course Name Hyperbolic Conservation Laws
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 515 Partial Differential Equations
1. L. C. Evans, Partial Differential Equations, American Mathematical So-

ciety, 2010.
2. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic

Systems of Conservation Laws, Springer, 1996.
Text Reference
3. A. Bressan, Hyperbolic Systems of Conservation Laws – The One-
Dimensional Cauchy Problem, Oxford University Press, 2000.
4. J. Smoller, Shock Waves and ReactionDiffusion Equations, Springer,

1994
Basic Concepts: Definition and examples, Loss of regularity, Weak solution,

Rankine-Hugoniot jump condition, Entropy solution. Scalar Conservation
Laws: Existence of an entropy solution, Uniqueness of the entropy solution,
Asymptotic behavior of the entropy solution, The Riemann problem.
Description
System of Conservation Laws: Linear hyperbolic system with constant coeffi-
cients, Nonlinear case, Simple waves and Riemann invariants, Shock waves and
contact discontinuities, Characteristic curves and entropy conditions, Solution
of the Riemann problem, The Riemann problem for the psystem.
69
Course Code MA5105

Course Name Coding Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 401 (Linear Algebra)
1. J. H. van Lint, Introduction to Coding Theory, Springer, 1999.
2. W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes,

Cambridge University Press, 2003.
Text Reference 3. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting
Codes, North-Holland, 1977.
4. S. Ling and C. Xing, Coding Theory: A First Course, Cambridge Uni-

versity Press, 2004.
Basic Concepts: Idea behind use of codes, block codes and linear codes, rep-
etition codes, nearest neighbour decoding, syndrome decoding, requisite basic
ideas in probability, Shannon’s theorem (without proof). Good linear and non-
linear codes: Binary Hamming codes, dual of a code, constructing codes by
various operations, simplex codes, Hadamard matrices and codes constructed
from Hadamard and conference matrices, Plotkin bound and various other
bounds, Gilbert-Varshamov bound. Reed-Muller and related codes: First or-
der Reed-Muller codes, RM code of order r, Decoding and Encoding using the
algebra of finite field with characteristic two. Perfect codes: Weight enumer-
ators, Kratchouwk polynomials, Lloyd’s theorem, Binary and ternary Golay
Description
codes, connections with Steiner systems. Cyclic codes: The generator and
the check polynomial, zeros of a cyclic code, the idempotent generators, BCH
codes, Reed-Solomon codes, Quadratic residue codes, generalized RM codes.
Optional topics; Codes over Z4 : Quaternary codes over Z4 , binary codes de-
rived from such codes, Galois rings over Z4 , cyclic codes over Z4 . Goppa codes:
the minimum distance of Goppa codes, generalized BCH codes, decoding of
Goppa codes and their asymptotic behaviour.Algebraic geometry codes: alge-
braic curves and codes derived from them, Riemann-Roch theorem (statement
only) and applications to algebraic geometry codes.
Course Code MA5106

Course Name Introduction to Fourier Analysis
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. R. S. Strichartz, A Guide to Distributions and Fourier Transforms, CRC

Press, 1994.
2. E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Prince-

Text Reference ton University Press, 2003.
3. I. Richards and H. Youn, Theory of Distributions: A Nontechnical In-

troduction, Cambridge University Press, 1990.
Basic Properties of Fourier Series: Uniqueness of Fourier Series, Convolutions,

Cesaro and Abel Summability, Fejer’s theorem, Poisson Kernel and Dirich-
let problem in the unit disc. Mean square Convergence, Example of Con-
tinuous functions with divergent Fourier series. Distributions and Fourier
Transforms: Calculus of Distributions, Schwartz class of rapidly decreas-
ing functions, Fourier transforms of rapidly decreasing functions, Riemann
Description Lebesgue lemma, Fourier Inversion Theorem, Fourier transforms of Gaus-
sians. Tempered Distributions: Fourier transforms of tempered distributions,
Convolutions, Applications to PDE’s (Laplace, Heat and Wave Equations),
Schrodinger-Equation and Uncertainty principle. Paley-Wienner Theorems,
Poisson Summation Formula: Radial Fourier transforms and Bessel‘s func-
tions. Hermite functions.
71
Course Code MA 5107

Course Name Continuum Mechanics
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
MA 417 (Ordinary Differential Equations) and MA 410 (Multivariable Calcu-
Prerequisite
lus)
1. O. Gonzalez and A. M. Stuart, A First Course in Continuum mechanics,

2. M. Gurtin, An Introduction to Continuum Mechanics, Academic press,

1981.
3. J. N. Reddy, An Introduction to Continuum Mechanics with Applica-

tions, Cambridge University Press, 2008.
Text Reference
4. J. N. Reddy, Principles of Continuum Mechanics: A Study of Conserva-
tion Principles with Applications, Cambridge University Press, 2010.
5. Y. R. Talpaert, Tensor analysis and Continuum Mechanics, Springer,

2003.
6. R. Temam and A. Miranville, Mathematical Modelling in Continuum

Mechanics, Cambridge University Press, 2005.
Preliminaries: Tensor algebra and calculus, Continuum mass and force con-
cepts. Kinematics of Continuous Media: Deformation, Changes in distance,
angles, volume, area, Particle derivatives, Measures of strain: Cauchy-Green
strain tensor. Balance Laws of motion: Lagrangean and Eulerian forms of
Description Conservation laws for mass, linear and angular momentum, and energy, Frame-
indifference. Constitutive relations: Constitutive laws for solids and fluids,
principle of material frame indifference, discussion of isotropy, linearized elas-
ticity, fluid mechanics. indifference, discussion of isotropy, linearized elasticity,
fluid mechanics.
Course Code MA 5108

Course Name Lie Groups and Lie Algebras
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 401 (Linear Algebra) and MA 403 (Real Analysis)
1. J.Stillwell, Naive Lie Theory, Springer, 2008.

Text Reference 2. A. Kirillov Jr., Introduction to Lie Groups and Lie Algebras, Cambridge
University Press, 2008.
Introduction, Examples: Rotations of the plane, Quaternions and space ro-

tations, SU(2) and SO(3), The Cartan-Dieudonné Theorem, Quaternions and
rotations in R4, SU(2)xSU(2) and SO(4). Matrix Lie groups: definitions and
examples. The symplectic, orthogonal and unitary groups, connectedness,
compactness. Maximal tori. centres and discrete subgroups The exponential
Description map, Lie algebras The matrix exponential, tangent spaces, the Lie algebra of a
Lie group. Complexification, the matrix logarithm, the exponential map, One
parameter subgroups, the functor from Lie groups to Lie algebras The adjoint
mapping, normal subgroups and Lie algebras The Campbell-Baker-Hausdorff
Theorem, simple connectivity, simply connected Lie groups and their charac-
terization by Lie algebras, covering groups.
73
Course Code MA5109

Course Name Graph Theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. D.B. West, Introduction to Graph Theory, Prentice Hall of India, 2001.
2. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications,

Text Reference Springer-Verlag, 2008.
3. R. Diestel, Introduction to Graph Theory, Springer-Verlag, 2010.
Basic Concepts: various kinds of graphs, simple graphs, complete graph, walk,
tour, path and cycle, Eulerian graph, bipartite graph (characterization), Havel-
Hakimi theorem and Erdos-Gallai theorem (statement only), hypercube graph,
Petersen graph, trees, forests and spanning subgraphs, distances, radius, di-
ameter, center of a graph, the number of distinct spanning trees in a complete
graph. Trees: Kruskal and Prim algorithms with proofs of correctness, Di-
jkstra’s a algorithm, Breadth first and Depth first search trees, rooted and
binary trees, Huffman’s algorithm Matchings: augmenting path, Hall’s match-
ing theorem, vertex and edge cover, independence number and their connec-
tions, Tutte’s theorem for the existence of a 1-factor in a graph, Connectiv-
ity k-vertex and edge connectivity, blocks, characterizations of 2- connected
graphs, Menger’s theorem and applications, Network flows, Ford- Fulkerson
algorithm, Supply-demand theorem and the Gale-Ryser theorem on degree
Description
sequences of bipartite graphs Graph Colourings chromatic number, Greedy
algorithm, bounds on chromatic numbers, interval graphs and chordal graphs
(with simplicial elimination ordering), Brook’s theorem and graphs with no
triangles but large chromatic number, chromatic polynomials. Hamilton prop-
erty Necessary conditions, Theorems of Dirac and Ore, Chvatal’s theorem and
toughness of a graph, Non-Hamiltonian graphs with large vertex degrees. Pla-
nar graphs Embedding a graph on plane, Euler’s formula, non-planarity of
K5 and K3,3, classification of regular polytopes, Kuratowski’s theorem (no
proof), 5-colour theorem. Ramsey theory Bounds on R(p, q), Bounds on
Rk(3): colouring with k colours and with no monochromatic K3, application
to Schur’s theorem, Erdos and Szekeres theorem on points in general position
avoiding a convex m-gon.
Course Code MA5110

Course Name Non-commutative Algebra
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 419 (Basic Algebra)
1. N. Jacobson, Basic Algebra, Vol. I and II, Dover Publications, 2009.
2. S. Lang, Algebra, 3rd Edition, Springer Verlag, 2002

Text Reference 3. T. Y. Lam, A First Course in Noncommutative Rings, 2nd edition,
Springer, 2001.
4. A. Knapp, Advanced Algebra, Birkhauser, 2007.
Wedderburn-Artin Theory: semisimple rings and modules, Weddereburn and

Artin’s structure theorem of semisimple rings.
Jacobson radical theory: Jacobson radical, Jacobson semisimple rings (or
semiprimitive rings), nilpotent ideal, Hopkins and Levitzki theorem, Jacob-
son radical under base change, semisimplicity of group rings.
Prime and primitive rings: prime and semiprime ideal (and ring), primitive
ring and ideal, Jacobson-Chevalley’s density theorem, Structure theorem for
left primitive rings, Jacobson-Herstein’s commutativity theorem.
Description
Introduction to division rings: Wedderburn’s (little) theorem, algebraic divi-
sion algebras over reals (Frobenius theorem), construction of division algebras,
polynomials over division rings.
Ordered structures in rings: orderings and preorderings in rings, formally real
ring, ordered division rings.
Local rings, semilocal rings and idempotents: Krull-Schmidt-Azumaya theo-
rem on uniqueness of indecomposable summands of a module, stable range of
a ring and cancellation of modules. Brauer group and Clifford algebras.
75
Course Code MA 5111

Course Name Theory of Finite Semigroups
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite
1. P. Grillet. Semigroups. an introduction to the structure theory, Marcel

Dekker Inc., 1995
2. J. Rhodes and B. Steinberg, The q-theory of finite semigroups. Springer,

2009
Text Reference
3. B. Steinberg.Representation theory of finite monoids. Springer, 2016
4. M. Aguiar and S. Mahajan. Topics in hyperplane arrangements. AMS,

2017
Monoids and their linearized algebras Bands, left regular bands Hyperplane ar-
rangements Birkhoff monoid, Tits monoid and Janus monoid Idempotents and
Description
simple modules Quivers of band algebras Noncommutative zeta and Mobius
functions Karoubi envelopes of semigroups.
Course Code MA5112

Course Name Introduction to Mathematical Methods
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite MA 515 (Partial Differential Equations)
1. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for

Scientists and Engineers, McGraw-Hill Book Co., 1978.
2. R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol. I &

II, Wiley Eastern, 1975.
3. J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathe-

matics, Springer Verlag, 1985.
4. S. G. Mikhlin, Variation Methods in Mathe-matical Physics, Pergaman

Press, Oxford 1964.
Text Reference
5. J. A. Murdock, Perturbations Theory and Methods, John Wiley and
Sons, 1991.
6. P. D. Miller, Applied asymptotic analysis, American Mathematical So-

ciety, 2006.
7. M. L. Krasnov et.al., Problems and exercises in the calculus of variations,

Mir Publishers, 1975.
8. M. Krasnov et. al., Problems and exercises in integral equations, Mir

Publishers, 1971.
Asymptotic expansions: Watson‘s lemma, method of stationary phase and

saddle point method. Applications to differential equations. Behaviour of
solutions near an irregular singular point, Stoke’s phenomenon. Method of
strained coordinates and matched asymptotic expansions, Lindstedt expan-
Description
sions. Calculus of variations: Classical methods.Integral equations: Volterra
integral equations of first and second kind. Iterative methods and Neumann
series.
77
Course Code MA 5113

Course Name Category Theory 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Aguiar and Mahajan, Monoidal functors, species and Hopf algebras,

American Mathematical Society, 2010.
2. Awodey, Category theory, Oxford University Press, 2010.
3. Borceau, Handbook of categorical algebra, Volumes 1, 2, 3, Cambridge

Text Reference 4. Leinster, Higher categories, Higher operads, Cambridge University Press,
2004.
5. Leinster, Basic category theory, Cambridge University Press, 2014.
6. Mac Lane, Categories for the working mathematician, Springer, 1998.
7. Riehl, Category theory in context, Aurora, Dover Publications, 2016.
Categories, functors, natural transformations. Limits and colimits. Adjoint

functors and universal constructions. Functor categories, comma categories,
quotient categories. Cauchy completeness, Karoubi envelopes. Cartesian cate-
Description
gories, group objects. The above concepts can be motivated and discussed by
connecting them to other areas of mathematics depending on the interests of
the instructor and students.
Course Code MA 5115

Course Name Hopf Algebras
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Sweedler, Hopf algebras, W.A.Benjamin, Inc, NewYork, 1969.
2. E. Abe, Hopf algebras, CUP, 1980.
3. C. Kassel, Quantum groups, Springer, 1995.
4. M.Aguiar, S. Mahajan. Bimonoids for hyperplane arrangements, CUP,

Text Reference 2020
5. D.Radford. Hopf algebras, World Scientific, 2012.
6. R. Underwood. Fundamentals of Hopf algebras, Springer, 2015.
7. M.Aguiar, S. Mahajan. Monoidal functors, species and Hopf algebras,

AMS, 2010
Algebras, coalgebras and bialgebras, Convolution algebra, antipode and Hopf

algebras, Universal constructions, Tensor algebra, Symmetric and exterior al-
gebras, universal enveloping algebras, Group-likes, primitives and coradical fil-
Description
tration, Structure results. Borel-Hopf and Carter-MilnorMoore theorems, Hopf
algebras for hyperplane arrangements, Connection to affine group schemes and
quantum groups.
79
Course Code MA 5116

Course Name Species and Operads
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. F.Bergeron, G.Labelle, P.Leroux. Combinatorial species and tree-like

structures. CUP, 1998.
2. M.Markl, S.Shnider, J. Stasheff. Operads in Algebra, Topology and

Physics, AMS, 2002.
3. M.Aguiar, S. Mahajan. Bimonoids for hyperplane arrangements, CUP,

Text Reference 2020.
4. J-L.Loday, B.Vallette. Algebraic operads, Springer, 2012
5. T.Leinster. Higher operads, Higher categories. CUP, 2004.
6. M.Aguiar and S.Mahajan. Monoidal Functors, species and Hopf alge-

bras, AMS, 2010.
Species. Exponential Species, species of linear orders and other examples.

Cauchy, Hadamard and substitution products on species and universal con-
Description structions. Power series/Generating function of species Operads. Commuta-
tive, associative and Lie operads and other examples, Algebras over operads,
Koszul theory of Operads, Species and operads for hyperplane arrangements
Course Code MA 5118

Course Name Category Theory 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 6
Half Semester N
Prerequisite Nil

3. Borceau, Handbook of categorical algebra, Volumes 1, 2, 3, Cambridge

Text Reference 4. Leinster, Higher categories, Higher operads, Cambridge University Press,
2004.
6. Mac Lane, Categories for the working mathematician, Springer, 1998.
7. Riehl, Category theory in context, Aurora, Dover Publications, 2016.
Monoidal categories, monoids, comonoids. Symmetric monoidal categories,

braidings, Hopf monoids. Higher monoidal categories. 2-categories, bicat-
Description egories, higher categories. Monads, distributive laws, higher monads. The
above concepts can be motivated and discussed by connecting them to other
areas of mathematics depending on the interests of the instructor and students.
81
Course Code MA 606

Course Name Coxeter Groups
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. J.Humphreys, Reflection groups and Coxeter groups, CUP, 1990
2. A.Bjorner and F.Brenti, Combinatorics of Coxeter groups, Springer,

2005.
Text Reference 3. P.Abramenko, K.Brown, Buildings, Springer, 2008
4. M.Davis. The geometry and topology of Coxeter groups, Princeton Uni-

5. M.Aguiar, S. Mahajan. Coxeter bialgebras, CUP, 2022
Reflection arrangement and reflection group, Coxeter diagram, Coxeter com-

plex, Bruhat order, Root system, Classification of Coxeter groups, Poincare
Description polynomial and related enumeration. Connection to related topics such as
buildings, geometric group theory, Coxeter bialgebras, depending on the inter-
est of the instructor and students.
Course Code MA 811

Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Dummit, Foote: Abstract algebra, second edition, Wiley student edi-

tions, 2005.
2. Jacobson: Basic algebra, I, Dover publications, 2009.

Text Reference
3. Jacobson: Basic algebra, II, Dover publications, 2009.
4. Lang: Algebra, third edition, Springer-Verlag, GTM 211, 2002
Review of field and Galois theory: solvable and radical extensions, Kummer
theory, Galois cohomology and Hilbert‘s Theorem 90, Normal Basis theorem.
Infinite Galois extensions: Krull topology, projective limits, profinite groups,
Fundamental Theorem of Galois theory for infinite extensions. Review of in-
tegral ring extensions: integral Galois extensions, prime ideals in integral ring
extensions, decomposition and inertia groups, ramification index and residue
Description class degree, Frobenius map, Dedekind domains, unique factorisation of ide-
als. Categories and functors: definitions and examples. Functors and natural
transformations, equivalence of categories. Products and coproducts, the hom
functor, representable functors, universals and adjoints. Direct and inverse
limits. Free objects. Homological algebra: Additive and abelian categories,
Complexes and homology, long exact sequences, homotopy, resolutions, de-
rived functors, Ext, Tor, cohomology of groups, extensions of groups.
83
Course Code MA 812

Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. (DF) Dummit, Foote: Abstract algebra, second edition, Wiley student

editions, 2005.
2. (J1)Jacobson: Basic algebra, I, Dover publications, 2009.

Text Reference
3. (J2) Jacobson: Basic algebra, II, Dover publications, 2009.
4. (L) Lang: Algebra, third edition, Springer-Verlag, GTM 211, 2002
A review of modules over a PID. [DF-12, J1-3, L-III.7] Noetherian modules

and rings: Primary decomposition, Nakayama’s lemma, filtered and graded
modules, the Hilbert polynomial, Artinian modules and rings. [DF-15, J2-3,
L-X]
Semisimple and simple rings: Semisimple modules, Jacobson density theorem,
semisimple and simple rings, Wedderburn-Artin structure theorems, Jacobson
radical, the effect of a base change on semisimplicity. [DF-18, J2-3, J2-4,
L-XVII]
Representations of finite groups: Basic definitions, characters, class func-
tions, orthogonality relations, induced representations and induced characters,
Description
Frobenius reciprocity, decomposition of the regular representation, supersolv-
able groups, representations of symmetric groups. [DF-18, DF-19, J2-5, L-
XVIII]
Categories and functors: Definitions and examples, functors and natural trans-
formations, the equivalence of categories, products and coproducts, the Hom
functor, representable functors, universals and adjoints, direct and inverse lim-
its, free objects. [DF-Appendix II, J2-1, L-I.11]
Homological algebra: Additive and abelian categories, complexes and homol-
ogy, long exact sequences, homotopy, resolutions, derived functors, Ext, Tor,
cohomology of groups, extensions of groups. [DF-17, J2-6, L-XX]
Course Code MA 813

Course Name Measure Theory
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. K. Chandrasekharan, A Course on Topological Groups, Hindustan Book

Agency, 1996.
2. L. Nachbin, The Haar Integral, van Nostrand, 1965.
Text Reference 3. I. K. Rana, An Introduction to Measure and Integration, 2nd Ed., Amer-
ican Mathematical Society, 2002.
4. H. L. Royden, Real Analysis, 3rd Ed., Prentice Hall of India, 1988.
5. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.
Review of measure theory: monotone convergence theorem, dominated con-

vergence theorem, complete measures. Borel measures: Riesz representation
theorem, Lebesgue measure on Rk , Lp -spaces, Complex measures: total vari-
ation, absolute continuity, Radon-Nikodym theorem, polar and Hahn decom-
positions, bounded linear functionals on Lp , generalised Riesz representation
theorem. Differentiation: Maximal function, Lebesgue points, absolute con-
Description
tinuity of functions, fundamental theorem of calculus, Jacobian of a differen-
tiable transformation, change of variable formula. Product measures: Fubini’s
theorem, completion of product measures, convolutions, Fourier transform,
Riemann-Lebesgue lemma, inversion theorem, Plancherel theorem, L1 as a
Banach algebra. Content on a locally compact Hausdorff space, existence and
uniqueness of the Haar measure on a locally compact group.
85
Course Code MA 814

Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1996.
2. S. Lang, Complex Analysis, 4th Ed., Springer, 1999.
3. D. H. Luecking and L. A. Rubel, Complex Analysis: A Functional Anal-

ysis Approach, Springer-Verlag, 1984.
Text Reference
4. R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable,
Birkhäuser, 2001.
5. R. Remmert, Theory of Complex Functions, Springer (India), 2005.
6. W. Rudin, Real and Complex Analysis, McGraw Hill, 1987.
Review of basic complex analysis: Cauchy’s theorem, Liouville’s theorem,

power series representation, open mapping theorem, calculus of residues.
Harmonic functions, Poisson integral, Harnack’s theorem, Schwarz reflection
principle. Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof
Description
method. Runge’s theorem, Mittag-Leffler theorem, Weierstrass theorem, con-
formal equivalence, Riemann mapping theorem, characterisation of simply con-
nected regions, Jensen’s formula. Analytic continuation, monodromy theorem,
Little Picard theorem.
Course Code MA 815

Course Name Differential Topology
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. R. Bott and L. W. Tu , Differential Forms in Algebraic Topology,

Springer-Verlag, New York, 1982.
Text Reference 2. L. Conlon, Differentiable manifolds, 2nd Ed., Birkhäuser, Boston, 2001.
3. G. E Bredon, Topology and Geometry, Springer-Verlag, New York, 1997.
Review of differentiable manifolds, tangent and cotangent bundles, tensors.

DeRham complex, Poincare’s Lemma, Mayer-Vietoris sequences, cohomology
Description with compact supports, degree of a map, Poincare duality. Vector bundles,
cohomology with vertical compact supports, Thom isomorphism, twisted DeR-
ham complex, Poincare duality for non-orientable manifolds.
87
Course Code MA 816

Course Name Algebraic Topology
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M.J. Greenberg and J. R. Harper, Algebraic Topology, Benjamin, 1981.
2. W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.
3. A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge,

2002.
4. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag,

Text Reference Berlin, 1991.
5. J.R. Munkres, Elements of Algebraic Topology, Addison Wesley, 1984.
6. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India),

2004.
7. H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press,

1980.
Paths and homotopy, homotopy equivalence, contractibility, deformation re-

tracts. Basic constructions: cones, mapping cones, mapping cylinders, suspen-
sion. Cell complexes, subcomplexes, CW pairs. Fundamental groups. Exam-
ples (including the fundamental group of the circle) and applications (including
Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-
Ulam Theorem, both in dimension two). Van Kampen‘s Theorem. Covering
spaces, lifting properties, deck transformations, universal coverings. Simplicial
Description complexes, barycentric subdivision, stars and links, simplicial approximation.
Simplicial Homology. Singular Homology. Mayer-Vietoris sequences. Long ex-
act sequence of pairs and triples. Homotopy invariance and excision. Degree.
Cellular Homology. Applications of homology: Jordan-Brouwer separation
theorem, Invariance of dimension, Hopf’s Theorem for commutative division
algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point Theorem.
Optional Topics: Outline of the theory of: cohomology groups, cup products,
Kunneth formulas, Poincare duality.
Course Code MA 817

Course Name Partial Differential Equations 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. Kesavan, Topics in Functional Analysis and Applications, New Age

International Pvt. Ltd., 1989.
2. L C. Evans, Partial Differential Equation, American Mathematical Soci-

ety, 1998.
3. M. Renardy and R. C. Rogers, An Introduction to Partial Differential

Text Reference Equations, Springer-Verlag, 2004.
4. G. B. Folland, Introduction to Partial Differential Equations, 2nd Ed.,

Prentice-Hall of India, 1995.
5. R. C. McOwen, Partial Differential Equations: Methods and Applica-

tions, 2nd Ed., Pearson Education, Inc., 2003.
Distribution Theory and Sobolev Spaces: Distributional derivatives, Defini-

tions and elementary properties of Sobolev Spaces, Approximations by smooth
functions, Traces, Imbedding Theorems (without proof), Rellich-Kondrachov
Compactness Theorem. Second Order Linear Elliptic Equations: Weak So-
Description lutions, Lax-Milgram Theorem, Existence and Regularity Results, Maximum
Principles, Eigenvalue Problems. Second Order Linear Parabolic Equations:
Existence of weak solutions and Regularity Results, Maximum Principles. Sec-
ond Order Linear Hyperbolic Equations: Existence of weak solutions and Reg-
ularity Results, Maximum Principles, Propagation of Disturbance
89
Course Code MA 818

Course Name Partial Differential Equations 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. L C. Evans, Partial Differential Equations, American Mathematical So-

ciety, 1998.
2. M. Renardy and R. C. Rogers, An Introduction to Partial Differential

Equations, Springer, 2004.
3. M. Defermos, Hyperbolic Conservation Laws in Continuum Physics,

Springer, 2000.
Text Reference
4. B. Dacorogna, Direct Methods in Calculus of Variation, Springer 1989.
5. P. Prasad and R. Ravindran, Partial Differential Equations, Wiley East-

ern, 1985.
6. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer,

1993.
Nonlinear First-Order Scalar Equations: Method of Characteristics, Weak So-

lutions and Uniqueness for Hamilton-Jacobi Equations, Scalar Conservation
Laws: shocks and entropy condition, weak solutions and uniqueness, and long
time behavior. Calculus of Variations: Euler-Lagrange Equation, Second Vari-
ations, Existence of Minimizers: Coercivity, Lower-Semicontinuity, Convexity,
Description
and Constrained Minimization Problems. Hamilton-Jacobi Equations: Vis-
cosity Solutions, Uniqueness, Applications to Control Theory and Dynamic
Programming. System of Conservation Laws: Theory of Shock Waves, Travel-
ling Waves, Entropy Criteria, Riemann Problem, Glimm Existence Result for
System of Two Conservation Laws.
Course Code MA 820

Course Name Stochastic Processes
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Norris J. R. Markov chains, Cambridge University press, Cambridge,

1997.
2. Hoel, Port and Stone, Introduction to Stochastic Processes, Houghton

Mifflin Company, USA, 1972.
3. David A. Levin, Yuval Peres and Elizabeth L. Wilmer, Markov chains

Text Reference and mixing times, AMS Providence, 2008.
4. William Feller, An introduction to probability and its applications, Vol.

I, 3 rd edition, John Wiley and Sons, Singapore, 1968.
5. K B Athreya and S N Lahiri, Probability Theory, TRIM 41, Hindustan

Book Agency, New Delhi, 2006.
Discrete time Markov Chains: Definition and basic properties, class structure,
hitting time and absorption probabilities, strong Markov property, recurrence
and transience, invariant distributions, convergence to equilibrium, time re-
versal, ergodic theorem. Markov chain mixing: Coupling and total variation
distance, Mixing time, upper bound and lower bound on mixing time. Contin-
uous time Markov chains- definition and examples, embedded Markov chain,
Description
Kolmogorov forward and backward equations, classification of states, limit
theorems. Random walk – in dimension one, two and three, The Reflection
Principle, hitting probabilities of a finite sets, Last visits and Long leads, Max-
ima and first passages, Duality, position of maxima. Poisson Process - defi-
nition and properties, inter arrival and waiting time distributions, conditional
distribution of arrival times.
91
Course Code MA 823

Course Name Probability
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. K L Chung, A course in probability theory, 3rd edition, Academic Press,

San Dieago, 2001.
2. P. Billingsley, Probability and measure, 3rd edition, John Wiely and

Sons, New York, 1995.
Text Reference
3. Robert B. Ash, Probability and measure theory, 2nd edition, Academic
Press, San Diego, 2008.
4. K B Athreya and S N Lahiri, Probability Theory, TRIM 41, Hindustan

Book Agency, New Delhi, 200
Review of probability space. Random variables in R and Rn , distribution of

random variables, Expectation of a R-valued random variable, Change of vari-
able formula, Fatou’s lemma, monotone convergence theorem, dominated con-
vergence theorem, Markov inequality, Jensen’s inequality, notion of indepen-
dence of sigma-fields and random variables, product of distributions, Fubini’s
theorem. Convergence almost surely, in probability, in law, convergence in
Description
moments, Borel-Cantelli lemma, Uniform integrability of sequence of random
variables. Characteristic functions, convolution of distributions, Uniqueness
theorem, inversion theorem. Weak law of large numbers, strong law of large
numbers, Lindberg-Feller central limit theorem, Law of iterated logarithms.
Radon Nikodym theorem (reading exercise), Condition expectation definition,
existence and its properties, regular conditional law
Course Code MA 824

Course Name Functional Analysis
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Ahues, A. Largillier and B. V. Limaye, Spectral Computations for

Bounded Operators, Chapman & Hall/CRC, 2001.
2. J. B. Conway, Functional Analysis, 2nd Ed., Springer-Verlag, 1990.
3. S. Lang, Complex Analysis, 4th Ed., Springer, 1999.
Text Reference 4. B. V. Limaye, Functional Analysis, 2nd Ed., New Age International Pub-
lishers, 1996.
5. F. Riesz and B. SzNagy, Functional Analysis, Dover Publications, 1990.
6. W. Rudin, Functional Analysis, Tata McGraw Hill, 1974.
7. K. Yosida, Functional Analysis, 5th Ed., Narosa, 1979.
Review of normed linear spaces, Hahn-Banach theorems, uniform boundedness

principle, open mapping theorem, closed graph theorem, Riesz representation
theorem on Hilbert spaces. Weak and weak* convergence, reflexivity in the
setting of normed linear spaces. Compact operators, Sturm-Liouville problems.
Description
Spectral projections, spectral decomposition theorem, spectral theorem for
a bounded normal operator, unbounded operators, spectral theorem for an
unbounded normal operator.
93
Course Code MA 833

Course Name Weak Convergance and Martingale Theory
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. P. Billingsley, Convergence of Probability Measures, Wiley, 1999.
2. R.J. Elliot, Stochastic Calculus and Applications, Springer-Verlag, 1982.
3. K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic

Press, 1967.
Text Reference
4. A.W. Van-der-Vaart and J.A. Wellner, Weak Convergence and Empirical
Processes: With Applications to Statistics, Springer-Verlag, 1996.
5. D. Williams, Probability with Martingales, Cambridge Mathematical

Textbooks, 1991.
Review of conditional expectations. Martingales in discrete and continuous

time. Square integrable Martingales. Weak convergence in metric spaces with
Description special reference to C([0,1]) space. Dependent variables. Diffusion processes
and mixing. Martingale Central Limit Theorem.
Course Code MA 839

Course Name Advanced Commutative Algebra
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. W.Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University

Text Reference Press, 1992. J. Herzog and T. Hibi, Monomial Ideals, Springer 2011.
Face rings of simplical complexes, rings of invariants of finite groups, local

Description cohomology of modules and its applications to Cohen-Macaulay Gorenstein
rings and face rings of simplicial complexes
95
Course Code MA 841

Course Name Topics in Algebra 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hacken-

sack, NJ, 2006.
2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition,

Text Reference Cambridge University Press, 1998
3. H. Matsumura, Commutative Ring Theory, Cambridge University Press,

1989.

Regular sequences, grade and depth. Projective dimension, Auslander-
Buchsbaum formula. Koszul complex. Rank of modules. Buchsbaum-
Eisenbud acyclicity criterion. Graded rings and modules. Basic properties
of graded modules: associated primes, dimension etc.
Hensel’s Lemma, Newton’ Theorem and Weierstrass Preparation Theorem.
Chevalley’s Theorem on invariants of a finite pseudo-reflection group acting
Description on the polynomial ring.
The Jacobian criterion for regularity. Divisor class group of a noetherian nor-
mal domain and its properties under ring extensions etc. Applications to
unique factorization.
Cohen-Macaulay rings. Homological characterization of regular local rings.
Injective hulls, Matlis Duality. Local cohomology. Basic properties. Invariance
under flat and finite base changes. Canonical module: Existence and basic
properties. Local duality and applications. Canonical module of graded rings.
Course Code MA 842

Course Name Topics in Algebra 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hacken-

sack, NJ, 2006.
2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition,

Text Reference Cambridge University Press, 1998
3. H. Matsumura, Commutative Ring Theory, Cambridge University Press,

1989.

Cohen-Macaulay rings and modules, Canonical Module, Gorenstein rings.
Hilbert functions and multiplicities, Macaulay’s Theorem
Description Stanley-Reisner rings, shellability.
Semigroup rings and rings of invariants
Determinantal rings, Straightening law.
Big Cohen-Macaulay modules, Hochster’s finiteness theorem.
97
Course Code MA 843

Course Name Topics in Analysis 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. R.C. Gunning, Introduction to holomorphic functions of several vari-

ables. Vol. I. Function theory, Wadsworth and Brooks/Cole, 1990.
2. A.W. Knapp, Advanced real analysis, Birkhauser, 2005.
3. S. Lang and W. Cherry, Topics in Nevanlinna theory, Springer-Verlag,

1990.
Text Reference 4. R. Narasimhan, Several complex variables, University of Chicago Press,
1995.
5. E.M. Stein, Harmonic Analysis: Real Variable Methods,Orthogonality,

and Oscillatory Integrals, Princeton University Press, 1993.
6. S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy’s

Theorem on Lie Groups, Birkhauser, 2004.

Singular Integrals (Calderon-Zygmund theory), the Kakeya problem, the Un-
certainty Principle, the almost everywhere convergence of Fourier series, mul-
tilinear operators between Lp spaces.
Pseudodifferential operators, Index theorems.
Description Advanced complex analysis in one variable: Nevanlina theory, the existence of
quasi-conformal maps, iterated polynomial maps, complex dynamics, compact
Riemann surfaces, the Corona theorem.
Holomorphic functions in several complex variables: elementary properties
of functions of several complex variables, analytic continuation, subharmonic
functions, Hartog’s theorem, automorphisms of bounded domains.
Course Code MA 844

Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Ahues, A. Largillier, B.V. Limaye, Spectral Computation for bounded

operators, Chapman and Hall/CRC, 2001.
2. K.E. Atkinson, The Numerical Solution of Integral Equations of the Sec-

ond Kind, Cambridge University Press, 1997.
3. G. Bachman, L. Narici and E. Beckenstein, Fourier and Wavelet Analysis,

Springer-Verlag, 2000.
4. S. K. Berberian, Lectures in Functional Analysis and Operator Theory,

Narosa Publishing House, 1979.
5. F. Chatelin, Spectral Approximation of Linear Operators, Academic

Press, 1983.
6. J.B. Conway, A Course in Functional Analysis, Springer-Verlag, 1985.

Text Reference
7. P.L. Duren, Theory of Hp spaces, Dover Publications, 2000.
8. W. Hackbusch, Integral Equations: Theory and Numerical Treatment,

Birkhauser, 1995.
9. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag,

1995.
10. R. Kress, Linear Integral Equations, Second Edition, Springer-Verlag,

1999.
11. P. Koosis, Introduction to Hp spaces, 2nd Edition, Cambridge University

Press, 1999.
12. C.S. Kubrusly, An Introduction to Models and Decompositions in Oper-

ator Theory, Birkhauser, 1997.
99

13. G.J. Murphy, C*-Algebras and Operator Theory, Academic Press Inc.,
1990.
14. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

Text Reference
15. W. Rudin, Functional Analysis, McGraw Hill, 1991.
16. A. Vretblad, Fourier Analysis and its Applications, Springer-Verlag,

2005.

Fourier Series and Fourier Transforms: Orthonormal Sequences in In-
ner Product Spaces, Fourier Series, Riemann-Lebesgue Lemma, Conver-
gence/Divergence of Fourier Series, Fejer Theory, Fourier Transform, Inversion
Theorem, Approximate Identities, Plancherel Theorem
Hp spaces: Harmonic and Subharmonic Functions, Hp spaces, Nevanlinna
Class of Functions, Boundary Values, Non-tangential Limits, F. and M. Riesz
Theorem, Inner Functions, Outer Functions, Factorization Theorems, Beurl-
ing’s Theorem
Banach Algebras: Examples of Banach Algebras, Spectrum, Gelfand Repre-
sentation, C*-Algebras, Positive Linear Functionals, Gelfand-Naimark Repre-
Description sentation
Elements of Operator Theory: Hilbert Space Operators, Parts of Spectrum,
Orthogonal Projections, Invariant Subspaces, Reducing Subspaces, Shifts, De-
compositions of Operators
Perturbation Theory for Linear Operators: Analyticity of the resolvent opera-
tor, spectral projection and the weighted mean of the eigenvalues, The method
of majorizing series, Spectral Decomposition Theorem.
Spectral Approximation: Norm and nu- convergence, Iterative refinement
methods such as the Rayleigh-Schrodinger series and methods based on the
fixed point techniques, error estimates.
Approximate solutions of Operator Equations: Galerkin, Iterated Galerkin
and Nystrom methods, Condition Numbers, Two Grid Methods.
Course Code MA 845

Course Name Topics in Combinatorics 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. C. Berge, Principles of Combinatorics, Academic Press, 1972.
2. I.G. Macdonald, Symmetric functions and Hall polynomials. Second

Text Reference edition, Oxford University Press, 1995.
3. R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and

Brooks/Cole, 1986.

Basic Combinatorial Objects : Sets, multisets, partitions of sets, partitions of
numbers, finite vector spaces, permutations, graphs etc.
Basic Counting Coefficients: The twelve fold way, binomial, q-binomial and
Description the Stirling coefficients, permutation statistics, etc.
Sieve Methods : Principle of inclusion-exclusion, permutations with restricted
positions, Sign-reversing involutions, determinants etc.
Combinatorial reciprocity.
Theory of Symmetric functions.
101
Course Code MA 846

Course Name Topics in Combinatorics 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979.
2. I. G. Macdonald, Symmetric functions and Hall polynomials. Second

edition, Oxford University Press, New York, 1995.
3. B.E. Sagan, The Symmetric Group: Representations, Combinatorial Al-

Text Reference gorithms and Symmetric Functions, Wadsworth and Brooks/Cole, 1991.
4. R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and

Brooks/Cole, Monterey, CA, 1986.
5. R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univer-

sity Press, Cambridge, 1999.

Partially ordered sets, Mobius inversion.
Rational generating functions: P-partitions and linear Diophantine equations.
Description Polya theory and representation theory of the symmetric group.
Combinatorial algorithms, and symmetric functions.
Generating functions: Single and multivariable Lagrange inversion.
Young tableaux and plane partitions
Course Code MA 847

Course Name Topics in Geometry 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. J. M. Lee, Riemannian Manifolds: An Introduction to Curvature,

2. W. M. Boothby, An Introduction to Differentiable Manifolds and Rie-

mannian Geometry, 2nd edition, Academic Press, 2002.
Text Reference 3. M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice

Hall, 1976.
4. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hin-

dustan Book Agency, 2002.
5. J. Milnor, Morse Theory, Princeton University Press, 1963.

Review of the theory of curves and surfaces in the Euclidean 3-space.
Differentiable manifolds, and Riemannian structures. Connections, and curva-
Description
ture tensor.
The theorems of Bonnet-Meyers and Hadamard. Manifolds of constant curva-
ture.
103
Course Code MA 848

Course Name Topics in Geometry 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, Amer-

ican Mathematical Society, Providence, RI, 1990.
2. D. Eisenbud and J. Harris, The Geometry of Schemes, Springer-Verlag,

2000.
Text Reference
3. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.
4. I. R. Shafarevich, Basic Algebraic Geometry, Vol. 1 and 2, Second edi-

tion, Springer-Verlag, 1994.

Affine and projective varieties, rational maps, nonsingularity.
Description Algebraic Curves, Riemann Roch Theorem.
Sheaves and Schemes. Basic properties. Divisors and Differentials.
Cohomology of sheaves, Serre Duality Theorem.
Course Code MA 849

Course Name Topics in Topology 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. B. Gray, Homotopy Theory, Academic Press, 1975.
2. A. Hatcher, Algebraic Topology, Cambridge University Press 2002.
3. G. W. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1978.
4. P. Hilton, Homotopy Theory and Duality, Gordon and Beach Sc. Pub-
Text Reference lishers, 1965.
5. N. Steenrod, The Topology of Fibre Bundles, 7th reprint, Princeton Uni-

6. R. M. Switzer, Algebraic topology: Homotopy and Homology, Springer

Verlag, 2002.
A selection of topics from the following: CW complexes, Homotopy groups,

Cellular Approximation. Whitehead‘s theorem, Hurewicz theorem. Excision,
Description Fibre bundles, Long exact sequences. Postnikov Towers, Obstruction Theory.
Stable homotopy groups. Spectral Sequences, Serre Class of abelian groups.
105
Course Code MA 850

Course Name Topics in Topology 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. G. E. Bredon, Introduction to Compact Transformation Groups, Aca-

demic Press 1972.
2. T. Brocker and T. tom Dieck, Representations of Compact Lie Groups,

Text Reference Springer-Verlag, New York, 1985.
3. W. Y. Hsiang, Cohomology Theory of Topological Transformation

Groups, Springer-Verlag, 1975.
Basics of Topological groups, Lie group. Group actions, homogeneous spaces

examples. G-spaces, existence of slice and tubes Covering homotopy theo-
rem, Classification of G-Spaces. Finite group actions, homology spheres G-
Description coverings, Cech theory Locally smooth actions, orbit types, principal orbits
Actions of tori. Cohomology structure of fixed point sets, Zp -actions, projec-
tive spaces and product of spheres.
Course Code MA 851

Course Name Topics in Number Theory 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New

York, 1994.
2. D. Bump, Automorphic forms and representations, Cambridge Univer-

Text Reference
3. H. Iwaniec and E. Kowalski, Analytic number theory, American Mathe-
matical Society, Providence, RI, 2004.
4. H. Hida, Modular forms and Galois cohomology, Cambridge University


Algebraic number theory, abelian and non-abelian reciprocity laws, the Lang-
lands programme, automorphic forms and representations.
The arithmetic of algebraic groups.
Description
Arithmetic algebraic geometry: counting rational points of varieties over finite
fields
Galois representations and galois cohomology.
Additive number theory: partitions, compositions, Goldbach problem.
107
Course Code MA 852

Course Name Topics in Number Theory 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New

York, 1994.
2. D. Bump, Automorphic forms and representations, Cambridge Univer-

Text Reference
3. H. Iwaniec and E. Kowalski, Analytic number theory, American Mathe-
matical Society, Providence, RI, 2004.
4. H. Hida, Modular forms and Galois cohomology, Cambridge University


Harmonic analysis on Lie groups, L-functions, l-adic representations and mo-
tives.
Description
Diophantine equations and the applications of K-theory to number theory.
Analytic number theory and transcendental methods.
Applications of ergodic theory to number theory.
Course Code MA 853

Course Name Topics in Differential Equations 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of

Second Order, Springer-Verlag, 1983.
2. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1984.
3. D. Serre, Systems of Conservation Laws, Vols. 1, 2, Cambridge Univer-

sity Press, 2000.
Text Reference 4. L. Evans, Weak Convergence Methods for Nonlinear PDEs, CBMS Re-
gional Conference series in Math., American Mathematical Society, Prov-
idence RI, 1990
5. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis

for Periodic Structures, North Holland, 1978.
6. M. Struwe, Variational Methods: Applications to nonlinear PDEs and

Hamiltonian systems, Springer-Verlag, 1990.
1. Schauder theory, regularity for second order elliptic equations. Nonlinear

analysis and its applications to nonlinear PDEs: Fixed point methods,
variational methods, monotone iteration, degree theory.
2. Evolution equations: Existence via semigroup theory

Description
3. Nonlinear Hyperbolic systems: Theory of well posedness, compensated
compactness,
4. Young measures; propagation of oscillations, weakly nonlinear geometric

optics.
109
Course Code MA 854

Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. D. K. Arrowsmith, C. M. Place: An Introduction to Dynamical Systems,

2. C. Chicone, Ordinary Differential Equations. Springer-Verlag, 1999.
3. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Sys-

tems, and Bifurcations of Vector Fields. Springer-Verlag, 2002.
4. P.Glendinning, Stability, instability and chaos: An Introduction to the

Theory of Nonlinear Differential Equations, Cambridge University Press,
1994.
Text Reference
5. J. Palis and W. C. de Melo, Geometric Theory of Dynamical Systems,
6. R. Grimshaw, Nonlinear Ordinary Differential Equations. CRC press,

1991.
7. N.A. Magnitskii and S.V. Sidorov, New Methods for Chaotic Dynamics,
World Scientific, 2006.
8. L. Perko, Differential Equations and Dynamical Systems, Springer-

Verlag, 2001.
Continued on next page ...

Description

A selection of topics from the following: Diffeomorphisms and flows: Elemen-
tary dynamics of diffeomorphisms, flows and differential equations, conjugacy,
equivalence of flows, Sternberg‘s theorem on smooth conjugacy (statement
only), Hamiltonian flows and Poincare maps. Local properties of flows and
diffeomorphisms: Hyperbolic fixed points, Hartman-Grobman theorems for
maps and flows, Normal forms for vector fields, Centre manifolds. Structural
stability and hyperbolicity: Structural stability for linear systems, Flows on
2-dimensional manifolds, Peixoto’s characterisation of structural stability on
unit disc, Anosov and Horseshoe diffeomorphisms, Homoclinic points, Mel-
nikov function. Bifurcations and Perturbations: Saddle-node and Hopf bifur-
Description
cations, Andronov-Hopf bifurcation, The logistic map, Arnold’s circle map;
Perturbation theory: Melnikov’s method for the study of perturbation of com-
pletely integrable systems. Floquet theory and Hill’s equation and some of its
applications. Two dimensional systems: Poincare-Bendixon theorem, Index of
planar vector fields and the Poincare Hopf index theorem for two dimensional
manifolds. Van der Pol’s equation, Duffing’s equation, Lorenz’s equation. First
integrals and functional independence of first integrals, notion of complete in-
tegrability, Jacobi multipliers, Liouville’s theorem on preservation of phase
volume, Jacobi’s last multiplier theorem and its applications.
111
Course Code MA 855

Course Name Topics in Numerical Analysis 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Axelsson, O. Iterative Solution Methods, Cambridge University Press,

1994.
2. Briggs, W. L., Henson, V. E. and McCormick, S. F. A Multigrid tutorial,

SIAM, 2000.
3. Godlewski, E. and Raviart, P. –A. Numerical Approximation of Hyper-

bolic Systems of Conservation Laws, Springer, 1995.
4. Kroner, D. Numerical Schemes for Conservation Laws. John Wiley, 1997.

Text Reference 5. LeVeque, R. J. Finite Volume Methods for Hyperbolic Problems, Cam-
bridge University Press, 2002.
6. LeVeque, R. J. Numerical Methods for Conservation Laws. Birkhauser,

1992.
7. Quarteroni, A. and Valli, A. Numerical Approximation of Partial Differ-

ential Equations, Springer, 1997.
8. Ueberrhuber, C. W. Numerical Computation: Methods, Software and

Analysis, Springer-Verlag, 1997.

Review of finite difference methods for elliptic, parabolic and hyperbolic prob-
lems. Stability, consistency and convergence theory.
Finite difference schemes for scalar conservation laws (Lax-Friedrichs, Upwind,
Lax-Wendroff, etc.), Conservative schemes and their numerical flux functions,
Consistency, Lax-Wendroff Theorem, CFL Condition, Nonlinear Stability and
TVD property, Monotone Difference schemes, Numerical entropy condition,
Description Convergence result.
Finite difference Schemes for one-dimensional system of conservation laws,
approximate Riemann solvers, Godunov’s method, High resolution methods,
Multidimensional approaches.
Large Scale Scientific Computing: Classical Iterative Methods for solving Lin-
ear systems, Large Sparse Linear Systems, Storage Schemes, GMRES algo-
rithm, Preconditioned Conjugate Gradient method and Multi-grid method,
Newton’s Method and some of its variations for solving nonlinear systems.
Course Code MA 856

Course Name Topics in Numerical Analysis II
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Z. Chen, Finite Element Methods And Their Applications, Springer-

Verlag, New York, 2005.
2. S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Ele-

ment Methods, 2nd Edition, Springer-Verlag, New York, 2002.
Text Reference
3. M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite
Element Analysis, John Wiley and Sons, 2000.
4. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems,

2nd Edition, Springer-Verlag, Berlin, 2006.

Mixed Finite Element Methods: Examples of mixed variational formulations-
primal, dual formulations; abstract mixed formulations, discrete mixed formu-
lations, existence-uniqueness of solutions, convergence analysis, implementa-
tion procedures.
Adaptive FEM: A study of -Explicit A posteriori error estimators, Implicit A
posteriori estimators, Recovery based error estimators, Goal Oriented adaptive
mesh refinement for second order elliptic boundary value problems.
Discontinuous Galerkin Methods for second order elliptic boundary value prob-
Description lems: Global element methods, Symmetric Interior Penalty Method, Discon-
tinuous hp- Galerkin Method, Non-symmetric interior penalty method: Con-
sistency, approximation properties, existence and uniqueness of solutions, error
estimates, implementation procedures.
FEM for parabolic problems: The standard Galerkin method, semi-
discretization in space. discretization in space and time, the discontinuous
Galerkin Method, a mixed method, implementation procedures.
Elements of Multigrid Methods: Multigrid Components - Interpolation, re-
striction Coarse-grid correction, V, W, and FMG cycles, Implementation, Con-
vergence analysis, Performance diagnostics.
113
Course Code MA 858

Course Name Topics in Probability II
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. V.S. Borkar, Optimal control of diffusion processes, Longman Scientific

and Technical, Harlow (copublished by John Wiley), 1989.
Text Reference
2. D. Nualart, The Malliavin calculus and related topics, Springer-Verlag,
1995.

Stochastic optimal control: compactness of laws, dynamic programming prin-
ciple.
Description
Malliavin calculus and applications to finance: Wiener-Ito chaos expansion,
Shorohod integral, Integration by parts formula, Clark- Ocone formula and
application to finance.
Course Code MA 859

Course Name Topics in Statistics I
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. J. George Shanthikumar and Moshe Shaked (1994) Stochastic Orders

and their Applications, Academic press.
Text Reference
2. C.D. Lai and M. Xie (2006) Stochastic Ageing and Dependence for Re-
liability, Springer Verlag.

Univariate Stochastic Orders-hazard rate order, likelihood ratio order, mean
residual rate order. Univariate variability orders- convex order, dispersive or-
Description der, peakedness order. Univariate monotone convex and related orders. Multi-
variate stochastic orders. Multivariate variability and related orders. Statisti-
cal Inference for stochastic ordering. Applications in reliability theory, biology,
economics and scheduling.
115
Course Code MA 860

Course Name Topics in Statistics II
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press,

2000.
2. U. Grenander, Abstract Inference, John Wiley, 1981.
3. P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd Edition,

Chapman and Hall/CRC, 1994.
Text Reference
4. L. Fahrmeir and G. Tutz, Multivariate Statistical Modeling based on
Generalized Linear Models, 2nd Edition, Springer-Verlag, 1994.
5. R. H. Myers, D. C. Montgomery and G. Geoffrey Vining, Generalized

Linear Models with applications in Engineering and Sciences, Wiley-
Interscience, 2001.

Inference in Semi-parametric models: Models with infinite imensional param-
eters, Efficient estimation and the delta method, Score and information oper-
ators, Estimating equations, Maximum Likelihood estimation, Testing.
Description Generalized linear models: Components of a GLM, estimation techniques,
diagnostics, continuous response models, Binomial response models, Poisson
response models, overdispersion, multivariate GLMs, quasi likelihoods, gener-
alized estimating equations, generalized linear mixed models, programming in
R and SAS.
Course Code MA 861

Course Name Combinatorics 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Enumerative Combinatorics - Stanley, Vol.1 (2nd Edition) and 2, Cam-

bridge University Press.
2. Extremal Combinatorics With Applications in Computer Science - Stasys

Jukna, Springer, 2nd Edition.
Text Reference 3. Computing the Continuous Discretely : Integer-point Enumeration in

Polyhedra - Beck and Robbins, Springer, 2nd edition.
4. Combinatorics of Finite Sets - Anderson, Dover Books on Mathematics.
5. Modern Graph theory - Bollobas, Graduate Texts in Mathematics,

Springer.
Extremal Set Theory: Sperner’s Theorem, Theorems of Erdos-Ko-Rado,

Kruskal-Katona, Dilworth‘s theorem, Kleitman’s lemma for ideals and corre-
lation inequalities. Graph theory: Matching theory, Hamiltonicity, Extremal
graph theory, Graph colorings, Ramsey theory. Basic Enumerative Combina-
Description
torics: Generating Functions, Quasi-polynomials and applications to Ehrhart
theory, Transfer Matrix Method, Stanley’s Reciprocity Theorem, Exponential
Structures, Trees, Lagrange inversion Theorem.
117
Course Code MA 862

Course Name Combinatorics 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Enumerative Combinatorics - Stanley, Vol.1 (2nd Edition) and 2, Cam-

bridge University Press.
2. Extremal Combinatorics With Applications in Computer Science - Stasys

Jukna, Springer, 2nd Edition
3. The Symmetric Group : Representations, Combinatorial Algorithms,

and symmetric functions - Bruce Sagan, Graduate texts in Mathematics,
Springer, 2nd ed.
4. Representation Theory : A combinatorial Viewpoint - A. Prasad, Cam-

bridge University Press
Text Reference
5. Combinatorics of Coxeter Groups : Bjorner and Brenti, Graduate Texts
in Mathematics, Springer.
6. Symmetric Functions and Hall Polynomials - Macdonald, Oxford Math-

ematical monographs.
7. Linear Algebra methods in Coombinatorics - Babai/Frankl, lecture notes.
8. The Polynomial method in Combinatorics - survey paper by T. Tao
9. Incidence Theorems and Their Applications - Z. Dvir, Foundations and

Trends in Theoretical Computer Science, Now Publishers Inc.
Advanced Enumeration: Permutation Statistics and generalizations to Cox-

eter groups, Enumeration with Symmetric Functions, RSK Algorithm, Frobe-
Description nius characteristic, The Jacobi-Trudi identity, Murnaghan-Nakayama Lemma,
Littlewood-Richardson rule. Linear algebra methods in Combinatorics, The
polynomial method, combinatorial Nullstellensatz and applications.
Course Code MA 863

Course Name Theoretical Statistics 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Main text: Jun Shao, Mathematical Statistics, 2nd Ed., Springer, 2003.
2. Additional Texts:
Text Reference (a) Theoretical Statistics, D.R. Cox, D.V. Hinkley CRC Press
(b) E. L. Lehmann, Theory of Statistical Inference, Wiley, 1983.
(c) E. L. Lehmann, Testing Statistical Hypotheses, Wiley, 1986.
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]
2. UMVUE, U-statistics, Asymptotic Unbiased estimator, V-statistics

[Chapter 3]
Description 3. Bayes Decision and Bayes estimators, Invariance, Minimaxity and ad-
missibility, MLE and efficient estimation method. [Chapter 4]
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.]
119
Course Code MA 864

Course Name Topics in Category Theory 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil

3. Borceau, Handbook of categorical algebra, Volumes 1, 2 and 3, Cam-

4. Goerss and Jardine, Simplicial homotopy theory, Birkhauser, 1997.

Text Reference
5. Hirschhorn, Model categories and their localizations, American Mathe-
matical Society, 2003.
6. Leinster, Higher categories, Higher operads, Cambridge University Press,

2004.
8. Mac Lane, Categories for the working mathematician, Springer, 1998
Categories, functors, natural transformations.

Limits, colimits, complete and cocomplete categories.
Adjoint functors, universal constructions, free and cofree objects.
Functor categories, comma categories, quotient categories, derived categories.
Representable functors, Yoneda lemma.
Description
Cauchy completeness, Karoubi envelopes.
Cartesian categories, group objects. The above concepts can be motivated and
discussed by connecting them to other areas of mathematics depending on the
interests of the instructor and students.
Course Code MA 865

Course Name Topics in Category Theory 2
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil

3. Borceau, Handbook of categorical algebra, Volumes 1, 2 and 3, Cam-

4. Goerss and Jardine, Simplicial homotopy theory, Birkhauser, 1997.

Text Reference
5. Hirschhorn, Model categories and their localizations, American Mathe-
matical Society, 2003.
6. Leinster, Higher categories, Higher operads, Cambridge University Press,

2004.
8. Mac Lane, Categories for the working mathematician, Springer, 1998
Monoidal categories, monoids, comonoids.

Symmetric monoidal categories, braidings, Hopf monoids.
Higher monoidal categories.
Description Enriched categories, 2-categories, bicategories, higher categories.
Monads, distributive laws, higher monads. The above concepts can be mo-
tivated and discussed by connecting them to other areas of mathematics de-
pending on the interests of the instructor and students.
121
Course Code MA 867

Course Name Statistical Modelling - 1
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Main Text: Linear Models by S.R. Searle (1971) Wiley & SonsOther
Text Reference 2. Additional Reference: Linear Model Methodology by A. I. Khuri (2009)
CRC Press
1. Full rank model (Chapters 3 and 4)
2. Models with rank deficiency (Chapter 5: Sections 5.1,5.2,5.3,5.4,5.5)
3. One-way classification model (Chapter 6: Sections 6.1,6.2,6.3,6.4)

Description
4. Two-way Crossed Classification model (Chapter 7: Sections 7.1,7.2)
5. Fixed, Random and Mixed models for Balanced Data (Chapter 9.1-9.5,
9.8, 9.9)
Course Code MA 899

Course Name Communication Skills
Total Credits 6
Type N
Lecture 1
Tutorial 2
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Alley, Michael The Craft of ScientificPresentations, Springer (2003).
2. Booth, Wayne C., Gregory G. Colomb, and JosephM. Williams, The

Craft of Research, The Universityof Chicago Press, 3rd edition, (2008).
3. Keshav. S, How to read a paper. ACM SIGCOMM Comp. Commun.

Rev., 37, 2007.
4. Monippally, M. M., Pawar, B.S. Academic Writing:A Guide for Manage-

ment Students and Researchers,Response Books, (2010).
Text Reference
5. Purdue Online Writing Lab (OWL),https://owl.purdue.edu/
6. Strunk Jr., William; E. B. White, The Elements of Style, Fourth Edition,

Longman; 4th edition (1999).
7. Truss, Lynne Eats, Shoots & Leaves: The ZeroTolerance Approach to

Punctuation Gotham;(2006).
8. Whitesides, George M. Whitesides Group: Writing a Paper, Advanced

Materials 16 (2004).
Continued on next page ...

Description
123

Course Name Communication Skills
Context of communication: Recognizing our capability and roles as profes-
sionals. Scientific Method: Question and answer aspects of technical commu-
nication; Scientific Methodology and its relationship to technical communi-
cation;Surveying literature: Categories; reading and organizing scientific lit-
erature; search engines and tools. Listening and Note taking: 5-R method
and mind-mapping.Technical writing: Report organization; Journal selec-
tion; Introduction, conclusion, and abstract writing. Speaking & Presentation
skills: Organization of presentation slides (number, content, and formatting);
Oral presentations; Audience/context dependent practices; Nonverbal aspects:
Description
body language, eye-contact, personal appearance, facing large audience. El-
evator pitch: Pitches for technical audience and policymakers. Workplace
communication: Sensitivity towards gender and diversity; Email communi-
cation and netiquettes.Ethics in academic communication: Intellectual Prop-
erty,copyrights and plagiarism; Authorship; Data ethics; Biases and balanced
criticism of literature;Suggested additional topics relevant to disciplines: Data
representation, Group discussion and interviews; accessible scientific writing,
report writing using LaTeX, Proofreading, etc
Course Code MAS801

Course Name Seminar
Total Credits 4
Type S
Lecture 0
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
Text Reference 1.
Description
Course Code MAS802

Course Name Seminar
Total Credits 4
Type S
Lecture 0
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
Text Reference 1.
Description
Chapter 6
LIST OF ALL STATISTICS COURSES
SI404 . . . . . . . . . . . . . . . . . . . . . . 126 SI515 . . . . . . . . . . . . . . . . . . . . . . 142

SI416 . . . . . . . . . . . . . . . . . . . . . . 127 SI526 . . . . . . . . . . . . . . . . . . . . . . 143
SI419 . . . . . . . . . . . . . . . . . . . . . . 128 SI527 . . . . . . . . . . . . . . . . . . . . . . 144
SI422 . . . . . . . . . . . . . . . . . . . . . . 129 SI534 . . . . . . . . . . . . . . . . . . . . . . 145
SI423 . . . . . . . . . . . . . . . . . . . . . . 130 SI536 . . . . . . . . . . . . . . . . . . . . . . 146
SI424 . . . . . . . . . . . . . . . . . . . . . . 131 SI537 . . . . . . . . . . . . . . . . . . . . . . 147
SI426 . . . . . . . . . . . . . . . . . . . . . . 132
SI539 . . . . . . . . . . . . . . . . . . . . . . 148
SI427 . . . . . . . . . . . . . . . . . . . . . . 133
SI541 . . . . . . . . . . . . . . . . . . . . . . 149
SI429 . . . . . . . . . . . . . . . . . . . . . . 134
SI543 . . . . . . . . . . . . . . . . . . . . . . 150
SI431 . . . . . . . . . . . . . . . . . . . . . . 135
SI503 . . . . . . . . . . . . . . . . . . . . . . 136 SI544 . . . . . . . . . . . . . . . . . . . . . . 151
SI505 . . . . . . . . . . . . . . . . . . . . . . 137 SI546 . . . . . . . . . . . . . . . . . . . . . . 152
SI507 . . . . . . . . . . . . . . . . . . . . . . 138 SI548 . . . . . . . . . . . . . . . . . . . . . . 153
SI509 . . . . . . . . . . . . . . . . . . . . . . 139 SI550 . . . . . . . . . . . . . . . . . . . . . . 154
SI513 . . . . . . . . . . . . . . . . . . . . . . 140 SI593 . . . . . . . . . . . . . . . . . . . . . . 155
SI514 . . . . . . . . . . . . . . . . . . . . . . 141 SI598 . . . . . . . . . . . . . . . . . . . . . . 156
125
126 CHAPTER 6. LIST OF ALL STATISTICS COURSES
Course Code SI 404

Course Name Applied Stochastic Processes
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. U. N. Bhat, Elements of Applied Stochastic Processes, Wiley, 1972.
2. V.G. Kulkarni, Modeling and Analysis of Stochastic Systems, Chapman

and Hall,London, 1995.
Text Reference 3. J. Medhi, Stochastic Models in Queuing Theory, Academic Press, 1991.
4. R. Nelson, Probability, Stochastic Processes,and Queuing Theory: The

Mathematics of Computer Performance Modelling, Springer-Verlag, New
York, 1995.
Stochastic processes: description and definition. Markov chains with finite

and countably infinite state spaces. Classification of states, irreducibility, er-
godicity. Basic limit theorems. Statistical Inference. Applications to queuing
models. Markov processes with discrete and continuous state spaces. Poisson
Description process, pure birth process, birth and death process. Brownian motion. Appli-
cations to queuing models and reliability theory. Basic theory and applications
of renewal processes, stationary processes. Branching processes. Markov Re-
newal and semiMarkov processes, regenerative processes.
127
Course Code SI 416

Course Name Optimization
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Beale, E.M.L., and Mackley, L. , Introduction to Optimization, John

Wiley & Sons, Hoboken, 1988.
2. Chavatal, V., Linear Programming, W.H. Reeman and Company, New

York, 1983.
3. Chong, E.P.K. and Zak, S.H., An Introduction to Optimization, 4th

Edition, John Wiley & Sons, Hoboken, 2013.
Text Reference
4. Joshi, M.C., and Moudgalya, K., Optimization: Theory and Practice,
Narosa, New Delhi, 2004.
5. Nocedal, J. and Wright, S. J., Numerical Optimization, 2nd Edition,

Springer, New York, 2006.
6. Vanderbei, R.J., Linear Programming Foundations and Extensions, 3rd

Edition, Springer, New York, 2008.
Unconstrained optimization using calculus (Taylor‘s theorem, convex func-

tions, coercive functions). Unconstrained optimization via iterative meth-
ods (Newton‘s method, Gradient/ conjugate gradient based methods, Quasi-
Newton methods). Constrained optimization (Penalty methods, Lagrange
multipliers, Karush-Kuhn-Tucker conditions). Introduction to Linear Pro-
Description gramming: Lines and hyperplanes, Convex sets, Convex hull, Formulation
of a Linear Programming Problem, Theorems dealing with vertices of feasible
regions and optimality, Graphical solution. Simplex method (including Big M
method and two-phase method), Dual problem, Duality theory, Dual simplex
method, Revised simplex method
Course Code SI 419

Course Name Combinatorics
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Bona, M. A walk through Combinatorics, 4th edition, World Scientific,

2017.
2. Nesetril, J,. and Matousek, J. Invitation to Discrete Mathematics, 2nd

Text Reference edition, Oxford University Press, 2009.
3. Lehman, E., Leighton, F. T., and Meyer, A. R. (2019) Mathematics for

Computer science, (Freely available online), 2019.
Counting Basic Combinatorial objects: Sets, Multisets, Partitions of sets,

Partitions of numbers, Permutations, Trees, Partially ordered sets.Generating
functions, Recurrence relations, Principle of Inclusion-Exclusion.Graph The-
Description
ory Graphs and Directed graphs, Paths, Walks, Connectivity, Matchings in
bipartite graphs, Network flows, Dilworths theorem.
129
Course Code SI 422

Course Name Regression Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite SI 427 (Probability 1)
1. Draper, N. and Smith,H. Applied Regression Analysis, 3rd Edition, John

Wiley and Sons Series in Probability and Statistics, New York, 1998.
2. Montgomery, D., Peck, E., Vining, G. Introduction to Linear Regression

Analysis, 5th Edition, John Wiley, New York, 2012.
Text Reference 3. Sen, A. and Srivastava, M. Regression Analysis Theory, Methods & Ap-
plications, 1st Edition, Springer-Verlag Berlin Heidelberg, New York,
1990.
4. Kutner, M., Nachtsheim, C., Neter, J. and Li, W. Applied Linear Sta-
tistical Models, 5th Edition, McGraw-Hill Companies, Boston, 2005.
Simple and multiple linear regression models – estimation, tests and confidence
regions. Simultaneous testing methods- Bonferroni method etc. Analysis of
Variance for simple and multiple regression models. Analysis of residuals. Lack
of fit tests. Checks (graphical procedures and tests) for model assumptions:
Normality, homogeneity of errors, independence, correlation of covariates and
errors. Multicollinearity, outliers, leverage and measures of influence. Model
Description selection (stepwise, forward and backward, best subset selection) and model
validation. Discussion of algorithms for model selection. Regression models
with indicator variables. Polynomial regression models. Regression models
with interaction terms. Transformation of response variables and covariates.
Variance stabilizing transformations, Box-Cox method. Ridge‘s regression.
Weighted Regression.
Course Code SI 423

Course Name Linear Algebra and its Applications
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Rao, A.R., and Bhimashankaram, P., Linear algebra, 2nd edition, Hin-
dustan book agencey, New Delhi, 2000.
2. Friedberg, S.H., Insel, A.J., and Spence, L.E., Linear algebra, 4th edition,
Text Reference PHI learning, New Delhi, 2011.
3. Strang, G., Linear algebra and its applications, 4th edition, Thomson
Learning, Toronto, 2006.
Vector spaces (with emphasis over R and C): Subspaces, linear dependence
and independence, basis and dimension. Linear transformations: Rank-nullity
theorem, matrix representation of a linear transformation, invertibility and
isomorphism, effect of change of basis on the matrix representation of a linear
transformation, dual spaces. Review of elementary properties of determinants,
Cramer‘s rule. Diagonalization: Eigenvalues and eigenvectors, algebraic and
Description geometric multiplicities of an eigenvalue, diagonalizability, invariant subspaces
and Cayley-Hamilton theorem. Inner product spaces: Gram-Schmidt orthog-
onalization, adjoint of a linear operator, normal and self-adjoint operator, or-
thogonal projections and the spectral theorem, singular value decomposition
and pseudo-inverse, bilinear and quadratic forms. Canonical forms: Jordan
canonical form (with emphasis on computation).
131
Course Code SI 424

Course Name Statistical Inference 1
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Casella, G. and Berger, R. Statistical Inference, 1st Edition, Duxbury

Press, Pacific Grove, 2002.
2. Hogg, R., McKean. J. and Craig, A., Introduction to Mathematical

Statistics, 8th Edition, Pearson, Boston, 2019.
3. Lehmann, E. Theory of Point Estimation, 1st Edition, John Wiley &

Text Reference Sons, New York, 1983.
4. Lehmann, E and Romano, J. Testing Statistical Hypotheses, 3rd Edition,

Springer-Verlag New York, 2005.
5. DeGroot, M. and Schervish, M. Probability and Statistics, 4th Edition,

Addison Wesley, Boston, 2002.
Distributions of functions of random variables, Sampling distributions, Order

statistics, Sufficiency and completeness, exponential family of distributions,
Methods of estimation (Method of Moments, MLE and Bayesian), Unbiased
Description estimators, Evaluating estimators, UMVUEs, Testing, Likelihood Ratio tests,
UMP tests, unbiased tests, Interval estimation, Consistent and efficient esti-
mators.
Course Code SI 426

Course Name Algorithms
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Dasgupta, S., Papadimitriou, C., and Vazirani, U. Algorithms, Tata

McGraw-Hill, 2008.
Text Reference 2. Kleinberg, J. and Tardos, E. Algorithm design, Pearson, 2006.
3. Cormen, T., Leiserson, C., Rivest, R., and Stein, C. Introduction to

Algorithms, 3rd edition, MIT Press, 2009.
Basics: Algorithm analysis and asymptotic notation, Linked lists. Graphs:

Breadth first search, Depth first search, Strongly connected components. Di-
vide and Conquer: Merge sort, Fast Fourier transform. Greedy Algorithms:
Dijkstra’s algorithm, Minimum spanning tree algorithms, Huffman codes and
Description data compression. Dynamic programming: Longest increasing sequences, edit
distance, shortest paths. Network flows: Maxflow Mincut theorem, max flow
algorithms, application to bipartite matchings. Introduction to Randomized
algorithms: randomized quick sort, global mincut, hashing.
133
Course Code SI 427

Course Name Probability 1
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Athreya K.B. and Lahiri S. N., Probability Theory, Hindustan Book

Agency, 2006.
2. Billingsley, P., Probability and Measure, 2nd edition, John Wiley & Sons,
New York, 1995.
3. Hoel, P.G., Port, S.C., and Stone, C.J., Introduction to Probability The-
Text Reference ory, Universal Book Stall, New Delhi, 1998.
4. Karr, A.F., Probability, Springer-Verlag, New York, 2003. Rosenthal,

J.S., A first look at rigorous Probability theory, 2nd edition, World Sci-
entific, 2006.
5. Ross, S., A first course in Probability, 9th Edition, Pearson, Delhi, 2019.
Random phenomena, sample spaces, events, sigma algebra, probability space,

properties of probability, conditional probability, independence, Bayes formula,
Polya’s urn model. Discrete random variable, probability mass function, inde-
pendent random variables, sum of random variables, random vector, expecta-
tion of discrete random variable, properties of expectation and variance. Con-
tinuous random variable, distribution function, density of a continuous random
variable, expectation, change of variable formula, random vector, joint distri-
Description bution of random variables, joint density, distribution of sums and products
of random variables, conditional density, conditional expectation, order statis-
tics, moment generating function, characteristic function, brief introduction
to moment problem. Inequalities: Markov, Chebyshev, Schwarz and Chernoff
bound. Convergence in probability, almost sure convergence, convergence in
distribution, relation between these three modes of convergences, weak law
of large numbers (WLLN), strong law of large numbers (SLLN), central limit
theorem (CLT).
Course Code SI 429

Course Name Real analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Ajit kumar, and Kumaresan, S., A basic course in Real analysis, CRC
Press, Boca Raton, 2014.
2. Apostol, T.M., Mathematical analysis, 2nd edition, Narosa Publishers,

New Delhi, 2002.
3. Bartle, R.G., and Sherbert, D. R., Introduction to Real analysis, 4th

edition, John Wiley, New York, 2011.
Text Reference
4. Ghorpade, S.R., and Limaye, B.V., A course in Calculus and Real anal-
ysis, Springer (India), New Delhi, 2006.
5. Ross, K.A., Elementary analysis: The theory of Calculus, 2nd edition,

Springer (India), New Delhi, 2013.
6. Tao T., Analysis I, 3rd Edition, Hindustan Book Agency, New Delhi,
2006.
Review of sequences and series of real numbers. Limit superior and limit
inferior, Cauchy sequences and completeness of R. Tests for convergence of
series of real numbers. Basic notions of Metric Spaces with emphasis on Rn .
Heine Borel Theorem. Continuity and Uniform continuity. Derivatives. Mean
Value Theorem and applications. Functions of bounded variation. Riemann-
Description Stieltjes integral. Improper integrals and Gamma function. Sequences and
series of functions. Uniform convergence, interchanging limits with integrals
and derivatives. Arzela-Ascoli theorem (statement only). Functions of sev-
eral variables: Partial derivative, directional derivative, total derivative; Mean
value theorem, Taylor’s theorem.
135
Course Code SI 431

Course Name Introduction to Data Analysis using R
Total Credits 6
Type T
Lecture 2
Tutorial 0
Practical 2
Selfstudy 0
Half Semester N
Prerequisite Nil
1. FOSSEE, Spoken tutorials at https://r.fossee.in/ James, G., Witten, D.,

Hastie, T., and Tibshirani, R., An introduction to statistical learning
with applications in R, Springer, New York, 2013.
Text Reference
2. Wickham, H., Advanced R, CRC press, New York, 2.Wickham, H., and
Grolemund, G., R for Data Science, O’Reilly Media Inc, Canada, 2017.
Overview of R software, Data Frames, R Scripts, creating, import-

ing/exporting and merging of data sets, creating matrices and basic matrix
operations in R, 2d/3d plotting, programming in R (for, if else, do and while
loops), functions, creating report using R markdown. Exploring data us-
ing R, Scatter plot, histogram, bar chart, pie chart, box plot, basic statis-
tics computation (mean, median, variance etc.) Generating random samples
from standard distributions (such as Bernoulli, Poisson, Normal, Exponential
etc.) and comparing theoretical pdfs/pmfs using histograms/frequency distri-
butions, quantiles of sampling distributions (t, chi and F distribution) Maxi-
mization/minimization of functions in R (some algorithm), MLE estimation.
Description
Polynomial fitting of scatter plot, introducing regression line, least squares
estimates, residual plots, testing normality of residuals (qqplot), goodness of
fit measures and tests, testing of regression parameters, simulation of regres-
sion model, empirical distribution of least square estimator and its compari-
son with theoretical distribution. Simulation of multivariate normal random
vectors, estimation of mean and covariance matrix, eigen values and eigen vec-
tor of variance covariance matrix, spectral decomposition covariance matrix.
Generating dependent random variables with some models like (random walk,
AR(1), MA(1) etc).
Course Code SI 503

Course Name Categorical Data Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. A. Agresti, Categorical Data Analysis, 3rd Edition, Wiley, November

2012.
2. A. Agresti, An Introduction to Categorical Data Analysis,2nd Edition,

Wiley, March 2007.
3. E.B. Andersen, The Statistical Analysis of Categorical Data, Springer

for Science, 1997.
Text Reference
4. R.F. Gunst and R.L. Mason, Regression Analysis and its Applications –
A Data Oriented Approach, Marcel Dekkar, 1980.
5. T.J. Santner and D. Duffy, The Statistical Analysis of Discrete Data,

6. A.A. Sen and M. Srivastava, Regression Analysis – Theory, Methods and

Applications, Springer-Verlag, 1990
Two-way contingency tables: Table structure for two dimensions. Ways of

comparing proportions. Measures of associations. Sampling distributions.
Goodness-of-fit tests, testing of independence. Exact and large sample infer-
ence. Models of binary response variables. Logistic regression. Logistic mod-
els for categorical data. Probit and extreme value models. Log-linear models
for two and three dimensions. Fitting of logit and log-linear models. Log-
Description linear and logit models for ordinary variables. Regression: Simple, multiple,
non-linear regression, likelihood ratio test, confidence intervals and hypothe-
ses tests, tests for distributional assumptions Collinearity, outliers, analysis
of residuals. Model building, Principal component and ridge regression. Lab
component: Relevant real life problems to be done using statistical Software
Packages such as SAS etc.
137
Course Code SI 505

Course Name Multivariate Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite SI 424 (Statistical Inference 1)
1. T.W. Anderson, An Introduction to Multivariate Statistical Analysis,

3rd Ed., Wiley, July 2003.
2. R. Gnanadesikan, Methods for Statistical Data Analysis of Multivariate

Observations, John Wiley, New York, 1997.
Text Reference
3. R.A. Johnson and D.W. Wicheran, Applied Multivariate Statistical
Analysis, 6th Edition, Wiley, April 2007.
4. M.S. Srivastava and E.M. Carter, An Introduction to Multivariate Statis-

tics, North Holland, 1983.
K-variate normal distribution. Estimation of the mean vector and dispersion

matrix. Random sampling from multivariate normal distribution. Multivariate
Description distribution theory. Discriminant and canonical analysis. Factor analysis.
Principal components.Distribution theory associated with the analysis.
Course Code SI 507

Course Name Numerical Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. K.E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989.
2. S.D. Conte and C. De Boor, Elementary Numerical Analysis An Algo-

rithmic Approach, McGraw-Hill, 1981.
3. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Dif-

ferential Equations, Cambridge Univ. Press, Cambridge, 1996.
Text Reference
4. G.H. Golub and J.M. Ortega, Scientific Computing and Differential
Equations: An Introduction to Numerical Methods, Academic Press,
1992.
5. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd ed.,

Texts in Applied Mathematics, Vol. 12, Springer Verlag, New York,
1993.
Principles of floating point computations and rounding errors. Systems of

Linear Equations: factorization methods, pivoting and scaling, residual er-
ror correction method. Iterative methods: Jacobi, Gauss-Seidel methods with
convergence analysis, conjugate gradient methods. Eigenvalue problems: only
implementation issues. Nonlinear systems: Newton and Newton like methods
and unconstrained optimization. Interpolation: review of Lagrange interpola-
tion techniques, piecewise linear and cubic splines, error estimates. Approxi-
mation : uniform approximation by polynomials, data fitting and least squares
Description
approximation. Numerical Integration: integration by interpolation, adaptive
quadratures and Gauss methods. Initial Value Problems for Ordinary Differ-
ential Equations: Runge-Kutta methods, multi-step methods, predictor and
corrector scheme, stability and convergence analysis. Two Point Boundary
Value Problems : finite difference methods with convergence results. Lab
Component: Implementation of algorithms and exposure to public domain
packages like LINPACK and ODEPACK.
139
Course Code SI 509

Course Name Time Series Analysis
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. Brockwell P. and Davis R., Introduction to Time Series and Forecasting,

2. Brockwell P. and Davis R., Time Series: Theory and Methods, Springer,
New York, 1991.
3. Box G.E.P., Jenkins G., Reinsel G. and Ljung, Time Series Analysis-
Forecasting and Control, 5th Edition, Wiley, New York, 2016.
Text Reference
4. Chatfield C., The Analysis of Time Series - An Introduction, 6th Edition,
Chapman and Hall / CRC, New York, 2016.
5. Shumway R.H. and Soffer D.S., Time Series Analysis and Its Applica-
tions, 4th Edition, Springer, New York, 2016.
6. Weiss C. H., An Introduction to Discrete-Valued Time Series Data, John

Wiley & Sons, Inc., Chichester, 2018.
Stationary processes – strong and weak, linear processes, estimation of mean

and covariance functions. Wald decomposition Theorem. Modeling using
ARMA processes, estimation of parameters testing model adequacy, Order es-
timation. Prediction in stationary processes, with special reference to ARMA
Description
processes. Frequency domain analysis – spectral density and its estimation,
transfer functions. Nonlinear ARCH and GARCH models. Discrete-Valued
time series models.
Course Code SI 513

Course Name Theory of Sampling
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Chaudhuri, A. and Stenger, H., Survey Sampling: Theory and Methods,

Chapman and Hall/CRC, Boca Raton, 2005.
2. Cochran, W.G., Sampling Techniques, 3rd Edition, John Wiley and Sons,
New York, 1977.
Text Reference
3. Des Raj, Sampling Theory, McGraw-Hill Book Co., New York, 1978.
4. Mukhopadhyay, P., Theory and Methods of Survey Sampling, Prentice-

Hall of India New Delhi, 1998.
Principals of sample survey, Probability sampling, Non-probability sampling,

Simple random sampling, Estimation of population total, Variance estimation,
finite population correction, Random sampling with replacement, linear esti-
mators of population mean, Sampling for proportions and percentages, sample
size estimation for proportion as well as continuous data in random sampling.
Stratified random sampling, Estimator of population total and its variance,
Optimum allocation, comparison between stratified and simple random sam-
pling, Stratified sampling for proportion and sample size estimation, construc-
tion of strata, Number of strata, Quota sampling. Ratio estimator, estimation
Description
of variance from sample, comparison between ratio estimator and best linear
unbiased estimator, bias of ratio estimates, ratio estimates in stratified sam-
pling. Regression estimators, Large sample comparison with ratio estimate.
Single stage cluster sampling with equal and unequal cluster sizes, Sampling
with probability proportion to size, selection with unequal probabilities with
and without replacement, the Horvitz Thompson estimator, Brewer’s method,
Murthy’s method, Rao, Hartley and Cochran method. Two stage sampling
with units of equal and unequal sizes. Introduction to randomized response
techniques with examples and estimation.
141
Course Code SI 514

Course Name Statistical Modeling
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. T. Hastie, and R. Tibshirani, Generalized Additive Models, Chapman

and Hall, London, 1990.
2. G.A.F. Seber, and C.J. Wild, Nonlinear Regression, John Wiley & Sons,
Text Reference 1989.
3. W. Hardle, Applied Nonparametric Regression, Cambridge University

Press, London, 1990.
Nonlinear regression, Nonparametric regression, generalized additive models,

Bootstrap methods, kernel methods, neural network, Artificial Intelligence, a
Description
few topics from machine learning.
Course Code SI 515

Course Name Statistical Techniques in Data Mining
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. L. Breiman, J.H. Friedman, R.A. Olschen and C.J. Stone, Classification

of Regresion Trees, Wadsowrth Publisher, Belmont, CA, 1984.
2. D.J. Hand, H. Mannila and P. Smith, Principles of Data Minng, MIT

Press,Cambridge, MA 2001.
3. M.H. Hassoun, Fundamentals of Artificial Neural Networks, Prentice-

Hall of India,New Delhi January 2003.
Text Reference 4. T. Hastie, R. Tibshirani & J. H. Friedman, The elements of Statistical
Learning: Data Mining, Inference & Prediction, 2nd Edition, Springer
Series in Statistics, Springer-Verlag, New York February 2009.
5. R.A. Johnson and D.W. Wichern, Applied Multivariate Statistical 6th

Edition Pearson April 2007.
6. S. James Press, Subjective and Objective Bayesian Statistics: Principles,

Models, and Applications, 2nd Edition, Wiley, 2002.
Introduction to Data Mining and its Virtuous Cycle. Cluster Analysis: Hierar-
chical and Non-hierarchical techniques. Classification and Discriminant Anal-
ysis Tools: CART, Random forests, Fisher‘s discriminant functions and other
related rules, Bayesian classification and learning rules. Dimension Reduction
and Visualization Techniques: Multidimensional scaling, Principal Component
Description
Analysis, Chernoff faces, Sun-ray charts.Algorithms for data-mining using mul-
tiple nonlinear and non-parametric regression. Neural Networks: Multi-layer
perceptron, predictive ANN model building using back-propagation algorithm.
Exploratory data analysis using Neural Networks self organizing maps. Ge-
netic Algorithms, Neuro-genetic model building. Discussion of Case Studies.
143
Course Code SI 526

Course Name Experimental Designs
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. A.M. Kshirsagar, A First Course in Linear Models, Marcel Dekker, 1983.
2. D.C. Montgomery, Design and Analysis of Experiments, 8th Ed., John

Text Reference Wiley & Sons, 2012.
3. C.F.J. Wu and M. Hamada, Experiments: Planning Analysis, and Pa-

rameter Design Optimization, John Wiley & Sons, 2nd Edition 2009.
Linear Models and Estimators, Estimability of linear parametric functions.

Gauss-Markoff Theorem. One-way classification and two-way classification
models and their analyses. Standard designs such as CRD, RBD, LSD, BIBD.
Analysis using the missing plot technique. Fctorial designs. Confounding.
Description Analysis using Yates‘ algorithm. Fractional factorial. A brief introduction
to Random Effects models and their analyses. A brief introduction to spe-
cial designs such as split-plot, strip-plot, cross-over designs. Response surface
methodology. Applications using SAS software.
Course Code SI 527

Course Name Introduction to Derivative Pricing
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite SI 427 (Probability 1) and SI 537 (Probability 2)
1. D. G. Luenberger, Investment Science, Oxford University Press, 1998.
2. J. C. Hull, Options, Futures and Other Derivatives, 4th Edition,

Prentice-Hall, 2000.
Text Reference 3. J. C. Cox and M. Rubinstein, Options Market, Englewood Cliffs, N.J.:
Prentice Hall, 1985.
4. C. P Jones, Investments, Analysis and Measurement, 5th Edition, John

Wiley and Sons, 1996.
Basic notions – Cash flow, present value of a cash flow, securities, fixed income
securities, types of markets. Forward and futures contracts, options, properties
Description of stock option prices, trading strategies involving options, option pricing using
Binomial trees, Black – Scholes model, Black – Scholes formula, Risk-Neutral
measure, Delta – hedging, options on stock indices, currency options.
145
Course Code SI 534

Course Name Nonparametric Statistics
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. W.W. Daniel, Applied Nonparametric Statistics, 2nd ed., Boston: PWS-

KENT, 1990.
2. M. Hollandor, and D.A. Wolfe, Non-parametric Statistical Inference,

McGraw-Hill, 1973.
3. E.L. Lehmann, Nonparametric Statistical Methods Based on Ranks,

McGraw-Hill, 1975.
4. J.D. Gibbons, Nonparametric Statistical Inference Marcel Dekker,

NewYork, 1985.
Text Reference
5. R.H. Randles and D.A. Wolfe, Introduction to the Theory of Nonpara-
metric Statistics,Wiley, New York, 1979.
6. P. Sprent, Applied Nonparametric Statistical Methods, Chapman and

Hall, London, 1989.
7. B.C. Arnold, N. Balakrishnan and H. N. Nagaraja, First Course in Order

Statistics. John Wiley, NewYork, 1992.
8. J.K. Ghosh and R.V. Ramamoorthi, Bayesian Nonparametrics, Springer

Verlag, NY, 2003.
Kolmogorov-Smirnov Goodness of Fit Test. The empirical distribution and its

basic properties. Order Statistics. Inferences concerning Location parameter
based on one-sample and two-sample problems. Inferences concerning Scale
parameters. General Distribution Tests based on Two or More Independent
Description
Samples. Tests for Randomness and equality of distributions. Tests for Inde-
pendence. The one-sample regression problem. Asymptotic Relative Efficiency
of Tests. Confidence Intervals and Bounds.
Course Code SI 536

Course Name Analysis of Multi-Type and Big Data
Total Credits 6
Type T
Lecture 3
Tutorial 0
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Bollen K.A. Structural Equations with Latent Variables, New York: John
Wiley, 1989.
2. Bollen K.A. Latent Curve Models: A Structural Equation Perspective.

Hoboken: John Wiley, 2006.
3. Hastie, T., Tibshirani, R. and Friedman, J. The Elements of Statistical

Learning. Berlin: Springer, 2009.
4. B Ühlmann, P. and van de Geer, S. Statistics for High-Dimensional Data:

Text Reference Methods, Theory and Applications. Berlin: Springer, 2011.
5. Cressie, N., Statistics for Spatial Data, Revised Edition. NJ: Wiley
Classics, 2015.
6. Gamerman, D., Hedibert, F. L. Markov Chain Monte Carlo: Stochastic

Simulation for Bayesian Inference, 2nd ed. FL: Chapman and Hall/CRC,
2006.
7. Lecture Notes based on selected recent papers on Big Data Modeling and
Analysis.
Overview of Spatial Data, Structured Data. Structural Equation Modeling. In-

troduction to Big Data. Large dimension small size multivariate data analysis;
tackling the problems of estimation and inference. Classification of Big Data;
Description Screening and Variable Selection. Lasso Regression; Projection Methods. In-
troduction to Markov Chain Monte Carlo (MCMC) Simulations; MCMC tech-
niques for Bayesian Modeling of Big Data.
147
Course Code SI 537

Course Name Probability 2
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Athreya, K.B. and Lahiri, S. N., Measure Theory and Probability Theory,
2. Ash, R. B., Probability and measure theory, Second edition, Academic

Press, Burlington, 2000.
3. Billingsley, P., Probability and Measure, Anniversary Edition, John Wi-

ley & Sons, Hoboken, 2012.
Text Reference
4. Chung, K. L., A Course in Probability Theory, Third edition, Academic
5. Durret, R., Probability: Theory and Examples, Fifth edition, Cambridge

University Press, Cambridge, 2019.
6. Pollard, D., A user’s guide to Measure Theoretic Probability, Cambridge

University Press, Cambridge, 2002.
Probability space, random variables (R, Rd ) valued, distributions of random

variables, change of variables formula, expectation of R valued random vari-
able, Jensen’s inequality, Holder’s inequality, Chebyshev’s inequality, Fa-
tou’s lemma, monotone convergence theorem, dominated convergence theorem,
product measure, Fubini’s theorem, notion of independence of sigma-fields and
random variables, Kolmogorov’s consistency theorem. Convergence in proba-
bility, almost sure convergence, convergence in distribution, convergence in Lp ,
Description
relation between different modes of convergence, Borel-Cantelli lemma, char-
acteristic function, inversion formula, continuity theorems, Scheffe’s lemma,
uniform integrability, tightness, Helly’s selection principle, moment problem.
Weak law of large numbers, strong law of large numbers, central limit theorem.
Radon Nikodym theorem (statement only), conditional expectation: definition
and its properties.
Course Code SI 539

Course Name Random graphs
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Bela Bollobas, Random Graphs, second edition, Cambridge University

2. Rick Durrett, Random Graph Dynamics, Cambridge University Press,

Cambridge, 2010.
3. Alan Frieze and Michal Karonski, Introduction to Random Graphs, Cam-

Text Reference bridge University Press, Cambridge, 2016.
4. Remco van der Hofstad, Random Graphs and Complex Networks, Vol 1,
Cambridge University Press, Cambridge, 2017.
5. Svante Janson, Tomasz Luczak and Andrzje Rucinski, Random Graphs,

Wiley, 2000.
Review of probabilistic tools: Markov inequality, Chebyshev’s inequality, con-

centration inequalities: Hoeffding, Efron-Stein, Azuma-Hoeffding, Mcdiarmid
(statements only). Modes of convergences. Some real life examples, two ba-
sic models of random graphs (Erdos-Renyi model G(n, M ) and Erdos-Renyi-
Gilbert model G(n, p)) and relationship between them, monotonicity, thresh-
olds and sharp thresholds, evolution: sub-critical phase, super-critical phase,
Description
phase transition, connectivity, threshold for connectivity, dense and sparse ran-
dom graphs, degree sequence and asymptotic distribution of degrees, sub-graph
counts, its asymptotic distribution and thresholds for sub- graph containment.
Introduction to other random graph models: Generalized binomial model, Ex-
ponential random graph models, Configuration model, Preferential attachment
model, Stochastic block model, examples.
149
Course Code SI 541

Course Name Statistical Epidemiology
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Lawson, A. Statistical Methods in Spatial Epidemiology, 2nd Edition,

Wiley, New York, 2006.
2. Gordis, L. Epidemiology, 5th Edition, Elsevier Saunders, Philadelphia,

2014.
Text Reference
3. Kalbfleisch, J. and Prentice, R. The Statistical Analysis of Failure Time
Data, 2nd Edition, Wiley, New York, 2002.
4. Lee, E. and Wang, J. Statistical methods for survival data analysis, 3rd
Edition, John Wiley & Sons., Hoboken, 2003.
Epidemiologic approach to clinical trials: observational studies, cross-sectional

studies, designing a case control study, bias in a case-control study, match-
ing issues, cohort studies, design of a cohort study, biases in a cohort study,
comparing case and cohort studies, randomized trials, selection of subjects,
crossover trials, issues on sample size, recruitment. Case studies to explore
above topics.
Spatial Epidemiology: Geographical Representation and Mapping, Spatial In-
Description
terpolation and Smoothing Methods, Estimation and Inference, Spatial Prox-
imity Indices, Disease Clustering, Spatial Regression, Infectious disease mod-
elling.
Survival Analysis in Epidemiology: Functions of survival time, censoring mech-
anisms, nonparametric estimators of survival function, Cox’s proportional haz-
ards model, Cases studies using survival analysis methods in health research.
Course Code SI 543

Course Name Asymptotic Statistics
Total Credits 8
Type T
Lecture 3
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite SI 427 (Probability 1) and SI 424 (Statistical Inference 1)
1. DasGupta A., Asymptotic Theory of Statistics and Probability, Springer,

New York, 2008.
2. Serfling R.J., Approximation Theorems of Mathematical Statistics, Wi-

Text Reference ley, New York, 2009.
3. van der vaart A. W. and Wellner J. A., Weak Convergence and Empirical
Processes, Springer, New York, 1996.
Review of modes of stochastic convergences: Almost sure convergence, con-

vergence in probability, convergence in the p-th moment and their relations.
Convergence in distribution Additional topics in stochastic convergence: Port-
manteau theorem (Statement only). Convergence in total variation (Scheffe’s
theorem). Skorohod representation theorem. HallyBray theorems (Statement
only). Uniform tightness and Prohorov’s theorem for random vectors. Char-
acteristic function. Levy’s continuity theorem (statement only). Strong law
of large numbers (i.i.d. random variables with finite mean). Weak law of
large numbers (finite variance). Levy-Lindeberg central limit theorem. Delta
method and variance stabilizing transformations. Asymptotic properties of
Description moment estimators, M-estimators and Z-estimators. Strong consistency and
asymptotic normality of the MLE. Berry-Essen Theorem (without proof).
Argmax theorem (statement without proof). Convergence of U-statistics
(without proof) and its applications to linear rank statistics. Glivenko-Cantelli
lemma. Convergence of the Kolmogorov-Smirnov statistics to the Brownian
bridge (statement only without proof). Convergence of the quantile process.
Almost sure and weak convergence results for maximum of i.i.d. random vari-
ables. Order statistics. Renyi’s representation theorem for the order statistics
of the i.i.d. exponential random variables. Bahadur-Rao representation theo-
rem for the sample quantiles. Efficiency of tests: Asymptotic power function,
consistency and asymptotic relative efficiency
151
Course Code SI 544

Course Name Martingale theory
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite SI 427 (Probability 1) and SI 537 (Probability 2)
1. Athreya, K.B. and Lahiri, S.N., Probability Theory, Hindustan Book

Agency, 2006.
2. Billingsley, P., Probability and Measure, Anniversary Edition, John Wi-

ley and Sons, Hoboken, 2012.
Text Reference
3. Chung, K. L., A Course in Probability Theory, Third edition, Academic
4. Williams, D., Probability with martingales, Cambridge University Press,

Cambridge, 1991.
Review of conditional expectation: Conditional expectation and conditional

probability, properties of conditional expectation, regular conditional distri-
butions, disintegration, conditional independence. Martingales and Stopping
Description times: Stopping times, random time change, martingale property, optional
sampling theorem, maximum and up-crossing inequalities, martingale conver-
gence theorem, Martingale central limit theorem.
Course Code SI 546

Course Name Statistical Inference II
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
1. Casella G. and Berger R.L., Statistical Inference, Wadsworth, a part of

Cengage Learning, Delhi, 2002.
2. Dasgupta A., Asymptotic Theory of Statistics and Probability, Springer,

New York, 2008.
3. Jurečková J., Sen P.K. and Picek J., Methodology in Robust and Non-
parametric Statistics, CRC press, Boca Raton, 2012.
Text Reference 4. Lehmann E.L., Theory of Point Estimation, Springer, New York, 1998.
5. Lehmann E.L. and Romano J.P. , Testing of Statistical Hypotheses,

6. Huber P. J. and Ronchetti E.M., Robust Statistics, Wiley, New York,

2009.
7. Shao J., Mathematical Statistics, Springer, New York, 2003.
Minimaxity and admissibility: Minimax estimation, admissibility and mini-

maxity in exponential families, admissibility and minimaxity in group families,
Simultaneous estimation. Maxmin tests and invariance, Hunt-stein Theorem,
Most stringent tests. Multiple testing via Maximin procedures and Scheffé’s
S-method.
U-statistics: Variance computation and projection method. Convergence of
U statistics (one sample and two samples). Linear rank statistics. Asymp-
Description
totic normality under null hypothesis. Pitman’s Asymptotic relative efficiency,
Noether’s theorem for evaluating asymptotic relative efficiency. Bahadur’s ef-
ficiency. Resampling techniques. Robust inference: Break-down point in finite
sample, Influence curve. M-estimator, L-estimator, Restimator, minimum dis-
tance estimator and Pitman’s estimator. Relations to minimax estimator and
equivariant estimators. Robust tests and confidence sets.
153
Course Code SI 548

Course Name Computational Statistics
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Efron B. and Tibshirani R.J., An Introduction to the Bootstrap, Chap-

man and Hall, New York, 1993.
2. Gentle J.E., Elements of Computational Statistics (ECS), Springer-

3. Gentle J.E., Computational Statistics, Statistics and Computing Series,

4. Gelman A., Carlin J. B., Stern H. S., and, Dunson D. B., Vehtari A.,
and Rubin D.B., Bayesian Data Analysis, 3rd Edition, CRC Press, Taylor
and Francis Group, Boca Raton, 2014.
Text Reference 5. Givens G. H. and Hoeting J. A., Computational Statistics, 2nd Edition,
John Wiley and Sons, Inc., Hoboken, New Jersey, 2013.
6. Lange K., Numerical Analysis for Statisticians, 2nd Edition, Springer-

7. Little R.J.A. and Rubin D.B., Statistical Analysis with Missing Data,
2nd Edition, Wiley, New York, 2019.
8. Liu J., Monte Carlo Strategies in Scientific Computing, Springer-Verlag,

New York, 2001.
9. Rice J.A., Mathematical Statistics and Data Analysis, 2nd Edition,

Duxbury Press, Belmont, California, 1995.
Introduction to Bayesian Theory and methods; non-informative priors and con-

jugate priors; posterior inference (with special reference to one parameter expo-
nential family)-credible intervals and hypothesis testing; hierarchical and em-
pirical Bayesian models; computational techniques for use in Bayesian analysis,
Description especially the use of simulation from posterior distributions, with emphasis on
the WinBUGS package as a practical tool. MCMC simulation (Markov chains;
Metropolis-Hastings algorithm; Gibbs sampling; convergence), EM algorithm,
Bootstrap (Bootstrapping; jackknife resampling; percentile confidence inter-
vals). Permutation tests.
Course Code SI 550

Course Name Weak convergence and empirical processes
Total Credits 6
Type T
Lecture 2
Tutorial 1
Practical 0
Selfstudy 0
Half Semester N
Prerequisite Nil
1. Billingsley, Patrick. Convergence of probability measures. John Wiley

& Sons, 2013.
2. Pollard, David. Empirical processes: theory and applications. Institute

of Mathematical Statistics, 1990.
Text Reference 3. Shorack, Galen R., and Jon A. Wellner. Empirical processes with ap-
plications to statistics. Society for Industrial and Applied Mathematics,
2009.
4. van der Vaart, Aad W. and Wellner, Jon. Weak convergence and em-
pirical processes: with applications to statistics. Springer Science &
Business Media, 2013.
Metric space topology: compactness, connectedness, separability, Arzela-

Ascoli theorem. Lévy-Prokohorov metric between two probability measures,
probability measures on Polish space, tightness, finite-dimensional conver-
gence. Weak convergence of probability measures and its consequences: the
Portmanteau theorem, the mapping theorem, Skorohod’s representation theo-
Description
rem. Brownian motion: Donsker’s theorem, continuity of sample path, modu-
lus of continuity, Brownian bridge. Empirical distribution and its functionals.
Kolmogorov-Smirnov’s statistic for goodness-of-fit and weak convergence to
the Brownian bridge. Statistical consequences and applications of the weak
convergence.
155
Course Code SI 593

Total Credits 4
Type
Lecture
Tutorial
Practical
Selfstudy
Half Semester
Prerequisite
Text Reference 1.
Description
Course Code SI 598

Total Credits 6
Type
Lecture
Tutorial
Practical
Selfstudy
Half Semester
Prerequisite
Text Reference 1.
Description

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1

Department of Mathematics, IIT Bombay

2 Integrated M.Sc. program in Mathematics 5

3 Two year M.Sc. program in Mathematics 11

4 Two year M.Sc. program in Statistics 15

5 LIST OF ALL MATHEMATICS COURSES 17

6 LIST OF ALL STATISTICS COURSES 125

B.S. program in Mathematics (for those who

First Semester Second Semester

Third Semester Fourth Semester

Fifth Semester Sixth Semester

Seventh Semester Eighth Semester

1.2 List of Electives

Integrated M.Sc. program in Mathematics

2.1 Minimum Requirements

(2) Compulsory Project in the 5th year.

(3) Number of Credits in core courses (counted as above): 92

(4) Number of credits for the project: 30

(5) Minimum number of Department electives: 9

(6) Minimum no of credits in department electives (including the Advanced Electives): 54

(7) Minimum no of Institute electives: 2

(8) Minimum No of credits in institute electives: 12

(9) Minimum No of credits: 330

2.2 Curriculum for students switching after first year

Third Semester Fourth Semester

Fifth Semester Sixth Semester

Seventh Semester Eighh Semester

Ninth Semester Tenth Semester

Total credits for eight semesters = 264.

2.3 Curriculum for students switching after second year

Fifth Semester Sixth Semester*

Seventh Semester Eighh Semester

Ninth Semester Tenth Semester

Total credits for six semesters = 220.

2.4 List of Electives

Two year M.Sc. program in Mathematics

First Semester Second Semester

Third Semester Fourth Semester

3.2 List of Electives for Mathematics programs

Odd Semester Electives Even Semester Electives

Odd Semester Electives Even Semester Electives

Two year M.Sc. program in Statistics

First Semester Second Semester

Third Semester Fourth Semester

4.2 List of Electives for Statistics programs

Odd Semester Electives Even Semester Electives

Odd Semester Electives Even Semester Electives

LIST OF ALL MATHEMATICS COURSES

MA812 . . . . . . . . . . . . . . . . . . . . . 83 MA849 . . . . . . . . . . . . . . . . . . . . . 104

Course Code MA 001

Text Reference 1. nil

Complex numbers as ordered pairs. Argand’s diagram. Triangle inequality.

Course Code MA 002

Course Code MA 106

1. H. Anton, Elementary Linear Algebra with Applications (8th Edition),

Vectors in Rn , linear independence and dependence, linear span of a set of

Course Code MA 108

1. E. Kreyszig, Advanced Engineering Mathematics (8th Edition), John