| Department of Mathematics

# B.Sc. (Research) in Mathematics

## Program Structure

A sample path through the B.Sc. (Research) in Mathematics program is shown below. Students may be allowed by the Department Undergraduate Committee to alter the sequence of core courses in order to ease goals such as obtaining a Minor from another department or to complete prerequisites for a summer program or internship. Similarly, each student will have individually guided choice in timing CCC, UWE and Mathematics Elective courses.
 Foundations 4 Calculus 1 4 Linear Algebra I 4 Mathematical Modelling 4 CCC 3
 Calculus II 4 Computing 4 Vector Calculus and Geometry 4 UWE 3 CCC 3
 Real Analysis I 4 Algebra I 4 Numerical Analysis I 4 Probability 4 UWE/CCC 3
 Real Analysis II 4 Ordinary Differential Equations 4 Algebra II 4 Introduction to Statistics 4 UWE/CCC 3
 Partial Differential Equations 4 Linear Algebra II 4 Undergraduate Seminar 4 UWE 3 CCC 3
 Major Elective 4 Major Elective 4 Major Elective 4 UWE 3 CCC 3
 Undergraduate Thesis I 4 Major Elective 4 Major Elective 4 UWE 3 UWE 3 CCC 3
 Undergraduate Thesis II 8 Major Elective 4 UWE 3 CCC 3
You must credit at least two courses from this category.
Course code
Title
Credit
MAT101
Calculus I
4

Core course for B.Sc. (Research) programs in Mathematics, Physics and Economics. Optional course for B.Sc. (Research) Chemistry.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII mathematics or MAT 020 (Elementary Calculus)

Overview:  This course covers one variable calculus and applications. It provides a base for subsequent courses in advanced vector calculus and real analysis as well as for applications in probability, differential equations, optimization, etc. One of the themes of the course is to bring more rigour to the formulas and techniques students may have learned in school.

Detailed Syllabus:

1. Real Number System: The axioms for N and R, mathematical induction.
2. Integration: Area as a set function, integration of step functions, upper and lower integrals, integrability of bounded monotone functions, basic properties of integration, polynomials, trigonometric functions.
3. Continuous Functions: Functions, limits, continuity, Intermediate Value Theorem, Extreme Value Theorem, integrability of continuous functions, Mean Value Theorem for integrals.
4. Differentiation: Tangent line, rates of change, derivative as function, algebra of derivatives, implicit differentiation, related rates, linear approximation, differentiation of inverse functions, derivatives of standard functions (polynomials, rational functions, trigonometric and inverse trigonometric functions), absolute and local extrema, First Derivative Test, Rolle's Theorem, Mean Value Theorem, concavity, Second Derivative Test, curve sketching.
5. Fundamental Theorem of Calculus: Antiderivatives, Indefinite Integrals, Fundamental Theorem of Calculus, Logarithm and Exponential functions, techniques of integration.
6. Polynomial Approximations: Taylor polynomials, remainder formula, indeterminate forms and L'Hopital's rule, limits involving infinity, improper integrals.
7. Ordinary Differential Equations: 1st order and separable, logistic growth, 1st order and linear.

References:

1. Calculus, Volume I, by Tom M Apostol, Wiley.
2. Introduction to Calculus and Analysis I by Richard Courant and Fritz John, Springer
3. Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition.
4. Calculus with Analytic Geometry by G F Simmons, McGraw-Hill

Past Instructors: Amber Habib, Debashish Bose

MAT102
Calculus II
4

Core course for B.Sc. (Research) programs in Mathematics, Physics. Optional course for B.Sc. (Research) Chemistry.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 101 (Calculus I)

Overview: The first part is an introduction to multivariable differential calculus. The second part covers sequences and series of numbers and functions. The concepts and techniques covered here are used extensively in the social and natural sciences as well as in engineering.

Detailed Syllabus:

1. Differential calculus in several variables: Functions of several variables, level curves and surfaces, limits and continuity, partial derivatives, tangent planes, chain rule, directional derivatives, gradient, Lagrange multipliers, extreme values and saddle points, 2nd derivative test
2. Sequences and Series: Limits of sequences, algebra of limits, series, divergence test, comparison and limit comparison tests, integral test, alternating series test, absolute convergence, root & ratio tests, Abel and Dirichlet tests, power series, Taylor series, Fourier Series.

References:

• Calculus, Volume II, by Tom M Apostol, Wiley.
• Essential Calculus – Early Transcendentals by James Stewart, Cengage, India Edition.
• Calculus and Analytic Geometry by G B Thomas and R L Finney, 9th edition, Pearson.
• Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1st edition, Springer (India), 2011.
• Calculus by Ken Binmore and Joan Davies, 1st edition, Cambridge, 2010.

Past Instructors: Amber Habib, Debashish Bose

MAT140
Discrete Structures
4

Major elective course for B.Sc. (Research) Mathematics. Not open to B.Tech. Computer Science majors or any other student who has taken CSD205.

Credits: 3:0:1 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII Mathematics

Overview: This course offers an in-depth treatment of Lattice theory which will be used in areas of algebra and analysis in the subsequent semesters. Special kinds of lattices known as Boolean algebras are studied in reasonable detail and their importance is demonstrated through real life applications involving digital circuits. This course builds on the Foundations course taken by the first year students and it provides an exposure to formal proof writing.

Detailed Syllabus:

1. Theory of Relations: Types of relations, Matrix representation of relations, Equivalence classes, Operations on relations, Closure of relations, Importance of transitive closure, Warshall’s algorithm.
2. Lattice Theory: Posets, Chains, Hesse diagram,  Extremal elements in a poset, Meet and Join operations, Lattices, General properties of lattices, isomorphism, modular lattice, distributive lattice, complements, atoms in a lattice, Boolean algebras.
3. Finite Boolean algebras: Functions on Boolean algebras, Karnaugh maps, Logic gates, Digital circuits.

References:

1. Thomas Donnellan, Lattice Theory, Pergamon Press, Oxford.
2. J.E. Whitesitts, Boolean Algebra and Its Applications, Addison-Wesley Publications.
3. G. Birkhoff, Lattice Theory, American Mathematical Society, 2nd Edition.
4. E. Mandelson, Schaum’s Outline of Boolean Algebra and Switching Circuits, McGraw Hill.
5. Bernard Kolman, Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Pearson Education, New Delhi.
6. Kenneth H. Rosen, Discrete Mathematics and its Applications, Tata McGraw-Hill, New Delhi.
7. C. L. Liu, D. P. Mohapatra, Elements of Discrete Mathematics, Tata McGraw-Hill, New Delhi.
8. J.P. Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, 1st edition, Tata McGraw-Hill, New Delhi, 2001.

Past Instructors: Niteesh Sahni

MAT220
Real Analysis I
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Calculus I (MAT 101)

Overview: This course provides a rigorous base for the geometric facts and relations that we take for granted in one-variable Calculus. The main ingredients include sequences; series; continuous and differentiable functions on R; their various properties and some highly applicable theorems. This is the foundational course for further study of topics in pure or applied Analysis, such as Metric Spaces, Complex Analysis, Numerical Analysis, and Differential Equations.

Detailed Syllabus:

1. Fundamentals: Review of N, Z and Q, order, sup and inf, R as a complete ordered field, Archimedean property and consequences, intervals and decimals. Functions: Images and pre-images, Cartesian product, Cardinality.
2. Sequences: Convergence, bounded, monotone and Cauchy sequences, subsequences, lim sup and lim inf.
3. Series: Infinite Series: Cauchy convergence criterion, Infinite Series of non-negative terms, comparison and limit comparison, integral test, p-series, root and ratio test, power series, alternating series, absolute and conditional convergence,  rearrangement.
4. Continuity: Limits of functions, continuous functions, Extreme Value Theorem, Intermediate Value Theorem, monotonic functions, uniform continuity.
5. Differentiation: Differentiable functions on R, local maxima, local minima, Mean Value Theorems, L'Hopital's Rule, Taylor's Theorem.

References:

1. A Basic Course in Real Analysis by Ajit Kumar and S Kumaresan. CRC Press, 2014.
2. Introduction to Real Analysis by R G Bartle & D R Sherbert, John Wiley & Sons, Singapore, 2/e (or later editions), 1994.
3. Elementary Analysis: The Theory of Calculus by Kenneth A Ross. Springer India, 2004.
4. Analysis I by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009.
5. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006.
6. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002.
7. Calculus, Volume 1, by Tom Apostol, Wiley India. 2nd Edition, 2011.

MAT240
Algebra I
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII Mathematics or MAT100 (Foundations)

Overview: Algebraic structures like groups, rings, integral domains, fields, modules and vector spaces are present in almost all mathematical applications as well as in development of more complicated structures in mathematics. The basic building block of these structures is the group. This, a first course in Abstract Algebra, concentrates mainly on groups and their basic properties. If there is time, we shall also take a brief look at Rings.

It is desirable that the student already has a basic understanding of sets, relations, functions, binary operations, equivalence relations, and sets of numbers.

Detailed Syllabus:

1. Groups: Definition, Examples and Elementary Properties.
Subgroups: Subgroup Tests, Subgroups Generated by Sets, Cyclic Groups, Classification of Subgroups of Cyclic Groups, Cosets and Lagrange's Theorem.
2. Normal Subgroups and Quotient Groups, Homomorphisms, Isomorphisms and Automorphisms of a Group.
Conjugates, centre, centralizer, normalizer.
Cayley’s Theorem.
Direct Products, Finite Abelian Groups.
3. Permutation Groups: Definition, Examples and Properties, Symmetric Group of n Letters (Sn), Alternating Group on n Letters (An).
4. (If time permits) Rings, Homomorphisms, Ideals and Quotient Rings, Integral Domains.

References:

1. Contemporary Abstract Algebra by Joseph A. Gallian, 4th  edition. Narosa, 1999.
2. Algebra by Michael Artin, 2nd Edition. Prentice Hall India, 2011.
3. Topics in Algebra by I.N. Herstein, 2nd Edition. Wiley India, 2006.
4. A First Course in Abstract Algebra by John B. Fraleigh, 7th Edition. Pearson, 2003.
5. Undergraduate Algebra by Serge Lang, 2nd Edition. Springer India, 2009.
6. Abstract Algebra by David S. Dummit and Richard M. Foote, 3rd Edition. John Wiley and Sons, 2011.

Past Instructors: Neha Gupta, Sanjeev Agrawal

MAT160
Linear Algebra
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII Mathematics

Overview: Linear Algebra provides the means for studying several quantities simultaneously. A good understanding of Linear Algebra is essential in almost every area of higher mathematics, and especially in applied mathematics. A CAS such as Maxima/Matlab will be used throughout the course for computational purposes.

Detailed Syllabus:

1. Matrices and Linear Systems
2. Vector Spaces and Linear Transformations
3. Inner Product Spaces
4. Determinant
5. Eigenvalues and Eigenvectors, Diagonalization
6. Quadratic Forms and Positive Definite Matrices
7. Applications chosen from: Numerical aspects, Difference equations, Markov matrices, Least squares.

References:

1. Linear Algebra by Jim Hefferon
2. Linear Algebra and its Applications by Gilbert Strang, 4th edition, Cengage.
3. Linear Algebra and its Applications by David C. Lay, 3rd edition, Pearson.
4. Linear Algebra: A Geometric Approach by S. Kumaresan, PHI, 2011.
5. Elementary Linear Algebra by Howard Anton and Chris Rorres, 9th edition, Wiley.
6. Linear Algebra: An Introductory Approach by Charles Curtis, Springer.
7. Matrix Analysis and Applied Linear Algebra by Carl D Meyer, SIAM.
8. Videos of lectures by Prof Gilbert Strang: 18.06 Linear Algebra, Spring 2010. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu

MAT280
Numerical Analysis I
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab) = 3:0:1 (3 lectures and 1 two-hour lab weekly)

Prerequisites: Class XII Mathematics

Overview: Numerical Analysis takes up the problems of practical computation that arise in various areas of mathematics, physics and engineering. The focus is on analyzing the numerical methods and algorithms for obtaining approximate solutions, error estimates and rate of convergence, and implementation of computer programs.

Detailed Syllabus:

1. Solving Equations: Iterative methods, Bisection method, Secant method, Newton-Raphson method, Rates of convergence, Roots of polynomials.
2. Interpolation: Lagrange and Hermite interpolation, Interpolating polynomials using difference operators.
3. Numerical Differentiation: Methods based on interpolation, methods based on finite difference operators.
4. Numerical Integration: Newton-Cotes formula, Gauss quadrature, Chebyshev’s formula.
5. Systems of Linear Equations: Direct methods (Gauss elimination, Gauss-Jordan method, LU decomposition, Cholesky decomposition), Iterative methods (Jacobi, Seidel, and Relaxation methods)
6. Labs: Computational work using C, Python or Matlab.

References:

1. E. Suli and D. Mayers, Introduction to Numerical Analysis, Cambridge University Press, 2003.
2. R.L. Burden and J.D. Faires, Numerical Analysis, Cengage Learning, 9th Edition, 2010.
3. M.K. Jain, S.R.K. Iyengar, and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, New Age International Ltd., 1999.
4. J.H. Mathews and K. Fink, Numerical Methods using Matlab, PHI Learning, 4th Edition, 2003.
MAT184
Probability
4

Core course for B.Sc. (Research) Mathematics, Economics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Any one of Calculus I (MAT 101) or Elementary Calculus (MAT 020) or Mathematical Methods I (MAT 103) or Basic Probability & Statistics (MAT 084)

Overview: Probability is the means by which we model the inherent randomness of natural phenomena. This course introduces you to a range of techniques for understanding randomness and variability, and for understanding relationships between quantities. The concluding portions on Statistics take up the problem of testing our theoretical models against actual data, as well as applying the models to data in order to make decisions. This course is a prerequisite for later courses in Advanced Statistics, Stochastic Processes, and Mathematical Finance, as well as for the Minor in Data Analytics.

Detailed Syllabus:

1. Probability: Classical probability, axiomatic approach, conditional probability, independent events, addition and multiplication theorems with applications, Bayes’ theorem.
2. Random Variables: Probability mass function, probability density function, cumulative density function, expectation, variance, standard deviation, mode, median, moment generating function.
3. Some Distributions and their Applications: Uniform (discrete and continuous), Bernoulli, Binomial, Poisson, Exponential, Normal.
4. Joint Distributions: Joint and marginal distributions, independent random variables, IIDs, conditional distributions, covariance, correlation, moment generating function.
5. Sequences of Random Variables: Markov’s Inequality, Chebyshev’s Inequality, Law of Large Numbers, Central Limit Theorem.

References:

1. A First Course in Probability by Sheldon Ross, 6th edition, Pearson.
2. Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press.
3. Theory and Problems of Probability and Statistics by Murray R Spiegel and Ray Meddis, Schaum’s Outlines.
4. John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011.
5. Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance by Kai Lai Chung and Farid Aitsahlia, 4th edition, Springer International Edition, 2004.

Past Instructors: Debashish Bose, Suma Ghosh

You must credit at least two courses from this category. The given list is not complete and any MAT course of level 300 or above is also in this category if it is available as UWE.
MAT484
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 184 Probability

Overview: Regression, the most widely used statistical technique, estimates relationships between independent (explanatory) variables and a dependent (outcome) variable. In this course you will learn different ways of estimating the parameter of the statistical models, criteria for these estimations, and then use them for deriving the coefficients of the regression models, use software  (R) to implement them, learn what assumptions underlie the models, learn how to test whether your data meet those assumptions and what can be done when those assumptions are not met, and develop strategies for building and understanding useful models.

Detailed Syllabus:

1. Review: Introduction, Descriptive Statistics; Sampling Distributions. Graphical representation of data, Basic distributions, properties, fitting, and their uses;
2. Estimation: Point and interval estimation, Histogram and Kernel density estimation, Sufficiency, Exponential family, Bayesian methods, Moment methods, Least squares, Maximum likelihood estimation;
3. Criteria for estimation: UMVUE, Large sample theory Consistency; asymptotic normality, Confidence intervals, Elements of hypothesis testing; Neyman-Pearson Theory, UMP tests, Likelihood ratio and related tests, Large sample tests;
4. Linear Models:  Simple and Multiple linear regression, Analysis and Inference.

References:

1. Mathematical Statistics: Basic Ideas and Selected Topics by Peter J. Bickel and Kjell A. Doksum
2. Testing Statistical Hypotheses by Erich L. Lehmann
3. Statistical Decision Theory: Foundations, Concepts and Methods by James O. Berger

Past Instructors: Charu Sharma

MAT584
Stochastic Processes
4
Stochastic Processes
MAT221
Real Analysis II
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Real Analysis I (MAT 220)

Overview: Continuing the work done in MAT 220 of  understanding  the rigor behind one-variable Differential Calculus, this course dwells on various aspects of Integration as well as functions on higher dimensional spaces. We discuss sequences and series of functions; uniform convergence and consequences; some important approximation theorems for continuous functions; rigorous discussions of some special functions; and finally the world of functions of several variables.

Detailed Syllabus:

1. Integration: Upper and lower Riemann integrals, basic properties of Riemann integral, Riemann integrability of continuous and monotone functions, non-Riemann integrable functions, Fundamental Theorem of Calculus and consequences.
2. Sequences and Series of Functions: Pointwise and uniform convergence, uniform convergence and continuity, series of functions, Weierstrass M-test,  uniform convergence and integration, uniform convergence and differentiation,  equicontinuous families of functions, Stone-Weierstrass Theorem.
3. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions.
4. Topology of Rn: Open and closed sets, continuous functions, completeness, compactness, connectedness.
5. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem.

References:

1. Analysis II by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009.
2. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006.
3. Real Mathematical Analysis by Charles C Pugh. Springer India. 2004.
4. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002.
5. Calculus, Volume 2, by Tom Apostol, Wiley India. 2nd Edition, 2011.

MAT241
Algebra II
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Algebra I (MAT 240)

Overview: The course continues the work done in MAT 240 on the one hand by extending the study of groups to include group actions and applications, and on the other by studying the algebraic structures of rings and fields.

Detailed Syllabus:

1. Groups – Definition,  subgroups, cyclic groups, homomorphisms, normal subgroups, semi-direct products, group actions, Sylow theorems.
2. Rings – Definition and examples of rings, ideals, quotient rings, maximal ideals, prime ideals, ring homomorphisms, integral domains, Euclidean domains, PID, UFD.
3. Polynomial Rings and Fields – Polynomial rings, irreducible polynomials, definition and examples of fields, characteristic, field extensions, finite fields.

References:

1. I. N. Herstein, Topics in Algebra, 2/e, Wiley Eastern, 1994.
2. Bhattacharya, Jain and Nagpaul, Basic Abstract Algebra, 2nd edition, CUP, 1995.
3. Joseph A. Gallian, Contemporary Abstract Algebra, 4th edition, Narosa, 1999.
4. M. Artin, Algebra, 2nd edition. Prentice Hall India, 2011.
5. Dummit and Foote, Abstract Algebra, 3rd edition, Wiley.
6. Serge Lang, Undergraduate Algebra, 2nd edition. Springer India, 2009.
7. Thomas W. Hungerford, Algebra, GTM 73, Springer India, 2004.
MAT360
Linear Algebra II
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Linear Algebra I (MAT 160)

Overview: In MAT 260 we studied real and complex linear transformations up to the diagonalizability of symmetric operators. In this course we take up vector spaces over arbitrary fields and more advanced results on expressing linear transformations by simple matrices.

Detailed Syllabus:

1. Linear Equations – Systems of linear equations, matrices, elementary row operations and row reduction.
2. Vector spaces – Abstract vector spaces, subspaces, dimension, coordinates.
3. Linear transformations – Matrix representations, change of basis, linear functionals and the double dual, transpose.
4. Determinants – Commutative rings, determinant function, permutations, properties.
5. Canonical Forms – Characteristic values, invariant subspaces, simultaneous diagonalization and triangulation, invariant direct sums, Primary Decomposition Theorem, cyclic subspaces, Rational Form, Jordan Form.
6. Inner Product Spaces – Linear functionals and adjoints, unitary and normal operators, spectral theory.

References:

1. Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd edition, PHI Learning.
2. Friedberg, Insel and Spence, Linear Algebra, 4th edition, PHI Learning
3. Sheldon Axler, Linear Algebra Done Right, 2nd edition, Springer International Edition
4. Paul Halmos, Finite Dimensional Vector Spaces, 2nd edition, Springer International Edition
5. Paul Halmos, Linear Algebra Problem Book, Mathematical Association of America, 1995.

Past Instructors: Neha Gupta

MAT388
Optimization I
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 160 (Linear Algebra) or MAT 104 (Mathematical Methods II)

Overview: Optimization deals with the problem of establishing the best & worst cases for a given situation. This course deals mostly with the special case of linear programming, which is commonly applied to problems of business and economics as well as industrial problems in transportation, energy and telecommunication.

Detailed Syllabus:

1. Mathematical modeling and optimization problem formulation
2. Application of optimization (linear case)
3. Geometry of linear optimization
4. Simplex method
5. Duality theory
6. Sensitivity analysis
7. Robust optimization
8. Graphs and Network flow problems
9. Discrete optimization or Integer programming formulations
10. Non-linear optimization – introduction and applications

Main References:

1. Linear Programming by G. Hadley, Narosa, 2000
2. Understanding and Using Linear Programming by J. Matousek and B. Gärtner, Springer, 2006
3. Introduction to Linear Optimization by D. Bertsimas and J. Tsitsiklis, Athena Scientific, 1997
4. Theory of Linear and Integer Programming by A. Schrijver, Wiley, 1998
5. Operations Research: An Introduction by H. Taha, Pearson, 2012

Past Instructors: Samit Bhattacharyya

MAT390
Introduction to Mathematical Finance
4

A Major Elective for B.Sc. (Research) Mathematics. Cross-listed as FAC201.

Credits (Lec:Tut:Lab): 3:0:1 (3 lectures and 1 two-hour lab weekly)

Prerequisites: MAT 184 Probability or MAT 205 Mathematical Methods III or CSD209.

Overview: Mathematical Finance is a modern study area where mathematical methods are used to create and add immense value in a practical environment. The aim of this course is twofold. First, to discuss the mathematical models that have driven the explosion of financial services and products over the last 30 years or so. Second, to use spreadsheet programs to work with actual data. This course is also the gateway to our Specialization in Mathematical Finance.

Detailed Syllabus:

1. Basic concepts: Bonds and shares, risk versus profit, return and interest, time value of money, arbitrage.
2. Fixed Income Securities: Net Present Value and Internal Rate of Return, price and yield of a bond, term structures, duration, immunization.
3. Mean-Variance Analysis: Random returns, efficient portfolios, feasible set, Markowitz model, Two Fund and One Fund Theorems, Capital Asset Pricing Model and applications.
4. Forwards, Futures and Swaps: Replicating portfolios, futures on assets without income, futures on assets with fixed income or dividend yield, hedging with futures, currency futures, stock index futures, forward rate agreements, interest rate swaps, currency swaps, equity swaps.
5. Stock Price Models: Geometric Brownian Motion, Binomial Tree.
6. Options: Call and put options, put-call parity, Binomial Options Pricing Model, dynamic hedging, risk-neutral valuation, Black-Scholes formula, trading strategies.
7. Labs: Microsoft Excel and VBA.

References:

1. Principles of Finance with Excel 2nd edition by Simon Benninga, Oxford University Press, 2010.
2. Mathematics for Finance by M Capinski and T Zastawniak, Springer (International Edition), 2003.
3. The Calculus of Finance by Amber Habib, Universities Press, 2011.
4. Options, Futures and Other Derivatives 7th edition by John C Hull and Sankarshan Basu, Pearson 2009.
5. Investment Science by David Luenberger, Oxford University Press (Indian Edition), 1997.
6. An Elementary Introduction to Mathematical Finance 2nd edition by Sheldon Ross, Cambridge University Press (Indian Edition), 2005.

Past Instructors: Amber Habib, Charu Sharma, Sunil Bowry

MAT420
Measure and Probability
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 220 Real Analysis I. And one of MAT 184 Probability or MAT 205 Mathematical Methods III or CSD209.

Overview: This sequel to an introductory course on probability provides a rigourous look at the subject which opens up many more applications, especially to stochastic processes. This course is compulsory for students opting for the Specialization in Mathematical Finance.

Detailed Syllabus:

1. Introduction to measures and probability triples
2. Random variables: independence, limit events, expectation
3. Inequalities and convergence, Laws of large numbers
4. Distributions, change of variables
5. Limit Theorems, Differentiation of expectation, Moment generating functions, Fubini’s theorem
6. Weak convergence
7. Characteristic functions, Central Limit Theorem and generalizations, Method of Moments
8. Lebesgue and Hahn decompositions
9. Conditional probability and expectation

References:

1. A First Look at Rigorous Probability Theory by J S Rosenthal, 2nd edition, World Scientific Publishing, 2006.
2. Measure, Integral and Probability by M Capinski and E Kopp, 2nd edition, Springer.
3. Probability and Random Processes by G R Grimmett and D R Stirzaker, 2nd edition, Oxford University Press.

Past Instructors: Debashish Bose

MAT230
Ordinary Differential Equations
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 101 Calculus I or MAT 103 Mathematical Methods II

Overview: Ordinary Differential Equations are fundamental to many areas of science. In this course we learn how to solve large classes of them, how to establish that solutions exist in others, and to find numerical approximations when exact solutions can’t be achieved. Further, many phenomena which undergo changes with respect to time or space can be studied using differential equations. In this course, we will also see many examples of mathematical modeling using differential equations.

Detailed Syllabus:

1. First Order ODEs: Modelling, Geometrical Meaning, Solution techniques
2. Second and Higher Order Linear ODEs: Modelling, Geometrical Meaning, Solution techniques
3. Numerical Techniques
4. Existence of Solutions of Differential Equations
5. Systems of ODEs: Phase Plane and Qualitative Methods
6. Laplace Transforms
7. Series Solutions

References:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 9th edition, Wiley India, 2012.
2. G.F. Simmons and S. Krantz, Differential Equations: Theory, Technique, and Practice, McGraw Hill Publishing Company, 2006.
3. J. Polking, D. Arnold, A. Boggess, Differential Equations, Pearson, 2005.
4. C. Henry Edwards and David E. Penney, Differential Equations and Boundary Value Problems: Computing And Modeling, 3rd edition, Pearson, 2010.
5. Hirsch, Morris W., Stephen Smale, and Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2012.

Past Instructors: Ajit Kumar

MAT332
Geometry of Curves & Surfaces
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. And MAT 160 Linear Algebra I.

Overview: This course combines the traditional approach to learn the basic concepts of curves and surfaces with the symbolic manipulative abilities of Mathematica. Students will learn and study the classical curves/surfaces as well as more interesting curves/surfaces using computer methods. For example, to see the effect of change of parameter, the student will explore and observe with the help of Mathematica and then the mathematical proof of the observation will be developed in the class.

Detailed Syllabus:

1- Curves in the plane: Length of a curve, Vector fields along curves. Famous plane curves: cycloids, lemniscates of Bernoulli, cardioids, catenary, cissoid of Diocles, tractrix, clothoids, pursuit curves.

2- Regular curve, curvature of a curve in a plane, curvature and torsion of a curve in R3. Determining a plane curve from given curvature.

3- Global properties of plane curves: Four vertex theorem, Isoperimetric inequality.

4- Curves on Sphere. Loxodromes on spheres, animation of curves on a sphere.

5- Review of calculus in Euclidean space.

6- Surfaces in Euclidean spaces: Patches in R3, local Gauss map, Regular surface, Tangent vectors.

7- Example of surfaces: Graphs of a function of two variables, ellipsoid, stereographic ellipsoid, tori, paraboloid, seashells.

8- Orientable and Non-orientable surfaces. Mobius strip, Klein Bottle.

9- The shape operator, normal curvature, Gaussian and mean curvature, fundamental forms.

10- Surfaces of revolution

References:

1. Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition by Elsa Abbena, Simon Salamon, Alfred Gray.
2. Elementary Differential Geometry by A.N. Pressley, Springer Undergraduate Mathematics Series.
3. Differential Geometry of Curves and Surfaces by Manfredo DoCarmo.

MAT440
Elementary Number Theory
4

Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII Mathematics

Overview: This introductory course to Number Theory is also the entry point to the specialization in applications of algebra.

Detailed Syllabus:

1. Divisibility: Definition and properties of Divisibility, Division Algorithm, Greatest Common Divisor, Least Common Multiple, Euclidean Algorithm, Linear Diophantine Equations.
2. Primes and their Distribution: Sieve of Eratosthenes, Euclid's Theorem, Prime Number Theorem (statement only), Goldbach Conjecture, Twin Primes, Fermat Primes, Mersenne Primes, The Fundamental Theorem of Arithmetic, Euclid's Lemma, Divisibility, gcd and lcm in terms of prime factorizations, Dirichlet's Theorem on primes in arithmetic progressions (statement only).
3. Theory of Congruences: Residue Classes, Linear congruences in one variable, Euclid's algorithm Chinese Remainder Theorem, Wilson's Theorem, Fermat's Theorem, Pseudoprimes and Carmichael Numbers, Euler's Theorem, Primality Testing, The Pollard Rho Factoring Method, Complete residue system.
4. Applications of Congruences: Divisibility tests, Modular Designs, Check Digits, The p-Queens Puzzle, Round-Robin Tournaments, The Perpetual Calendar.
5. Arithmetic Functions: Multiplicative Functions, Moebius function, Moebius inversion formula, The number-of-divisors and sum-of-divisors functions, Euler phi function, Greatest Integer Function, Carmichael conjecture,Perfect numbers, characterization of even perfect numbers, Dirichlet product, Riemann Zeta function.
6. Group of Units and Quadratic Residues: Primitive roots, Group of units, Quadratic Residues and Non-Residues, Legendre symbol, Euler's Criterion, Gauss' Lemma,  Law of Quadratic Reciprocity.
7. Sums of Squares: Sums of Two squares, Sums of Three squares and Sums of Four squares.

References:

• David M. Burton Elementary Number Theory, Tata McGraw-Hill.
• Gareth A. Jones and J. Mary Jones Elementary Number Theory, Springer Undergraduate Mathematics Series.
• Thomas Koshy Elementary Number Theory with Applications, 2nd  Edition, Academic Press.
• Kenneth Rosen Elementary Number Theory and its Applications, 5th Edition, McGraw Hill.
• G. H. Hardy and E. M. Wright An Introduction to the Theory of Numbers, 5th edition, Oxford University Press.

Past Instructors: A Satyanarayana Reddy

MAT442
Graph Theory
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 160 Linear Algebra I

Overview: Graphs are fundamental objects in combinatorics. The results in graph theory, in addition to their theoretical value, are increasingly being applied to understand and analyze systems across a broad domain of enquiry, including natural sciences, social sciences and engineering. The course does not require any background of the learner in graph theory. The emphasis will be on the axiomatic foundations and formal definitions, together with the proofs of some of the central theorems. Few applications of these results to other disciplines would be discussed.

Detailed Syllabus:

Unit 1

• Definitions of Graph, Digraph, Finite and Infinite Graph, Degree of a Vertex, Degree Sequence, Walk, Path, Cycle, Clique.
• Operations on graphs, Complement of a graph, Subgraph, Connectedness, Components, Isomorphism.
• Regular graph, Complete graph, Bipartite graph, Cyclic graph, Euler graph, Hamiltonian path and circuit, Tree, Cut set, Spanning tree.

Unit 2

• Planar graph, Colouring, Covering, Matching, Factorization, Independent sets.

Unit 3

• Graphs and relations, Adjacency matrix, Incidence matrix, Laplacian matrix, Spectral properties of graphs, Matrix tree theorem, Automorphism group of a graph.

Unit 4

• DFS, BFS for minimal spanning tree, Kruskal, Prim and Dijkstra algorithms.

References:

• D. West, Introduction to Graph Theory, 2nd ed., PHI Learning, New Delhi, 2009.
• N. Deo, Graph Theory: With Application to Engineering and Computer Science, PHI Learning, New Delhi, 2012.
• C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, New Delhi, 2013.
• B. Kolman, R.C. Busby, S.C. Ross, Discrete Mathematical Structures, 6th ed., PHI Learning, New Delhi, 2012.
• F. Harary, Graph Theory, Narosa, New Delhi, 2012.
• J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, New Delhi, 2013.
• R.J. Wilson, Introduction to Graph Theory, 4th ed., Pearson Education, New Delhi, 2003.

Past Instructors: Sudeepto Bhattacharya

MAT490
Discrete Time Finance
4

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial/lab weekly)

Prerequisites: MAT 184 Probability (MAT 390 Introduction to Mathematical Finance is recommended but is not a compulsory requirement.)

Overview: This course serves two purposes. On the one hand, it introduces various theoretical notions in the simpler setting of discrete time and sets the stage for continuous time finance. On the other, it has a strong computational aspect and the student learns to implement models using Excel or Matlab.

Detailed Syllabus:

1. Binomial pricing models
2. Conditional expectation, Martingales, Markov Processes
3. Risk-neutral probability measure
4. American derivatives
5. Random Walks
6. Interest rate models and derivatives
7. Implementation of models in Excel/Matlab

References:

1. Steven E Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer 2004.
2. Les Clewlow and Chris Strickland, Implementing Derivatives Models, Wiley 1998.
3. John C Hull, Options, Futures and Other Derivatives, 8th edition, Pearson, 2013.
4. Rudiger Seydel, Tools for Computational Finance, 5th edition, Springer, 2012.

Past Instructors: Sunil Bowry

MAT330
Partial Differential Equations
4

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. MAT 230 Ordinary Differential Equations or MAT 104 Mathematical Methods II.

Overview: Many physics principles like conservation of mass, momentum, energy, when applied to real life scenarios, take the form of PDEs. In this course we will learn how basic physics concepts together with simple calculus translate into mathematical models of many engineering problems in the form of PDEs. We will learn some well-known techniques to solve these problems in simple settings. We will also learn approximation techniques which will be needed in cases where it is impossible to get analytical solutions.

Detailed Syllabus:

Essentially Chapters 1, 2, 4, 5, 6, and 8 of the book by Strauss. This material will be supplemented with exercises from other prescribed texts and Matlab exercises. The list of topics covered is:

1. Definition of PDEs, well-posedness, initial value and boundary value problems
2. Examples of PDEs, classification of PDEs
3. Wave equation, diffusion equation
4. Source terms
5. Boundary conditions and their impact on solution
6. Fourier Series and their use in solving PDEs
7. Harmonic equations and their solution
8. Numerical methods

References:

• Partial Differential Equations, an Introduction, Second Edition, by Walter A. Strauss
• Applied Partial Differential Equations by Paul DuChateau, David Zachmann
• Partial Differential Equations for Scientists and Engineers, by Stanley J. Farlow

Past Instructors: Ajit Kumar, Samit Bhattacharyya, Srinivas VVK