Course description not available.

MAT350

Mathematical Modelling in Biotechnology

4.00

Undergraduate

Course description not available.

MAT552

Homological Algebra

4.00

Undergraduate

Course description not available.

MAT594

Bayesian Network Learning

4.00

Undergraduate

Course description not available.

MAT588

Convex Optimization

4.00

Undergraduate

Course description not available.

MAT494

Deep Learning

4.00

Undergraduate

Course description not available.

MAT560

1st Course in Operator Theory

4.00

Undergraduate

Overview: Hermitian, Unitary and Normal operators are used widely. To understand the
questions (and the answers) asked about these operators, it is essential to understand the
situation in finite dimensional linear spaces. This course aims to do just that. The spectral
theorems for Self-adjoint and Normal operators will be proved. If time permits, we shall look at
either tensor products or quadratic forms.
.
Detailed Syllabus
1. Review: Vector spaces, subspaces, linear span, linear independence, quotient spaces,
basis, dimension, linear transformations, linear functionals, dual spaces, annihilators,
transpose, matrix of a linear transformation, determinants, characteristic polynomial,
Cayley-Hamilton theorem, minimal polynomial, characteristic roots, eigenvalues,
eigenvectors.
2. Linear Transformations: Range, null space, rank, nullity, row rank, column rank,
projections.
3. Inner Product Spaces: Inner products, Cauchy-Schwarz Inequality, Gramian, orthogonal
and orthonormal sets, Gram-Schmidt orthogonalisation, complete orthonormal sets, orthogonal complements, perpendicular projections, Complexification.
4. Adjoints: Adjoint of a linear transformation, existence in finite dimensional inner
product spaces, inner products on dual space, orthogonal projections.
5. Spectral Theorem: Self-adjoint operators, matrix of self-adjoint operators, Hermitian,
Unitary and Normal operators, Positive operators, Spectral theorems for self-adjoint and
Normal operators, orthogonal linear transformations, polar decomposition, other
consequences of the spectral theorem.
6. If time permits – Tensor Products or Quadratic Forms

MAT550

Algebraic Topology

4.00

Undergraduate

Course description not available.

MAT394

Machine Learning through R.

4.00

Undergraduate

Course description not available.

MAT161

Applied Linear Algebra

4.00

Undergraduate

Course description not available.

MAT551

Topics in Algebraic Topology

4.00

Undergraduate

Course Description: The course will cover topics such as differential forms, spectral sequences and characteristic classes. The exact choice of topics will depend on the instructor and students. Students should have prior exposure to the basic concepts of topology and algebraic topology.

MAT591

Field Theory

4.00

Undergraduate

Course description not available.

MAT492

Data Mining & its Applications

4.00

Undergraduate

Course description not available.

MAT491

Game Theory

4.00

Undergraduate

Course description not available.

MAT150

Mathematical Modelling

4.00

Undergraduate

Mathematical modeling is the science and art of addressing real-world problems many other scientific disciplines. The practice of mathematical modeling inherently captures the interdisciplinary nature of the real-world phenomena, and thus appropriate for students from all disciplines. This course is designed to introduce students to fundamental concepts and methods of mathematical modeling, through a hands on, project-oriented approach. The applications studied will motivate the mathematics covered, contrary to traditional math courses.
Core course for B.Sc. (Research) programs in Mathematics.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: Class XII mathematics

Mat132

Vector Calculus and Geometry

4.00

Undergraduate

Core course for B.Sc. (Research) programs in Mathematics.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 101 (Calculus I)
Overview: Analytic Geometry: Double and triple integrals: Double integrals over rectangles, double integrals over general regions, double integrals in polar coordinates, center of mass, triple integrals, triple integrals in cylindrical coordinates, triple integrals in spherical coordinates, change of variables. Vector Integration: Vector fields, line integrals, fundamental theorem, independence of path, Green's theorem, surface integrals, Stokes' theorem, Gauss theorem.
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.

MAT544

Combinatorial Design Theory

4.00

Undergraduate

(a) Introduction to Design Theory: Basic deﬁnitions and properties, Incidence matrices, Fisher’s Inequality.
(b) Symmetric BIBD’s
i. Intersection Property, Residual and Derived BIBD’s, Projective planes and Geometries ii. The Bruck-Ryser-Chowla Theorem
(c) Diﬀerence Sets and Automorphisms: Quadratic residue diﬀerence sets, Singer diﬀerence sets.
(d) Hadamard Matrices and Designs An equivlance between Hadamard matrices and BIBD’s, Conference matrices and Hadamard matrices, Bent Functions
(e) Latin Squares: Steiner Triple systems, Orthogonal Latin Squares, MOL’s, Orthogonal arrays
(f) PBIBD’s: Connection of PBIBD’s to Association Schemes and Distance regular graphs.
(g) Applications of Combinatorial Design Theory: Medicine, Agriculture, Visual cryptography, Information Security, Statistical designs.

MAT543

Error Correcting Codes

4.00

Undergraduate

Error Correcting Codes

MAT542

Cryptography

4.00

Undergraduate

Cryptography

MAT528

Frame Theory

4.00

Undergraduate

Major Elective for BSc (Research) Mathematics. Available as UWE.
Prerequisites: MAT360 Linear ALgebra II
Overview: The course is an introduction to finite frames. Frames play a fundamental role in signal processing, image processing, data compression, sampling theory and more.
Detailed Syllabus:
1. Linear Algebra Review: Vector spaces, bases of a vector space, linear operators and matrices, rank of a linear operator and a matrix, determinant and trace of a matrix, inner products and orthonormal bases, orthogonal direct sum.
2. Finite-Dimensional Operator Theory: Linear functionals and dual spaces, Riesz representation theorem and adjoint operators, self-adjoint and unitary operators, the Moore-Penrose inverse, eigenvalues of an operator, square roots of a positive operator, trace of operators, the operator norm, the spectral theorem.
3. Introduction to Finite Frames: ????-frames, Parseval frames, reconstruction formula, frames and matrices, similarity and unitary equivalence of frames, frame potential.

MAT522

Fourier Analysis

4.00

Undergraduate

Fourier Analysis

MAT499

Undergraduate Thesis II

8.00

Undergraduate

Compulsory final semester course for BSc (Research) Mathematics.
Prerequisites: MAT498 Undergraduate Thesis I.
Overview: The course can take a variety of forms, from a reading course on advanced topics to computational work in an application of mathematics. The work will be presented in a public seminar at the end of the academic year.
Details:
The student must select and gain the consent of a project supervisor (from the faculty of the Department of Mathematics) by the end of the first week of his/her seventh semester. If the student is unable to fix a supervisor, the Department Undergraduate Committee will arrange one. The student will begin work during the seventh semester by registering for MAT498 Undergraduate Thesis I.
The supervisor will help the student in deciding the structure of the project and identifying reading material and other resources.
While the supervisor will help and encourage the student, successful completion of the project is the student’s responsibility. Credit will be given for independence and initiative in identifying resources and topics.
The project will conclude with the submission of a written report (dissertation) and a public presentation of the work done.
The dissertation should be a detailed exposition of the work done, including a literature review as well as specific investigations. It should be typeset using LaTeX/MS Word.

MAT490

Discrete Time Finance

4.00

Undergraduate

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: Binomial pricing models Conditional expectation, Martingales, Markov Processes Risk-neutral probability measure American derivatives Random Walks Interest rate models and derivatives Implementation of models in Excel/Matlab
References: Steven E Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer 2004. Les Clewlow and Chris Strickland, Implementing Derivatives Models, Wiley 1998. John C Hull, Options, Futures and Other Derivatives, 8th edition, Pearson, 2013. Rudiger Seydel, Tools for Computational Finance, 5th edition, Springer, 2012.
Past Instructors: Sunil Bowry

MAT484

Advanced Statistics

4.00

Undergraduate

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: Review: Introduction, Descriptive Statistics; Sampling Distributions. Graphical representation of data, Basic distributions, properties, fitting, and their uses; Estimation: Point and interval estimation, Histogram and Kernel density estimation, Sufficiency, Exponential family, Bayesian methods, Moment methods, Least squares, Maximum likelihood estimation; 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; Linear Models: Simple and Multiple linear regression, Analysis and Inference.
References: Mathematical Statistics: Basic Ideas and Selected Topics by Peter J. Bickel and Kjell A. Doksum Testing Statistical Hypotheses by Erich L. Lehmann Statistical Decision Theory: Foundations, Concepts and Methods by James O. Berger
Past Instructors: Charu Sharma

MAT452

Introduction to Differential Manifolds

4.00

Undergraduate

Major elective for BSc (Research) Mathematics. Available as UWE.
Prerequisites: MAT332 Geometry of Curves and Surfaces or MAT221 Real Analysis II (These were previously numbered 432 and 420, respectively)
Overview
This course is an introductory course, which starts from several variable calculus and aimed to discuss classical integrability theorems for example Frobenius theorem etc. After these course, students will be able to do any next level course for example Riemannian geometry, Riemann surface, Complex geometry, Symplectic geometry etc.
Detailed Syllabus
1- Several variable calculus: Local immersion and submersion theorems, Inverse and Implicit function theorems.
2- Differential manifolds: Differential structure, Smooth functions on manifolds, critical points.
3- Tangent Bundle: Tangent space of R^n, Taylor theorem, Tangent space of an imbedded manifold, Tangent bundle. Vector field, orientation.
4- Vector field and flow: Integral curves, flow, one parameter group of diffeomorphism.
5- Introduction and particular cases of Frobenius theorem (integrability theorems)
Text Books
1- A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition by Michael Spivak
2- Foundations of Differentiable Manifolds and Lie Groups Authors: Warner, Frank W.
3- An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition by William M. Boothby

MAT444

Basic Category Theory

4.00

Undergraduate

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 and MAT 240 Algebra I
Overview: Category theory is a branch of mathematics which studies and isolates the fundamental structures, underlying constructions and techniques appearing across different areas of mathematics, physics and computer science. The goal of this course is to exhibit the power of category theory as a language for understanding and formalizing common concepts occurring in various branches of mathematics and computers.
Detailed Syllabus:
Unit 1: The category of sets
Sets and functions, Commutative diagrams, Products and coproducts, Finite limits in Set, Finite colimits in Set, other notions in Set.
Unit 2: Categories and functors, without admitting it
Monoids, Groups, Graphs, Orders, Databases: schemas and instances.
Unit 3: Basic Category theory
Categories and functors, natural transformations, Categories and schemas are equivalent (Cat ~ Sch), Yoneda's lemma, Limits and colimits.
Unit 4: Categories at work
Adjoint functors, Categories of functors, Monads.
References: Steven Awodey, Category Theory. Oxford University Press, 2006. Lawvere & Schanuel, Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press. Michael Barr and Charles Wells, Category Theory for Computing Science, Centre de Recherches Mathématiques, 1999. Benjamin Pierce, Basic Category Theory for Computer Scientists. MIT Press Cambridge Saunders Mac Lane, Categories for the Working Mathematician. (the standard reference) David I Spivak, Category Theory for Scientists. MIT Press; 1st edition
Past Instructors: Neha Gupta

MAT442

Graph Theory

4.00

Undergraduate

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

MAT440

Elementary Number Theory

4.00

Undergraduate

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: Divisibility: Definition and properties of Divisibility, Division Algorithm, Greatest Common Divisor, Least Common Multiple, Euclidean Algorithm, Linear Diophantine Equations. 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). 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. Applications of Congruences: Divisibility tests, Modular Designs, Check Digits, The p-Queens Puzzle, Round-Robin Tournaments, The Perpetual Calendar. 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. 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. 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

MAT434

Computational PDE

4.00

Undergraduate

Major Elective for BSc (Research) Mathematics. Available as UWE.
Prerequisites: MAT330 PDE.

MAT433

Computational Fluid Dynamics

4.00

Undergraduate

Outline: Many physics laws like laws of motion, mass conservation law, energy conservation law,
when applied to engineering problems, come in form of Partial Differential Equations (PDE).
There are several softwares available for solving PDEs but they all require enough human
intervention to make if necessary to understand background theory. In this course we will learn
about Finite Volume Method (FVM), the most common method of solving PDEs computationally.
Though the course is titled based on Fluid Dynamics but materials discussed here should be
accessible to and useful for any audience who deals with PDE.
Learning objectives:
• Software: OpenFOAM (http://openfoam.org/)
• Physics to PDE: derivation of PDEs
• Finite Volume Method (FVM): PDEs to linear algebraic equations
• Implementations of FVM on OpenFOAM: C++ programming
References:
• The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction
with OpenFOAM and Matlab by F Moukalled, L Mangani, and M Darwish
• OpenFOAM user and programmer guide
Pedagogy: This course will be problem/project driven. This means that for each learning objectives
we will pose a list of small projects. Using the given references and internet, students will need to
figure out solutions on their own. The instructor will function as mentor and the contact hours will
be spent only on discussion, and not too much instructions.
Prerequisites:
Mathematics : Multi-variable Calculus,
Physics : Newton's laws of motion, work, energy, momentum
Computers : C++
Other courses (helpful but not mandatory):
ODE, PDE, Numerical Analysis, Linear Algebra, Fluid Dynamics
Assessment: Entirely on project reports (written and oral)

MAT432

Geometry of Curves & Surfaces

4.00

Undergraduate

Geometry of Curves and Surfaces

MAT431

Partial Differential Equations

4.00

Undergraduate

Partial Differential Equations

MAT430

Ordinary Differential Eqns.

4.00

Undergraduate

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.

MAT425

Advanced Complex Analysis

4.00

Undergraduate

Course description not available.

MAT424

Complex Analysis

4.00

Undergraduate

A Major Elective for B.Sc. (Research) Mathematics.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 221 Real Analysis II
Overview: This course covers the basic principles of differentiation and integration with complex numbers. Topics will be taught in a computational and geometric way. Knowledge of topology of euclidean space and calculus of several real variables will be assumed.
Detailed Syllabus: Algebraic properties of complex numbers, modulus, complex conjugate, roots of complex numbers, regions. Functions of a complex variable, limits, continuity. Differentiation, Cauchy-Riemann equations, harmonic functions, polar coordinates. Exponential function, logarithm, branch and derivative of logarithm, complex exponents, trigonometric functions, hyperbolic functions, inverse hyperbolic functions. Derivatives of curve w(t) in complex plane, Definite integral of functions w(t), Contours, Contour Integrals, Antiderivatives, Modulus of Contour integrals, Cauchy Goursat theorem. Simply and multiply connected domain, Cauchy Integral Formula and applications, Liouville's theorem, maximum modulus principle. Convergence of series, Power Series, Laurent series, Residues, Cauchy's Residue theorem, Singularities, Zeroes of analytic functions, Behaviour of function near singularities.
References: James W Brown and Ruel V Churchill, Complex Variables and Applications, 8th edition, Tata McGraw-Hill, 2009. H A Priestley, Introduction to Complex Analysis, 2nd edition, Oxford University Press. 2003. J Bak and D J Newman, Complex Analysis, 2nd edition, Springer, 2008. M J Ablowitz and A S Fokas, Complex Variables: Introduction and Applications, 2nd edition, Cambridge University Press India, 2006.

MAT422

Metric Spaces

4.00

Undergraduate

Metric Spaces

MAT420

Measure and Probability

4.00

Undergraduate

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: Introduction to measures and probability triples Random variables: independence, limit events, expectation Inequalities and convergence, Laws of large numbers Distributions, change of variables Limit Theorems, Differentiation of expectation, Moment generating functions, Fubini’s theorem Weak convergence Characteristic functions, Central Limit Theorem and generalizations, Method of Moments Lebesgue and Hahn decompositions Conditional probability and expectation
References: A First Look at Rigorous Probability Theory by J S Rosenthal, 2nd edition, World Scientific Publishing, 2006. Measure, Integral and Probability by M Capinski and E Kopp, 2nd edition, Springer. Probability and Random Processes by G R Grimmett and D R Stirzaker, 2nd edition, Oxford University Press.
Past Instructors: Debashish Bose

MAT399

Undergraduate Seminar

4.00

Undergraduate

Core course for BSc (Research) Mathematics.
The Undergraduate Seminar is an introduction to the activity of research in mathematics. One aim is to help students prepare for their Undergraduate Thesis by practicing, on a smaller scale, the skills of literature survey, public presentations, and mathematical writing.

MAT390

Introduction to Mathematical Finance

4.00

Undergraduate

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: Basic concepts: Bonds and shares, risk versus profit, return and interest, time value of money, arbitrage. Fixed Income Securities: Net Present Value and Internal Rate of Return, price and yield of a bond, term structures, duration, immunization. Mean-Variance Analysis: Random returns, efficient portfolios, feasible set, Markowitz model, Two Fund and One Fund Theorems, Capital Asset Pricing Model and applications. 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. Stock Price Models: Geometric Brownian Motion, Binomial Tree. Options: Call and put options, put-call parity, Binomial Options Pricing Model, dynamic hedging, risk-neutral valuation, Black-Scholes formula, trading strategies. Labs: Microsoft Excel and VBA.
References: Principles of Finance with Excel 2nd edition by Simon Benninga, Oxford University Press, 2010. Mathematics for Finance by M Capinski and T Zastawniak, Springer (International Edition), 2003. The Calculus of Finance by Amber Habib, Universities Press, 2011. Options, Futures and Other Derivatives 7th edition by John C Hull and Sankarshan Basu, Pearson 2009. Investment Science by David Luenberger, Oxford University Press (Indian Edition), 1997. 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

MAT388

Optimization I

4.00

Undergraduate

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: Mathematical modeling and optimization problem formulation Application of optimization (linear case) Geometry of linear optimization Simplex method Duality theory Sensitivity analysis Robust optimization Graphs and Network flow problems Discrete optimization or Integer programming formulations Non-linear optimization – introduction and applications
Main References: Linear Programming by G. Hadley, Narosa, 2000 Understanding and Using Linear Programming by J. Matousek and B. Gärtner, Springer, 2006 Introduction to Linear Optimization by D. Bertsimas and J. Tsitsiklis, Athena Scientific, 1997 Theory of Linear and Integer Programming by A. Schrijver, Wiley, 1998 Operations Research: An Introduction by H. Taha, Pearson, 2012
Past Instructors: Samit Bhattacharyya

MAT386

Dynamical Systems

4.00

Undergraduate

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 104 Mathematical Methods II or MAT 160 Linear Algebra I.
Overview:
Detailed Syllabus:
Basic Concepts: Discrete and continuous dynamical systems. Linear and nonlinear systems and principle of superposition. Linear and nonlinear forces. Concepts of evolution, iterations, orbits, fixed points, periodic and aperiodic (chaotic) orbits. Basics of Linear Algebra: Symmetric & Skew-symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Canonical forms; simple and non-simple canonical systems. System of Equations.
Stability Analysis:
Stability of a fixed point and classification equilibrium states (for both discrete and continuous systems). Concept of bifurcation and classification of bifurcations. Concepts of Lyapunov stability & Asymptotic stability of orbits. Phase Portraits of various Linear and Nonlinear systems. Hopf bifurcation. Concept of attractors and repellers, limit cycles and torus.
Phenomena of Bifurcation:
Definition of bifurcation. Bifurcations in one, two and higher dimensional systems. Hopf, Period doubling, Saddle node, Transcritical bifurcations. Feigenbaum’s number. Local and Global bifurcations. Homoclinic & Hetero-clinic points and orbits. Poincaré-Bendixson Theorem. Conservative and Dissipative Systems.
Investigation Tools & Chaos Theory:
Time Series and Phase Plane Analysis, Poincaé Map & Section. Lyapunov Characteristic Exponents. Hamiltonian Systems and concept of iIntegrability and non-integrability. Concept of Chaos and Chaotic evolution of a Dynamical System. Measure of Chaos. Routs to Chaos.
Applications:
Applications of Dynamical Systems, (to Physics, Biology, Economics with Examples). Mathematical Models. Population Dynamics. Investigation of Evolutionary Phenomena in Logistic Map, Lotka-Volterra System, Duffing Oscillator, Oscillation of Nonlinear Pendulum, Predator-Prey Systems etc.
Tutorial:
Drawing orbits of a system for given initial values. Clear Demonstration of Linear and Nonlinear Systems. Calculation of fixed points for given system and examine their stabilities (discrete and continuous). Drawing time series graphs, phase portraits for regular and chaotic systems. Cobweb Plots. Calculations of Eigenvalues and Eigenvectors corresponding to any fixed point. Plotting Bifurcation diagrams of 1 and 2 dimensional systems. Calculations of Lyapunov exponent.
Software such as MATHEMATICA / MATLAB will be used as needed.
References: Nonlinear Systems, by P. G. Drazin, Cambridge University Press India. An Introduction to Chaotic Dynamical Systems by R. L. Devaney, Addison Wesley, 1989. Chaos in Dynamical Systems, by Edward Ott, Cambridge University Press, 2002 Chaotic Dynamics – An Introduction, by G. L. Baker and J. P. Gollub, Cambridge University Press, 1996. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, by J. Guckenheimer and P. Holmes, Springer, 1983.
Past Instructors: L M Saha

MAT384

Introductory Econometrics

4.00

Undergraduate

Introductory Econometrics

MAT380

Numerical Analysis II

4.00

Undergraduate

Course description not available.

MAT360

Linear Algebra II

4.00

Undergraduate

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: Linear Equations – Systems of linear equations, matrices, elementary row operations and row reduction. Vector spaces – Abstract vector spaces, subspaces, dimension, coordinates. Linear transformations – Matrix representations, change of basis, linear functionals and the double dual, transpose. Determinants – Commutative rings, determinant function, permutations, properties. Canonical Forms – Characteristic values, invariant subspaces, simultaneous diagonalization and triangulation, invariant direct sums, Primary Decomposition Theorem, cyclic subspaces, Rational Form, Jordan Form. Inner Product Spaces – Linear functionals and adjoints, unitary and normal operators, spectral theory.
References: Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd edition, PHI Learning. Friedberg, Insel and Spence, Linear Algebra, 4th edition, PHI Learning Sheldon Axler, Linear Algebra Done Right, 2nd edition, Springer International Edition Paul Halmos, Finite Dimensional Vector Spaces, 2nd edition, Springer International Edition Paul Halmos, Linear Algebra Problem Book, Mathematical Association of America, 1995.
Past Instructors: Neha Gupta

MAT341

Commutative Algebra

4.00

Undergraduate

Major Elective for BSc (Research) Mathematics. Available as UWE.
Prerequisites: MAT240 Algebra I

MAT340

Algebra II

4.00

Undergraduate

Overview: The course continues the work done in MAT 240 and MAT 260 by studying the algebraic structures of rings and fields on the one hand and abstract linear algebra and module theory on the other. After laying the groundwork in these topics, diverse applications - such as finite fields, the structure of abelian groups and the Jordan canonical form of a matrix - are studied.
Detailed Syllabus:
1. Review - definition and examples of rings, ideals, quotient rings, maximal ideals, prime ideals, ring homomorphisms
2. Types of rings - integral domains, Euclidean domains, PIDs, UFDs, polynomial rings, factorization of polynomials, irreducibility criteria
3. Vector spaces - abstract vector spaces, examples, dimension, subspaces, linear transformations, matrix representations, change of basis, rank of linear transformations
4. Modules - definition and examples of modules, submodules, finitely generated modules, free modules, quotient modules and module homomorphisms
5. Fields – definition and examples of fields, characteristic, field extensions, finite extensions, zeroes of an irreducible polynomial, algebraic extensions, splitting fields, algebraic closures, finite fields - Definition, constructions and properties
6. Modules over PID - rank of matrices over PID, Smith normal form of a matrix, structure theorem for modules over PID, application to finitely generated abelian groups, applications to linear algebra - rational, Jordan canonical forms of a matrix

MAT332

Geometry of Curves & Surfaces

4.00

Undergraduate

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: Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition by Elsa Abbena, Simon Salamon, Alfred Gray. Elementary Differential Geometry by A.N. Pressley, Springer Undergraduate Mathematics Series. Differential Geometry of Curves and Surfaces by Manfredo DoCarmo.
Past Instructors: Pradip Kumar

MAT320

Real Analysis II

4.00

Undergraduate

Overview: Continuing the work done in MAT 220 of understanding the rigor behind one-variable Calculus, this course dwells on various aspects of functions on more general spaces, namely, metric spaces. A brief introduction to the generalities of metric spaces leads to discussions on functions on metric spaces; sequences and series of functions on metric spaces; uniform convergence and consequences; some important approximation theorems for continuous functions; rigorous discussions of some special functions; and then finally to 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. Topology of Rn: Open and closed sets, continuous functions, completeness, compactness, connectedness.
3. 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.
4. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions.
5. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem.

MAT299

Undergraduate Seminar

3.00

Undergraduate

A major elective for BSc (Research) Mathematics 2nd and 3rd year students only.

MAT284

Probability and Statistics

4.00

Undergraduate

Core course for BSc (Research) Economics. Students of BSc (Research) Mathematics or any B.Tech. program are not allowed to credit this course.
Prerequisites: Calculus I (MAT 101)
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.
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. Sequences of Random Variables: Chebyshev’s Inequality, Law of Large Numbers, Central Limit Theorem, random walks.
5. Joint Distributions: Joint and marginal distributions, covariance, correlation, independent random variables, least squares method, linear regression.
6. Sampling: Sample mean and variance, standard error, sample correlation, chi square distribution, t distribution, F distribution, point estimation, confidence intervals.
7. Hypothesis Testing: Null and alternate hypothesis, Type I and Type II errors, large sample tests, small sample tests, power of a test, goodness of fit, chi square test.
Main References:
• A First Course in Probability by Sheldon Ross, 6th edition, Pearson.
• John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011.
Other References:
• 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.
• Introduction to the Theory of Statistics by Alexander M. Mood, Franklin A. Graybill and Duane C. Boes, 3rd edition, Tata McGraw-Hill, 2001.

MAT283

Introduction to Statistics

4.00

Undergraduate

Course description not available.

MAT280

Numerical Analysis I

4.00

Undergraduate

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: Solving Equations: Iterative methods, Bisection method, Secant method, Newton-Raphson method, Rates of convergence, Roots of polynomials. Interpolation: Lagrange and Hermite interpolation, Interpolating polynomials using difference operators. Numerical Differentiation: Methods based on interpolation, methods based on finite difference operators. Numerical Integration: Newton-Cotes formula, Gauss quadrature, Chebyshev’s formula. Systems of Linear Equations: Direct methods (Gauss elimination, Gauss-Jordan method, LU decomposition, Cholesky decomposition), Iterative methods (Jacobi, Seidel, and Relaxation methods) Labs: Computational work using C, Python or Matlab.
References: E. Suli and D. Mayers, Introduction to Numerical Analysis, Cambridge University Press, 2003. R.L. Burden and J.D. Faires, Numerical Analysis, Cengage Learning, 9th Edition, 2010. M.K. Jain, S.R.K. Iyengar, and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, New Age International Ltd., 1999. J.H. Mathews and K. Fink, Numerical Methods using Matlab, PHI Learning, 4th Edition, 2003.

MAT260

Linear Algebra

4.00

Undergraduate

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. Detailed Syllabus: 1. Matrices and Linear Systems 2. Vector Spaces and Linear Transformations 3. Inner Product Spaces 4. Determinant 5. Eigenvalues and Eigenvectors 6. Positive definite matrices 7. Linear Programming and Game Theory Matlab will be used in tutorial for computational purposes. Main References: • Linear Algebra and its Applications by Gilbert Strang, 4th edition. • Algebra by Michael Artin, Second edition. Other References: • Linear Algebra: A Geometric Approach by S. Kumaresan, PHI, 2011. • Advance engineering Mathematics by E. Kreyszig, 10th Edition

MAT246

Combinatorics

4.00

Undergraduate

Course description not available.

MAT241

Algebra II

4.00

Undergraduate

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: Groups – Definition, subgroups, cyclic groups, homomorphisms, normal subgroups, semi-direct products, group actions, Sylow theorems. Rings – Definition and examples of rings, ideals, quotient rings, maximal ideals, prime ideals, ring homomorphisms, integral domains, Euclidean domains, PID, UFD. Polynomial Rings and Fields – Polynomial rings, irreducible polynomials, definition and examples of fields, characteristic, field extensions, finite fields.
References: I. N. Herstein, Topics in Algebra, 2/e, Wiley Eastern, 1994. Bhattacharya, Jain and Nagpaul, Basic Abstract Algebra, 2nd edition, CUP, 1995. Joseph A. Gallian, Contemporary Abstract Algebra, 4th edition, Narosa, 1999. M. Artin, Algebra, 2nd edition. Prentice Hall India, 2011. Dummit and Foote, Abstract Algebra, 3rd edition, Wiley. Serge Lang, Undergraduate Algebra, 2nd edition. Springer India, 2009. Thomas W. Hungerford, Algebra, GTM 73, Springer India, 2004.

MAT240

Algebra I

4.00

Undergraduate

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: 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. Normal Subgroups and Quotient Groups, Homomorphisms, Isomorphisms and Automorphisms of a Group. Conjugates, centre, centralizer, normalizer. Cayley’s Theorem. Direct Products, Finite Abelian Groups. Permutation Groups: Definition, Examples and Properties, Symmetric Group of n Letters (Sn), Alternating Group on n Letters (An). (If time permits) Rings, Homomorphisms, Ideals and Quotient Rings, Integral Domains.
References: Contemporary Abstract Algebra by Joseph A. Gallian, 4th edition. Narosa, 1999. Algebra by Michael Artin, 2nd Edition. Prentice Hall India, 2011. Topics in Algebra by I.N. Herstein, 2nd Edition. Wiley India, 2006. A First Course in Abstract Algebra by John B. Fraleigh, 7th Edition. Pearson, 2003. Undergraduate Algebra by Serge Lang, 2nd Edition. Springer India, 2009. Abstract Algebra by David S. Dummit and Richard M. Foote, 3rd Edition. John Wiley and Sons, 2011.
Past Instructors: Neha Gupta, Sanjeev Agrawal

MAT232

Fractal Geometry

4.00

Undergraduate

Course description not available.

MAT230

Ordinary Differential Equations

4.00

Undergraduate

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: First Order ODEs: Modelling, Geometrical Meaning, Solution techniques Second and Higher Order Linear ODEs: Modelling, Geometrical Meaning, Solution techniques Numerical Techniques Existence of Solutions of Differential Equations Systems of ODEs: Phase Plane and Qualitative Methods Laplace Transforms Series Solutions
References: Erwin Kreyszig, Advanced Engineering Mathematics, 9th edition, Wiley India, 2012. G.F. Simmons and S. Krantz, Differential Equations: Theory, Technique, and Practice, McGraw Hill Publishing Company, 2006. J. Polking, D. Arnold, A. Boggess, Differential Equations, Pearson, 2005. C. Henry Edwards and David E. Penney, Differential Equations and Boundary Value Problems: Computing And Modeling, 3rd edition, Pearson, 2010. 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

MAT221

Real Analysis II

4.00

Undergraduate

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: 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. 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. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions. Topology of Rn: Open and closed sets, continuous functions, completeness, compactness, connectedness. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem.
References: Analysis II by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006. Real Mathematical Analysis by Charles C Pugh. Springer India. 2004. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002. Calculus, Volume 2, by Tom Apostol, Wiley India. 2nd Edition, 2011.

MAT220

Real Analysis I

4.00

Undergraduate

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: 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. Sequences: Convergence, bounded, monotone and Cauchy sequences, subsequences, lim sup and lim inf. 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. Continuity: Limits of functions, continuous functions, Extreme Value Theorem, Intermediate Value Theorem, monotonic functions, uniform continuity. Differentiation: Differentiable functions on R, local maxima, local minima, Mean Value Theorems, L'Hopital's Rule, Taylor's Theorem.
References: A Basic Course in Real Analysis by Ajit Kumar and S Kumaresan. CRC Press, 2014. Introduction to Real Analysis by R G Bartle & D R Sherbert, John Wiley & Sons, Singapore, 2/e (or later editions), 1994. Elementary Analysis: The Theory of Calculus by Kenneth A Ross. Springer India, 2004. Analysis I by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002. Calculus, Volume 1, by Tom Apostol, Wiley India. 2nd Edition, 2011.
Past Instructors: Pradip Kumar

MAT210

Programming

2.00

Undergraduate

Programming

MAT205

MATHEMATICAL METHODS III ? Probability and Statistics

3.00

Undergraduate

Core course for B.Tech. except Computer Science. Not available as UWE.
Credits (Lec:Tut:Lab)= 3:0:0 (3 lectures weekly)
Prerequisites: MAT 103 (Mathematical Methods I)
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.
Detailed Syllabus: Probability: sample space and events, classical and axiomatic probability, permutations and combinations, conditional probability, independence, Bayes’ formula Random Variables: discrete and continuous probability distributions, mean and variance, binomial and Poisson, normal, joint distributions, covariance, correlation and regression (linear) Mathematical Statistics: exploring data, random samples, point estimation, Central limit theorem, Maximum likelihood, chi-square, t and F-distributions, confidence intervals, hypothesis testing
References: Advanced Engineering Mathematics by Erwin Kreyszig, Wiley. Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press. Theory and Problems of Beginning Statistics by L. J. Stephens, Schaum’s Outline Series, McGraw-Hill John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011.
Past Instructors: Charu Sharma, Niteesh Sahni, Suma Ghosh

MAT203

Mathematical Methods II

4.00

Undergraduate

Mathematical Methods II

MAT202

Mathematical Methods

3.00

Undergraduate

Mathematical Methods

MAT201

Mathematical Methods I

4.00

Undergraduate

Mathematical Methods I
COURSE DESCRIPTION :
In this course we study multi variable calculus. Concepts of derivatives and integration will be developed for higher dimensional space. This course has direct applications in most of engineering applications.
ASSESSMENT SCHEME :
• Midterm 1 (20 %)
• Midterm 2 (20 %)
• End term ( 30 % )
• Tutorial quizzes ( 20 %)
• HW 10 %

MAT199

Project I (2nd Part)

3.00

Undergraduate

Project I

MAT184

Probability

4.00

Undergraduate

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: Probability: Classical probability, axiomatic approach, conditional probability, independent events, addition and multiplication theorems with applications, Bayes’ theorem. Random Variables: Probability mass function, probability density function, cumulative density function, expectation, variance, standard deviation, mode, median, moment generating function. Some Distributions and their Applications: Uniform (discrete and continuous), Bernoulli, Binomial, Poisson, Exponential, Normal. Joint Distributions: Joint and marginal distributions, independent random variables, IIDs, conditional distributions, covariance, correlation, moment generating function. Sequences of Random Variables: Markov’s Inequality, Chebyshev’s Inequality, Law of Large Numbers, Central Limit Theorem.
References: A First Course in Probability by Sheldon Ross, 6th edition, Pearson. Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press. Theory and Problems of Probability and Statistics by Murray R Spiegel and Ray Meddis, Schaum’s Outlines. John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011. 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

MAT160

Linear Algebra

4.00

Undergraduate

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: Matrices and Linear Systems Vector Spaces and Linear Transformations Inner Product Spaces Determinant Eigenvalues and Eigenvectors, Diagonalization Quadratic Forms and Positive Definite Matrices Applications chosen from: Numerical aspects, Difference equations, Markov matrices, Least squares.
References: Linear Algebra by Jim Hefferon Linear Algebra and its Applications by Gilbert Strang, 4th edition, Cengage. Linear Algebra and its Applications by David C. Lay, 3rd edition, Pearson. Linear Algebra: A Geometric Approach by S. Kumaresan, PHI, 2011. Elementary Linear Algebra by Howard Anton and Chris Rorres, 9th edition, Wiley. Linear Algebra: An Introductory Approach by Charles Curtis, Springer. Matrix Analysis and Applied Linear Algebra by Carl D Meyer, SIAM. Videos of lectures by Prof Gilbert Strang: 18.06 Linear Algebra, Spring 2010. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu

MAT140

Discrete Structures

4.00

Undergraduate

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: 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. 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. Finite Boolean algebras: Functions on Boolean algebras, Karnaugh maps, Logic gates, Digital circuits.
References: Thomas Donnellan, Lattice Theory, Pergamon Press, Oxford. J.E. Whitesitts, Boolean Algebra and Its Applications, Addison-Wesley Publications. G. Birkhoff, Lattice Theory, American Mathematical Society, 2nd Edition. E. Mandelson, Schaum’s Outline of Boolean Algebra and Switching Circuits, McGraw Hill. Bernard Kolman, Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Pearson Education, New Delhi. Kenneth H. Rosen, Discrete Mathematics and its Applications, Tata McGraw-Hill, New Delhi. C. L. Liu, D. P. Mohapatra, Elements of Discrete Mathematics, Tata McGraw-Hill, New Delhi. 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

MAT110

Computing

4.00

Undergraduate

Core course for B.Sc. (Research) Mathematics. Optional for B.Sc. (Research) Economics. Not available for B.Tech. students. Others may credit it as UWE with permission from Department of Mathematics. Does not count towards Minor in Mathematics.
Credits (Lec:Tut:Lab)= 3:0:1 (Three lecture hours and two lab hours weekly)
Prerequisites: Class XII Mathematics
Overview: This course aims to empower the students in data abstraction, algorithm design and performance estimation. In the process they shall learn the art of programming – a pretty useful skill to have! Programming in C and Matlab will be taught.
Detailed Syllabus: Basic programming constructs: conditional statements, functions, loops, arrays, structures, pointers. Linear data structures: Linked list, queue, stack Trees and Graphs: basic operations Searching and Hashing: Linear search, Binary search, tree search, hash tables Sorting: Insertion sort, bubble sort, merge sort, heap sort Introduction to MATLAB programming.
References: B. Kolman, R. Busby, and S. Ross, Discrete Mathematical Structures, PHI, 2012 Jeri R.Hanly, Eklliot B.Koffmain, Problem Solving and Program Design in C,Pearson,2009 A. Aho, J. Hopcroft, D. Ullman, Data structures and Algorithms, Addison-Wesley, 1983 A. Aho, and D. Ullman, Foundations of Computer Science, Comp. Sci. Press, 1992 T. Cormen and C. Leiserson, Introduction to Algorithms, MIT Press, 2009 N. Kalicharan, Data Structures in C, CreateSpace Independent Publishing, 2008 A. Tenenbaum, Data Structures using C, PHI, 2003
Past Instructors: Charu Sharma, Niteesh Sahni

MAT105

Programming in Excel VBA

2.00

Undergraduate

Programming in Excel VBA

MAT104

Mathematical Methods II

4.00

Undergraduate

Core course for all B.Tech. Programs. Optional for B.Sc. (Research) Chemistry. Not available as UWE.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: Class XII Mathematics
Overview: We will study Ordinary Differential Equations which are a powerful tool for solving many science and engineering problems. This course also covers some basic linear algebra which is needed for systems of ODEs.
Detailed Syllabus: First order ODEs: separable, exact, linear Second order ODEs: homogeneous and nonhomogeneous linear, linear with constant coefficients, Wronskian, undetermined coefficients, variation of parameters Laplace transform: definition and inverse, linearity, shift, derivatives, integrals, initial value problems, time shift, Dirac’s delta function and partial fractions, convolution, differentiation and integration of transform Matrices: operations, inverse, determinant, eigenvalues and eigenvectors, diagonalization Systems of ODEs: superposition principle, Wronskian, constant coefficient systems, phase plane, critical points, stability
References: James Stewart, Essential Calculus – Early Transcendentals, Cengage. Erwin Kreyszig, Advanced Engineering Mathematics, Wiley.
Past Instructors: Ajit Kumar, Neha Gupta

MAT103

Mathematical Methods I

4.00

Undergraduate

Core course for all B.Tech. Optional for B.Sc. (Research) Chemistry. Not open as UWE.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: Class XII Mathematics.
Overview: In this course we study multi-variable calculus. Concepts of derivatives and integration will be developed for higher dimensional spaces. This course has direct applications in most engineering applications.
Detailed Syllabus: Review of high school calculus. Parametric curves (Vector functions): plotting, tangent, arc-length, polar coordinates, derivatives and integrals. Functions of several variables: level curves and surfaces, differentiation of functions of several variables, gradient, unconstrained and constrained optimization. Double and triple integrals: integrated integrals, polar coordinates, cylindrical and spherical coordinates, change of variables. Vector fields, divergence and curl, Line and surface integrals, Fundamental Theorems of Green, Stokes and Gauss.
References: A Banner, The Calculus Lifesaver, Princeton University Press. James Stewart, Essential Calculus – Early Transcendentals, Cengage. G B Thomas and R L Finney, Calculus and Analytic Geometry, Addison-Wesley. Erwin Kreyszig, Advanced Engineering Mathematics, Wiley.
Past Instructors: Ajit Kumar, Sneh Lata

MAT102

Calculus II

4.00

Undergraduate

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: 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 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

MAT101

Calculus I

4.00

Undergraduate

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: Real Number System: The axioms for N and R, mathematical induction. 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. Continuous Functions: Functions, limits, continuity, Intermediate Value Theorem, Extreme Value Theorem, integrability of continuous functions, Mean Value Theorem for integrals. 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. Fundamental Theorem of Calculus: Antiderivatives, Indefinite Integrals, Fundamental Theorem of Calculus, Logarithm and Exponential functions, techniques of integration. Polynomial Approximations: Taylor polynomials, remainder formula, indeterminate forms and L'Hopital's rule, limits involving infinity, improper integrals. Ordinary Differential Equations: 1st order and separable, logistic growth, 1st order and linear.
References: Calculus, Volume I, by Tom M Apostol, Wiley. Introduction to Calculus and Analysis I by Richard Courant and Fritz John, Springer Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition. Calculus with Analytic Geometry by G F Simmons, McGraw-Hill
Past Instructors: Amber Habib, Debashish Bose

MAT100

Foundations

4.00

Undergraduate

Core course for B.Sc. (Research) Mathematics. Not available as UWE.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: None
Overview: Introduction to modern mathematical language and reasoning: Sets and Logic, Proof strategies, Functions, Induction.
Detailed Syllabus: Sentential Logic: Deductive reasoning, negation of a sentence, conjunction and disjunction of sentences, equivalence of sentences, truth tables, logical connectives. Sets: Operations on sets, Venn diagrams, cartesian product, quantifiers. Proof Strategies: Direct proofs, proofs involving negations, conditionals, conjunctions, and disjunctions, existence and uniqueness proofs, proofs involving equivalence. Relations and Functions: Ordered pairs, equivalence relations, equivalence classes, partitioning of a set, functions as many-one relations, graphs of functions, one-one functions, onto functions, inverse of a function, images and inverse images of sets. Mathematical Induction: Division algorithm, principle of mathematical induction, well ordering principle, strong induction, principle of recursive definition. More on Sets: Finite and infinite sets, countable and uncountable sets.
References: Book of Proof by Richard Hammack, 2nd edition, Richard Hammack. Mathematical Thinking by Keith Devlin, Lightning Source. How to Prove It by Daniel J. Velleman, Cambridge University Press. Mathematical Writing by Franco Vivaldi, Springer. Proofs and Fundamentals by Ethan D. Bloch, Springer. Introduction to Logic and to the Methodology of Deductive Sciences, Alfred Tarski, Oxford University Press.
Past Instructors: Amber Habib, Priyanka Grover

MAT084

Basic Probability and Statistics

4.00

Undergraduate

Core course for B.Sc. (Research) Biotechnology. Only available as UWE with prior permission of Department of Mathematics. Does not count towards Minor in Mathematics.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures +1 tutorial weekly)
Prerequisites: Class XII Mathematics or MAT 020 (Elementary Calculus) or MAT 101 (Calculus I)
Overview: Probability is the means by which we model the inherent randomness of natural phenomena. This course provides an introduction 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 will act as an introduction to probability and statistics for students from natural sciences, social sciences and humanities.
Detailed Syllabus: Describing data: scales of measurement, frequency tables and graphs, grouped data, stem and leaf plots, histograms, frequency polygons and ogives, percentiles and box plots, graphs for two characteristics Summarizing data: Measures of the middle: mean, median, mode; Measures of spread: variance, standard deviation, coefficient of variation, percentiles, interquartile range; Chebyshev’s inequality, normal data sets, Measures for relationship between two characteristics; Relative risk and Odds ratio Elements of Probability: Sample space and events, basic definitions and rules of probability, conditional probability, Bayes’ theorem, independent events Sampling: Population and samples, reasons for sampling, methods of sampling, standard error, Population parameter and sample statistic Special random variables and their distributions: Bernoulli, Binomial, Poisson, Uniform, Normal, Exponential, Gamma, distributions arising from the Normal: Chi-‐square, t, F Distributions of Sampling statistics: Sampling distribution of the mean, The central limit theorem, Determination of sample size, standard deviation versus standard error, the sample variance, sampling distributions from a normal population, sampling from a finite population Estimation: Maximum likelihood estimator; Interval estimates; Estimating the confidence interval for population mean, variance and proportions; Confidence intervals for the difference between independent means Hypothesis testing: Null and alternate hypothesis; Significance levels; Type I and Type II errors; Tests based on Normal, t, F and Chi-‐Square distributions for testing of mean, variance and proportions, Tests for independence of attributes, Goodness of fit; Non-‐parametric tests: the sign test, the Signed Rank test, Wilcoxon Rank-‐Sum Test. Analysis of variance: Comparing three or more means: One-‐way analysis of variance, Two-‐factor analysis of variance, Two-‐way analysis of variance with interaction Correlation and Regression: Correlation, calculating correlation coefficient, coefficient of determination, Spearman’s rank correlation; Linear regression, Least square estimation of regression parameters, distribution of the estimators, assumptions and inferences in regression; analysis of residuals: assessing the model; transforming to linearity; weighted least squares; polynomial regression
Main References: Introduction to Probability and Statistics for Engineers and Scientists by Sheldon Ross, 2nd edition, Harcourt Academic Press.
Other References: Basic and Clinical Biostatistics by Beth Dawson-‐Saunders and Robert G. Trapp, 2nd edition, Appleton and Lange. John E. Freund’s Mathematical Statistics with Applications by I. Miller & M. Miller, 7th edition, Pearson, 2011.
Past Instructors: Sneh Lata, Suma Ghosh

MAT020

Elementary Calculus

4.00

Undergraduate

Core course for B.Sc. (Research) Biotechnology. Only available as UWE with prior permission of Department of Mathematics. Does not count towards a Minor in Mathematics.
Credits (Lec:Tut:Lab): (3:1:0)
Overview: This course is targeted at undergraduates who did not take Mathematics at +2 level in school, and now need to quickly acquire basic calculus skills in order to satisfy their major requirements. For example, the purpose may be to enable them to take a Probability course which requires basic concepts from calculus. The course will emphasize geometric meaning rather than formal proof.
Students who need greater rigour, as well as more computational skills, should take MAT101 Calculus I.
Detailed Syllabus: Functions: Real line and its subsets, real functions, graphs, polynomials, rational functions, real powers, trigonometric functions, roots, boundedness, monotonicity, composition of functions, inverse functions. Limits and Continuity: Algebra of limits, left and right limits, limits involving infinity, continuity, left and right continuity, types of discontinuity. Differentiation: Rates of change, tangents to graphs, first and higher derivatives, algebra of differentiation, chain rule, exponentials and logarithms. Applications of differentiation: Exponential growth and decay, intervals of increase and decrease, first and second derivative tests, curve sketching. Integration: Definite and indefinite integrals, Fundamental Theorem of Calculus, substitution, integration by parts, trigonometric integrals, improper integrals. First-Order Differential Equations: Separable differential equations, logistic growth.
References: Short Calculus, by Serge Lang, Springer. Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition.
Past Instructors: L M Saha

MAT000

Tutorial

3.00

Undergraduate

Tutorial

MAT330

Partial Differential Equations

4.00

Undergraduate

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: Definition of PDEs, well-posedness, initial value and boundary value problems Examples of PDEs, classification of PDEs Wave equation, diffusion equation Source terms Boundary conditions and their impact on solution Fourier Series and their use in solving PDEs Harmonic equations and their solution 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

MAT652

Advanced Homological Algebra

4.00

Graduate

Course description not available.

MAT678

Numerical optimization

4.00

Graduate

Course description not available.

MAT734

Lie Groups and Riemannian Geometry

4.00

Graduate

Course description not available.

MAT794

Theory of Copulas

4.00

Graduate

Course description not available.

MAT834

Topology and Geomtery

4.00

Graduate

Course description not available.

MAT811

Evolutionary Game Theory and A

4.00

Graduate

Evolutionary Game Theory and Applications

MAT785

Optimal control of systems gov

4.00

Graduate

Optimal control of systems governed by PDEs

MAT681

Finite Volume method

4.00

Graduate

Finite Volume method

MAT646

Commutative Algebra

4.00

Graduate

Commutaive Algebra

MAT661

Advanced Linear Algebra

4.00

Graduate

Advanced Linear Algebra

MAT641

Advanced Algebra

4.00

Graduate

Advanced Algebra

MAT832

Analysis and Geometry

4.00

Graduate

Analysis and Geometry

MAT810

Reading Course: Probabilistic

4.00

Graduate

Reading Course: Probabilistic models and Statistical inference

MAT694

Numerical PDE

4.00

Graduate

Numerical PDE

MAT725

Matrix Analysis

4.00

Graduate

Matrix Analysis

MAT692

Numerical Linear Algebra

4.00

Graduate

Numerical Linear Algebra

MAT732

Fractal Geometry

4.00

Graduate

Fractal Geometry.

MAT898

Research Methodology

4.00

Graduate

Research Methodology

MAT809

Non Negative Matrices

4.00

Graduate

Non Negative Matrices

MAT808

Algebraic Number Theory

4.00

Graduate

Algebraic Number Theory

MAT807

Algebraic Graph Theory

4.00

Graduate

Algebraic Graph Theory

MAT806

Reading Course - Math of Inf.

4.00

Graduate

Reading Course - Mathematics of Infectious Disease

MAT805

Application of complex network

4.00

Graduate

Application of Complex Networks to Landscape Ecology

MAT804

Complex Networks

4.00

Graduate

Complex Networks

MAT803

Reading Course - Biomathematic

4.00

Graduate

Reading Course - Biomathematics

MAT802

Reading Course - Lie Groups

4.00

Graduate

Reading Course - Lie Groups: Representations and Invariants

MAT801

Reading Course I - Appro

4.00

Graduate

Reading Course - Approximation Problems in Normed Linear Spaces

MAT800

Reading course

4.00

Graduate

Reading course

MAT799

Project

8.00

Graduate

Project

MAT792

Signal and Image Processing

4.00

Graduate

Signal and Image Processing

MAT791

Evolutionary Game Theory

4.00

Graduate

Evolutionary Game Theory

MAT790

Game Theory

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 660 Linear Algebra
Overview:
Detailed Syllabus: Simple Decision Models: Ordinal Utility, Linear Utility, Modelling Rational Behaviour, Modelling Natural Selection, Optimal Behaviour, Strategic Behaviour, Randomizing Strategies, Optimal Strategies Strategic Games: Interactive Decision Problems, Describing Static Games, Games in Normal Form, Describing Strategic Games, Solving Games Using Dominance, Nash Equilibrium in Strategic Games, Existence of Nash Equilibria, The Problem of Multiple Equilibria, Classifying Games, Two- Player Zero-Sum Games, Mixed Strategies in Finite Games, Matrix and Bimatrix Games, Games with n-Players Infinite Dynamic Games: Repeated Games, The Iterated Prisoner’s Dilemma, Folk Theorems Population Games: Evolutionary Game Theory, Evolutionarily Stable Strategies, Games against the Field, Pairwise Contest Games
References: K. Binmore Playing for Real: A Text on Game Theory, Oxford University Press. J. N. Webb Game Theory, Decisions, Interaction and Evolution, Springer. J. G-Diaz, I. G-Jurado, M. G.F-Janeiro An Introductory Course on Mathematical Game Theory, Graduate Studies in Mathematics 115, American Mathematical Society.

MAT786

Dynamical Systems

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 621 for graduate students, MAT 320 for undergraduates.
Overview:
Detailed Syllabus: Introduction: Definition of Dynamical Systems, Discrete and Continuous Systems, Fixed points, Iterations, Classification of orbits, Stability of a fixed point, Classification of fixed points. Bifurcation Analysis: Definition of bifurcation, Classification of bifurcation, Period doubling phenomena, Hopf bifurcation. Chaos Theory: Definition of chaos, Regular and chaotic evolution, How chaos appears in a system. Tools for identification of regular and chaotic motions: Time series, Phase plot, Poincarè map, surface of section. Applications: Applications of Dynamical Systems to Population Dynamics, Predator-Prey evolution, Spread of Epidemic, Food Chain systems and in other areas. Use of Software MATHEMATICA or MATLAB for exploring above topics.
References: R. L. Devaney: Introduction to Chaotic Dynamical Systems. Benjamin – Cummings, 1986 R. L. Devaney : A First Course in Chaotic Dynamical Systems: Theory and Experiment. Westview Press, 1992 P. G. Drazin: Nonlinear Systems. Cambridge Texts in Applied Mathematics, 1992 Stephen Lynch: Dynamical Systems with Applications using MATHEMATICA. Birkhäuser, 2007 D. K. Arrowsmith and C. M. Place: An Introduction to Dynamical Systems. Cambridge University Press, 1990

MAT782

Numerical Differential Equations

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 680 (MAT 280 for undergraduates)
Overview:
Detailed Syllabus: Review of numerical techniques for Linear System of equations, Review of Numerical Differentiation and Integration: Mid-point rule, Trapezoidal rule, Simpson's rule, Richardson improvement, variable steps, errors and convergence of above methods. Numerical ODE: Initial Value Problems (Euler methods, Heun's Method, Taylor Series Method, Runge Kutta method), Boundary value problems (Shooting Method). Numerical PDE: Finite difference methods for 2 dimension parabolic, hyperbolic, and elliptic PDEs. Eigenvalue Problems: Power Method, Jacobi's method, Householder's method Programming: Matlab, C++
References: Numerical Methods using Matlab, John H. Mathews and Kurtis D. Fink, 4th edition, PHI Learning, 2005. Introduction to Numerical Analysis, J. Stoer and R. Bulirsch, 3rd edition, Springer, 2002.

MAT744

Representation Theory

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 634 (Differential Geometry) and MAT 640 Graduate Algebra I
Overview: Representations of groups realize the group elements as linear transformations on vector spaces, or even more concretely, as matrices. This enables the use of linear algebra to study algebra, and connects group theory with other areas such as geometry, harmonic analysis and number theory. In this course, we will first study the representation theory of finite groups, and then that of compact groups.
Detailed Syllabus:
Part 1: Representations of Finite Groups
Review of group actions, representations, unitarizability and unitary equivalence of finite dimensional representations of finite groups, complete reducibility, group algebra as a *-algebra, regular representations, matrix coefficients, Schur’s lemmas, tensor products of representations, orthogonality of matrix coefficients, orthogonality of characters, direct sum decompositions, projection formulas, dimension theorem, character tables, Frobenius-Schur theorem on real and quaternionic representations, Fourier analysis on finite groups, subgroups of index 2, induced representations, Frobenius character formula, Frobenius reciprocity, Mackey irreducibility criterion.
Part 2: Representations of Compact Groups
Review of manifolds and Lie groups, the classical compact groups, topological properties of G and G/H, invariant forms and integration, Haar measure, examples of Haar measure for matrix groups, matrix coefficients, characters, Schur orthogonality, review of spectral theory, Schur’s lemma, regular representations, Frobenius reciprocity, Peter-Weyl theorem, representations and harmonic analysis of SU(2), Fourier theory.
References: Representation Theory of Finite Groups by Benjamin Steinberg, Springer. Representations of Finite and Compact Groups by Barry Simon, Graduate Studies in Mathematics, American Mathematical Society. Representation Theory – A First Course by William Fulton and Joe Harris, Springer. A First Course on Representation Theory and Linear Lie Groups by S C Bagchi, S Madan, A Sitaram, and U B Tewari, Universities Press. Compact Lie Groups by Mark R Sepanski, Springer. Short Courses in Mathematics by S Kumaresan, Universities Press.

MAT742

Complex Networks

4.00

Graduate

Complex Networks

MAT741

Analytic Number Theory

4.00

Graduate

Analytic Number Theory

MAT740

Number Theory

4.00

Graduate

Number Theory

MAT726

Topics in Complex Analysis

4.00

Graduate

Topics in Complex Analysis

MAT724

Hardy-Hilbert Spaces & Apps.

4.00

Graduate

Hardy-Hilbert Spaces and Applications

MAT723

Algebras of Operators

4.00

Graduate

Algebras of Operators

MAT722

Algebraic Topology

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 640, MAT 622. Undergraduates can substitute MAT 640 with MAT 240.
Overview: Algebraic topology is a tool which captures key information about geometrical properties of a topological space. We shall focus on two basic algebraic invariants of topological spaces: the fundamental group and the simplicial homology groups. If time permits, we shall also get a glimpse of singular homology.
Detailed Syllabus: The Fundamental Group Homotopy, Fundamental Group, Introduction to Covering Spaces, The Fundamental Group of the circle S 1 , Retractions and fixed points, Application to the Fundamental Theorem of Algebra, The Borsuk-Ulam Theorem, Homotopy Equivalence and Deformation Retractions, Fundamental group of a product of spaces, torus, n-sphere, and the real projective n-space. van Kampen’s Theorem: Free Products of Groups, The van Kampen Theorem, Fundamental Group of a Wedge of Circles, Definition and construction of Cell Complexes, Application of Van Kampen Theorem to Cell Complexes, Statement of the Classification Theorem for Surfaces. Covering Spaces: Universal Cover and its existence, Unique Lifting Property, Homomorphisms and automorphisms of Covering Spaces, Action of the fundamental group on the fibers, Deck Transformations, Group Actions, Covering Space Actions, Normal or Regular Covering Spaces. Simplicial Homology (If time permits): Finite Simplicial complexes, Polyhedra and Triangulations, Simplicial approximation, Barycentric subdivision. Orientation of simplicial complexes, Simplical chain complex and homology. Invariance of homology groups. Computations and applications.
References: A.Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002. S. Deo, Algebraic Topology, Hindustan Book Agency, 2006. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991. W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995. J.R. Munkres, Elements of Algebraic Topology, Addison Wesley, 1984. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004. J.W. Vick, Homology Theory, Springer-Verlag, 1994. E. H. Spanier, Algebraic Topology, Springer, 1994.

MAT721

General Measure Theory

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 623 Analysis II
Overview:
Detailed Syllabus: Measure and Integration Measure Spaces Measurable functions Integration – Fatou’s Lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem. General Convergence Theorems Signed Measures – Hahn Decomposition Theorem, Jordan decomposition of a measure, Radon-Nikodym Theorem, Lebesgue Decomposition Theorem. The Lp spaces – Riesz Representation Theorem. Measure and Outer Measure Outer measure and measurability The extension theorem – Caratheodory Theorem The Lebesgue-Stieltjes Integral Product measures – Fubini’s Theorem, Tonelli Theorem, Lebesgue Integral on Rn, change of variable. Inner Measure Measure and Topology Baire sets and Borel sets Regularity of Baire and Borel measures Construction of Borel Measure Positive linear functionals and Borel Measures – Riesz Markov Theorem (Dual of Cc(X)). Bounded linear functionals on C(X) – Riesz Representation Theorem
References: Real Analysis by H. L. Royden and P. Fitzpatrick. 4th edition, Prentice-Hall India, 2010. Measure Theory and Integration by G. de Barra, New Age International, reprint 2006. Real Analysis: Modern Techniques and their Applications by G. B. Folland, Wiley, 2nd edition, 1999. Measure Theory by Paul Halmos, Springer, 1974.

MAT713

Formal Lang. & Automata Th II

4.00

Graduate

Formal Languages and Automata Theory II

MAT712

Automata Theory

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites:
Overview:
Detailed Syllabus: Formal Logic: Statements and notation, Connectives, Normal Forms, Predicate Logic and Inference Theory, Propositional Logic, Proof in Propositional Logic Syntax of First-Order Logic: First-Order Languages, Formulas of a Language, First-Order Theories Semantics of First-Order Languages: Structures of First-Order Languages, Truth in a Structure, Model of a Theory, Embeddings and Isomorphisms Regular Languages and Regular Grammars: Regular Expressions, Regular Languages, Properties of Regular Languages, Regular Grammars Finite Automata: Finite State machines, Finite Automata, Deterministic Finite Automata, Nondeterministic Finite Automata Context-Free Grammars and Languages: Context-Free Grammars, Parsing, Ambiguity in Grammars and Languages, Pushdown Automata, Chomsky Normal Forms Computability: Turing Machines, Decidable Languages, the Halting Problem, Undecidability, Reducibility Time Complexity: Measuring Complexity, the Complexity Class P, the Complexity Class NP, NP-Completeness
References: S.M. Srivastava A Course on Mathematical Logic, Springer. J. E. Hopcroft, R. Motwani, J. D. Ullman Introduction to Automata Theory, Languages and Computations, Pearson. J. P. Tremblay, R. Manohar Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw-Hill.

MAT690

Time Series And Forecasting

3.00

Graduate

Time Series And Forecasting

MAT689

Operations Research

3.00

Graduate

Operations Research

MAT688

Optimization

4.00

Graduate

Optimization

MAT687

Graph Theory and Complex Net.

4.00

Graduate

Graph Theory and Complex Networks

MAT686

Mathematics for Data Analytics

5.00

Graduate

Mathematics for Data Analytics

MAT685

Business Statistics

3.00

Graduate

Business Statistics

MAT684

Statistics I

4.00

Graduate

Core course for M.Sc. Mathematics.
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 184 (Probability) for undergraduates.
Overview: This course builds on a standard undergraduate probability and statistics course in two ways. First, it makes probability more rigourous by using the concept of measure. Second, it discusses more advanced topics such as multivariate regression, ANOVA and Markov Chains.
Detailed Syllabus: Probability: Axiomatic approach, conditional probability and independent events Random Variables – Discrete and continuous. Expectation, moments, moment generating function Joint distributions, transformations, multivariate normal distribution Convergence theorems: convergence in probability, Weak law of numbers, Borel- Cantelli lemmas, Strong law of large numbers, Central Limit Theorem Random Sampling & Estimators: Point Estimation, maximum likelihood, sampling distributions Hypothesis Testing Linear Regression, Multivariate Regression ANOVA Introduction to Markov Chains
References: Statistical Inference by Casella and Berger. Brooks/Cole, 2007. (India Edition) An Intermediate Course in Probability by Allan Gut. Springer, 1995. Probability: A Graduate Course by Allan Gut. Springer India. Measure, Integral and Probability by Capinski and Kopp. 2nd edition, Springer, 2007.

MAT683

Computational Stat. using R

4.00

Graduate

Computational Statistics using R

MAT682

Computational Economics

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites:
Overview: This is a joint offering with Department of Economics. The objective of the course is to introduce graduate students to computational approaches for solving mathematical problems and economic models. The first half of the course will be devoted in learning (i) the core of the Python programming language, including the main scientific libraries, (ii) a number of mathematical topics central to economic modeling, such as finite and continuous Markov chains, filtering and state space models, Fourier transforms and spectral analysis and (iii) related numerical analysis methods like function approximation, numerical optimization, simulation based techniques and Monte Carlo, recursion. The second half of the course will be devoted in applying these techniques to solve economic problems like growth models, optimal savings problem, and optimal taxation problems. We will pay particular attention to methods for solving dynamic optimization problems.
Detailed Syllabus:
(a) Programming .Basics of Python: Input and output statements, formatting output, copy and assignment, arithmetic operations, string operations, lists and tuples, control statements, user defined functions, call by reference, variable number of arguments, one dimensional arrays, two dimensional arrays, random number generation. The NUMPY and SCIPY packages: Numpy numerical types, data type objects, character codes, dtype constructors. Mathematical libraries, plotting 2D and 3D functions, ODE integrators, charts and histograms, image processing functions, solving models involving difference equations, differential equations, finding limit at a point, approximation using Taylor series, interpolation, definite integrals.
(b) Numerical analysis. Solution of equations in one variable and two variables - Bisection, Newton-Raphson, General iterative scheme, Solution of systems of linear equations – Gauss-Jordan, LU decomposition, QR factorization, Lagrange and Hermite interpolation, Orthogonal polynomials, Gaussian quadrature.
(c) Deterministic dynamic programming. Understanding the fundamentals of dynamic programming and applying to solve the models for equipment replacement, Shortest path, and resource allocation.
(d) Growth Models. As an application we will study the neoclassical growth model.
(e) Other Applications. Other applications that we may study include the optimal savings problem, heterogeneous agents problem, etc.
References:
[1] John Stachurski and Thomas J. Sargent, Quantitative Economics, http://quant-econ.net.
[2] John Stachurski, Economic Dynamics: Theory and Computation, MIT Press, 2009.
[3] M. Miranda and P. Fackler, Applied Computational Economics and Finance, MIT Press, 2002.
[4] K. Judd, Numerical Methods in Economics, MIT Press.
[5] N. L. Stokey and R. E. Lucas with E. C. Prescott, Recursive Methods in Economic Dynamics, Harvard University Press, 1989.
[6] J. Adda and R. Cooper, Dynamic Economics: Quantitative Methods and Applications, MIT Press, 2003.
[7] S. E. Dreyfus, and A. M. Law, The Art and Theory of Dynamic Programming, Academic Press, 1977.
[8] John Zelle, Python Programming: An Introduction to Computer Science, Franklin, Beedle & Associates Inc., 2010.
[9] Ivan Idris, Numpy 1.5 Beginner’s Guide, Packt Publishing, 2011.
[10] Hans Petter Langtangen, A Primer on Scientific Programming on Python, Springer, 2011.
[11] E. Sulli, and D. Mayers, Introduction to Numerical Analysis, Cambridge University Press, 2003.

MAT680

Numerical Analysis & Computer programming

4.00

Graduate

Credits: 4 (3 lectures and 2 lab hours weekly)
Prerequisites: None. Not open for undergraduates.
Overview: This course takes up the problems of practical computation that arise in various areas of mathematics such as solving algebraic or differential equations. The focus is on algorithms for obtaining approximate solutions, and almost half of the course will be devoted to their implementation by computer programs in MATLAB.
Detailed Syllabus: Solving equations: Iterative methods, Bisection method, Secant method, and Newton-Raphson method. Solving Linear systems: Gaussian Elimination and pivoting Computing eigenvalue and eigenvector: Jacobi method Curve fitting Solution of ODEs and systems: Runge-Kutta method, Boundary value problems, Finite Difference Method Solutions of PDEs
References: Numerical Methods using Matlab, by John H. Mathews and Kurtis D. Fink, 4th edition, PHI, 2009. An Introduction to Numerical Analysis, by E. Suli and D. Mayers, Cambridge University Press. Numerical Analysis, by Rainer Kress, Springer, 2010. Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch, 3rd edition, Springer, 2009.

MAT670

Biomathematics

4.00

Graduate

Biomathematics

MAT660

Linear Algebra

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: None. Not open to undergraduates.
Overview: The theory of vector spaces is an indispensable tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications of the language of vector spaces and morphisms among such spaces.
Detailed Syllabus: Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem; statements of various versions of Axiom of Choice. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces. Systems of linear equations: Elementary matrix operations and systems of linear equations. Determinants: Definition, existence, properties, characterization. Diagonalization: Eigenvalues and eigenvectors; diagonalizability; invariant subspaces; Cayley-Hamilton Theorem. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form.
References: Friedberg, Insel and Spence: Linear Algebra, 4th edition, Prentice Hall India Hoffman and Kunze: Linear Algebra, 2nd edition, Prentice Hall India Paul Halmos: Finite Dimensional Vector Spaces, Springer India Sheldon Axler: Linear Algebra Done Right, 2nd edition, Springer International Edition S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India

MAT645

Topological Graph Theory

4.00

Graduate

Topological Graph Theory

MAT644

Discrete Mathematics

4.00

Graduate

Discrete Mathematics

MAT643

Algebraic Graph Theory

4.00

Graduate

Algebraic Graph Theory

MAT642

Graph Theory

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: For undergraduates: MAT 140 (Discrete Structures), MAT 360 (Linear Algebra II). For MSc students: MAT 660 (Linear Algebra)
Overview: Combinatorial graphs serve as models for many problems in science, business, and industry. In this course we will begin with the fundamental concepts of graphs and build up to these applications by focusing on famous problems such as the Traveling Salesman Problem, the Marriage Problem, the Assignment Problem, the Network Flow Problem, the Minimum Connector Problem, the Four Color Theorem, the Committee Scheduling Problem , the Matrix Tree Theorem, and the Graph Isomorphism Problem. We will also highlight the applications of matrix theory to graph theory.
Detailed Syllabus: Fundamentals: Graphs and Digraphs, Finite and Infinite graphs, Degree of a vertex, Degree Sequence, Walk, Path, Cycles, Clique, Operations on Graphs, Complement, Subgraph, Connectedness, Components, Isomorphism, Special classes of graphs: Regular, Complete, Bipartite, Cyclic and Euler Graphs, Hamiltonian Paths and Circuits. Trees and binary trees. Connectivity: Cut Sets, Spanning Trees, Fundamental Circuits and Fundamental Cut Sets, Vertex Connectivity, Edge Connectivity, Separability. Planar graphs, Coloring, Ramsey theory, Covering, Matching, Factorization, Independent sets, Network flows. Graphs and Matrices: Incidence matrix, Adjacency matrix, Laplacian matrix, Spectral properties of graphs, Matrix tree theorem, Automorphism group of a graph, vertex, edge and distance transitive graphs, Cayley graphs. Algorithms and Applications: Algorithms for connectedness and components, spanning trees, minimal spanning trees of weighted graphs, shortest paths in graphs by DFS, BFS, Kruskal's, Prim's, Dijkstra's algorithms.
References: D. West, Introduction to Graph Theory, Prentice Hall. Narsingh Deo, Graph Theory: With Application to Engineering and Computer Science, PHI, 2003. Chris D. Godsil and Gordon Royle, Algebraic Graph Theory, Springer-Verlag, 2001. Norman Biggs, Algebraic Graph Theory, 2nd edition, Cambridge Mathematical Library. Frank Harary, Graph Theory, Narosa Publishing House. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Addison Wesley. R. J. Wilson, Introduction to Graph Theory, 4th Edition, Pearson Education, 2003. Josef Lauri, Raffaele Scapellato, Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts.

MAT640

Graduate Algebra I

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: Linear Algebra, Group Theory
Overview: An overview of graduate algebra with an emphasis on commutative algebra. The instructor may choose 3 or 4 topics from the following, depending on the background and interest of the students.
Detailed Syllabus: Group Actions: Orbit-stabilizer theorem, Centralizers, Normalizers, Class Equation, Sylow Theorems. Rings: Homomorphisms and ideals, quotient, adjunction, integral domains and fraction fields, maximal ideals, factorization, unique factorization domains, principal ideal domains, Euclidean domains, factoring polynomials. Modules: Submodules, quotient modules, homomorphisms, isomorphism theorems, direct sums, simple and semisimple modules, free modules, finitely generated modules, Schur’s Lemma, Jordan-Holder Theorem, Modules over a matrix algebra. Fields: Algebraic and transcendental elements, degree of field extension, adjunction of roots, finite fields, function fields, transcendental extensions, algebraic closure. Galois Theory: Galois group, Galois extension, cubic equations, symmetric functions, primitive elements, quartic equations, Kummer extensions, quintic equations. Homological Algebra: Categories, monomorphisms and epimorphisms, projective and injective modules, left/right exact functors, additive functors, the Hom functor, diagram chasing, push-outs and pull-backs, tensor product, natural transformations, adjoint functors, flat modules.
References: Algebra by M Artin, 2nd edition, Prentice-Hall India, 2011. Abstract Algebra by D S Dummit and R M Foote, 2nd edition, Wiley, 1999. Algebra by S. Lang, 3rd edition, Springer, 2005. Algebra by T. W. Hungerford, Springer India, 2005.

MAT634

Differential Geometry

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 621 (Analysis I) or MAT 332/432 (Geometry of Curves and Surfaces)
Overview: Differential Geometry generalizes the calculus of several variables on Euclidean spaces to `differential manifolds’. This enables the use of analysis and linear algebra to study geometry. The two highlights of this course are the study of Lie groups and of differential forms, with the latter leading to the general Stokes’ theorem in integration.
Detailed Syllabus:
Part 1: Calculus in Rn
Rn as a normed linear space, derivative, chain rule, mean value theorem, directional derivatives, inverse mapping theorem, implicit function theorem, immersions and submersions, integration, higher derivatives, maxima and minima, existence of solutions of ODE.
Part 2: Differential Manifolds and Lie Groups
Differential manifolds, smooth maps and diffeomorphisms, Lie groups, tangent spaces, derivatives, immersions and submersions, submanifolds, vector fields, Lie algebras, flows, exponential map, Frobenius theorem, Lie subgroups and subalgebras.
Part 3: Differential Forms and Integration
Multilinear algebra, exterior algebra, tensor fields, exterior derivative, Poincare lemma, Lie derivative, orientable manifolds, integration on manifolds, Stokes' theorem.
References: An Introduction to Differentiable Manifolds and Riemannian Geometry, by William M Boothby, 2nd edition, Academic Press. A Course in Differential Geometry and Lie Groups, by S Kumaresan, TRIM Series, Hindustan Book Agency. Analysis on Manifolds, by James Munkres, Addison-Wesley. Calculus on Manifolds, by Michael Spivak, Addison-Wesley.

MAT632

Geometry

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 360 Linear Algebra II for undergraduates, MAT 660 Linear Algebra for graduate students
Overview: This course provides a bridge to modern geometry. It provides a unified axiomatic approach leading to a coherent overview of the classical geometries (affine, projective, hyperbolic, spherical), culminating in a treatment of surfaces that sets the stage for future study of differential geometry.
Detailed Syllabus: Affine geometry – finite planes, planes over fields, affine transformations, collineations, affine coordinates, triangles and parallelograms, classical theorems of Menelaus and others. Projective geometry – finite planes, projective completion of affine planes, homogeneous coordinates, projective transformations, collineations, projective line, poles and polars. Conics – affine and projective classifications, group actions. Euclidean geometry – isometries, triangles, parallelograms, length minimizing curves, geometry of plane curves. Hyperbolic geometry – Poincare upper half plane, Poincare metric, distance function, triangles and area, two-point homogeneity. Spherical geometry – Sphere, tangent space, great circles, triangles and area, two-point homogeneity Surfaces – Level surfaces, parametrized surfaces, curvature, Gauss theorem, introduction to manifolds.
References: An Expedition to Geometry by S Kumaresan and G Santhanam. Hindustan Book Agency, 2005. Geometry by M. Audin. Springer International Edition, Indian reprint, 2004. Geometry by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray. Cambridge University Press, 2nd edition, 2012. Geometry by Roger A Fenn, Springer International Edition, 2005.

MAT630

Differential Equations

4.00

Graduate

Core course for M.Sc. Mathematics
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 230/430 ODE (for undergraduates)
Overview:
Detailed Syllabus
(a) Review of Solution Methods for first order and second order linear equations.
(b) Existence and Uniqueness of Initial Value Problems: Lipschitz and Gronwall's inequality, Picard’s Theorem, dependence on initial conditions, continuation of solutions and maximal interval of existence.
(c) Higher Order Linear Equations and Linear Systems: fundamental solutions, Wronskian, variation of constants, matrix exponential solution, behaviour of solutions.
(d)* Two Dimensional Autonomous Systems and Phase Space Analysis: critical points, proper and improper nodes, spiral points, saddle points, Limit cycles, and periodic solutions.
(e)* Asymptotic Behavior: Stability (linearized stability and Lyapunov methods).
(f) Sturm-Liouville Boundary Value Problems: Sturm-Liouville problem for 2nd order equations, Green's function, Sturm comparison theorems and oscillations, eigenvalue problems.
Sections (d) and (e) will also be explored by computer implementation using MATLAB or other software.
References: M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and Introduction to Chaos, Academic Press, 2004. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol. 7, 2nd edition, Springer Verlag, New York, 1998. G. F. Simmons and S. G. Krantz, Differential Equations, Theory, Technique, and Practice, 4th edition, McGraw Hill Education, New Delhi, 2013. William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Wiley, New York, 1992.
Past Instructors: Ajit Kumar, Samit Bhattacharyya

MAT626

Functional Analysis

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 621 for graduate students, MAT 231/320 and MAT 360 for undergraduates.
Overview: This course introduces the tools of Banach and Hilbert Spaces, which generalize linear algebra and geometry to infinite dimensions. It is a prerequisite for advanced topics like Spectral Theory, Operator Algebras, Operator Theory, Sobolev Spaces, and Harmonic Analysis. Functional Analysis is a vital component of applications of mathematics to areas like Quantum Physics and Information Theory.
Detailed Syllabus: Banach Spaces Spaces: Some inequalities, Banach Spaces, finite dimensional spaces, compactness and dimension, quotient spaces, bounded operators, sums of normed spaces. Theorems: Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, Principle of Uniform Boundedness. Spaces: Hahn-Banach Theorem, Spaces in Duality, Adjoint operator. Topologies: Weak topology induced by seminorms, weakly continuous functionals, Hahn-Banach separation theorem, weak*-topology, Alaouglu's Theorem, Goldstine's Theorem, reflexivity, extreme points, Krein-Milman Theorem. Hilbert Spaces Inner products: Inner product spaces, Hilbert spaces, orthogonal sum, orthogonal complement, orthonormal basis, orthonormalization, Riesz Representation Theorem. Operators on Hilbert spaces: Adjoint operators and involution in B(H), Invertible operators, Self adjoint operators, Unitary operators, Isometries. Spectrum: Spectrum of an operator, Spectral mapping theorem for polynomials.
Main References: E. Kreyszig: Introductory Functional Analysis with Applications, Wiley India. G. F. Simmons: Topology and Modern Analysis, Tata McGraw-Hill, 2004. Gert K. Pedersen: Analysis Now, Springer, 1988. John B. Conway: A Course in Functional Analysis, Springer International Edition, 2010. V. S. Sunder: Functional Analysis - Spectral Theory, Hindustan Book Agency, 1997. S. Kesavan: Functional Analysis, Hindustan Book Agency, 2009. G. Bachman and L. Narici: Functional Analysis, 2nd edition, Dover, 2000. Sterling K. Berberian: Lectures in Functional Analysis and Operatory Theory, Springer, 1974.

MAT624

Complex Analysis

4.00

Graduate

Core course for M.Sc. Mathematics
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 621 (Analysis I). Undergraduates not allowed.
Overview: A graduate course of one variable complex analysis.
“The shortest path between two truths in the real domain passes through the complex domain” – Jacques Hadamard.
Detailed Syllabus: The complex number system: The field of Complex numbers, the complex plane, Polar representation and roots of complex numbers, Line and Half planes in the Complex plane, the extended plane and its Stereographic representation. Metric spaces and Topology of complex plane. Open sets in Complex plane, Few properties of metric topology, Continuity, Uniform convergence Elementary properties of Analytic functions. Analytic functions as mapping. Exponential and Logarithm Complex Integration: Basic review of Riemann-Stieltjes integral (without proof), Path integral, Power series representation of an analytic function, Liouville’s theorem and Identity theorem, Index of a closed curve, Cauchy theorem and Integral Formula, Open mapping theorem. Singularities: Removable singularity and Pole, Laurent series expansion, Essential singularity and Casorati-Weierstrass theorem Residues, Solving integral, Argument Principle, Rouche’s Theorem, Maximum modulus theorem. Harmonic Functions: Basic properties, Dirichlet problem, Green function.
References: Functions of One Complex Variable by John B Conway, 2nd edition, Narosa. Complex Analysis by Lars Ahlfors, 3rd edition, McGraw Hill Education India. Introduction to Complex Analysis by H A Priestley, Oxford University Press. Complex Function Theory by D Sarason, 2nd edition, TRIM Series, Hindustan Book Agency. Complex Analysis by T W Gamelin, Springer. Complex Variables by M J Ablowitz and A S Fokas, 2nd edition, Cambridge University Press.

MAT623

Analysis II

4.00

Graduate

Core course for M.Sc. Mathematics
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 621 (Analysis I)
Overview: This course puts the concept of integration of a real function in its most appropriate setting. It is also a prerequisite for the study of general measures, which is the foundation for a large part of pure and applied mathematics – such as spectral theory, probability, stochastic differential equations, harmonic analysis, and partial differential equations.
Detailed Syllabus: Lebesgue Measure: Outer measure, measurable sets, Lebesgue measure, measurable functions, pointwise convergence, almost everywhere convergence. The Lebesgue Integral: Riemann integral, Lebesgue integral of a bounded measurable function over a set of finite measure, Lebesgue integral of a non-negative and a general measurable function. Differentiation and Integration: Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolute continuity. The Classical Banach spaces: Lp spaces, Minkowski and Hölder inequalities, completeness of Lp spaces, bounded linear functions on Lp spaces. Introduction to General Topology: Open and closed sets, bases, separation properties, countability and separation, continuous maps, compactness, connectedness.
References: Real Analysis by H. L. Royden and P. Fitzpatrick. 4th edition, Prentice-Hall India, 2010. Measure Theory and Integration by G. de Barra, New Age International, reprint 2006. Real Analysis by N. L. Carothers, Cambridge University Press.

MAT622

Topology

4.00

Graduate

Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 231/320 Real Analysis II for undergraduates, MAT 621 for graduate students.
Overview: This course concerns 'General Topology' which can be characterized as the abstract framework in which the notion of continuity can be framed and studied. Thus topology provides the basic language and structure for a large part of pure and applied mathematics.
We will take up the following topics: Open and closed sets, continuous functions, subspaces, product and quotient topologies, connected and path connected spaces, compact and locally compact spaces, Baire category theorem, separability axioms.
Detailed Syllabus: Review: Operations with infinite collections of sets, axiom of choice, Zorn's lemma, real line, metric spaces. Topological Spaces: Definition and examples of topological spaces, Hausdorff property, fine and coarse topologies, subspace topology, closed sets, continuous functions, homeomorphisms, pasting lemma, product topology, quotient topology. Connectedness and Compactness: Connected spaces and subsets, path connectedness, compact spaces and subsets, tube lemma, Tychonoff theorem, local compactness, one-point compactification, Baire category theorem. Separation Axioms: First and second countability, separability, separation axioms (T1 etc.), normal spaces, Urysohn lemma, Tietze extension theorem. Topics for Student Presentations: Order topology, quotients of the square, locally (path) connected spaces, sequential and limit point compactness, topological groups, nets, applications of Baire category theorem.
Main Reference: Topology by James R. Munkres, 2nd Edition. Pearson Education, Indian Reprint, 2001. Basic Topology by M. A. Armstrong. Springer-Verlag, Indian Reprint, 2004. Topology by K. Jänich. Undergraduate Texts in Mathematics, Springer-Verlag, 1984. Introduction to Topology and Modern Analysis by G. F. Simmons. International Student Edition. McGraw-Hill, Singapore, 1963. Topology of Metric Spaces by S. Kumaresan. 2nd edition, Narosa, 2011.

MAT621

Analysis I

4.00

Graduate

Core course for M.Sc. Mathematics
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 231/320 Real Analysis II (for undergraduates)
Overview: The aim of this course is to build a rigorous base for advanced topics such as Complex Analysis, Measure & Integration, Numerical Analysis, Functional Analysis, and Differential Equations. The course design also attempts to take into account the diverse backgrounds of our students.
Detailed Syllabus:
Topics under each section are divided in two parts. Part (a) contains topics that will be covered only briefly whereas topics in part (b) will be done in detail. Real number system Archimedean property, density of rationals, extended real numbers, countable sets, uncountable sets. Cauchy completeness of reals, Axiom of Choice, Zorn’s Lemma, equivalence of AC & ZL. Metric spaces Definitions and examples, open sets, closed sets, limit points, closure,equivalent metrics, relative metric, product metric, convergence, continuity, connectedness, compactness. Uniform continuity, completion of a metric space, Cantor’s intersection property, finite intersection property, totally bounded spaces, characterization of compact metric spaces. Sequences and Series of Functions Pointwise and uniform convergence, uniform convergence and continuity, uniform convergence and integration, differentiation, Weierstrass M-test. Power series, exponential and logarithmic functions, Fourier series, equicontinuous family of functions, Stone-Weierstrass approximation theorem, Arzela-Ascoli theorem.
References: Principles of Mathematical Analysis by Walter Rudin, Tata McGraw-Hill Mathematical Analysis by Tom M. Apostol, Narosa Topology of Metric Spaces by S. Kumaresan, Narosa Introduction to Topology & Modern Analysis by G. F. Simmons, Tata McGraw-Hill Real Analysis by N. L. Carothers, Cambridge University Press

MAT620

Measure and Integration

4.00

Graduate

Measure and Integration

MAT606

Topics in Mathematics

4.00

Graduate

Topics in Mathematics

MAT605

Fundamentals of Mathematics

2.00

Graduate

Fundamentals of Mathematics

MAT601

Mathematical Computing

3.00

Graduate

Mathematical Computing

MAT600

Basic Tools in Mathematics

4.00

Graduate

Basic Tools in Mathematics

MAT590

Computational Finance

4.00

Graduate

Course description not available.

MAT584

Stochastic Processes

4.00

Graduate

Stochastic Processes