Syllabus for
Master of Science (Mathematics)
Academic Year  (2022)

The MSc course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician’s habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself.

Course Design: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters focus is on the basic courses in mathematics such as Algebra, Topology, Analysis and Graph Theory along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skill-based courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory, Numerical Analysis  and courses on Data Science . Important feature of the curriculum is that students can specialize in any one of areas (i) Fluid Mechanics, (ii) Graph Theory (and (iii) Data Science with a project on these topics in the fourth semester, which will help the students to pursue research in these topics or grab the opportunities in the industry. To gain proficiency in software skills, four Mathematics Lab papers are introduced one in each semester. viz. Python Programming for Mathematics, Computational Mathematics using Python, Numerical Methods using Python and Numerical Methods for Boundary Value Problem using Python / Network Science with Python and NetworkX / Programming for Data Science in R  respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning and Teaching Technology and Service learning.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: Identify the need and scope of the Interdisciplinary research

PO3: Enhance research culture and uphold the scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO2: Demonstrate problem solving, analytical and logical skills to provide solutions for the scientific requirements

PSO3: Develop critical thinking with scientific temper

PSO4: Communicate the subject effectively and express proficiency in oral and written communications to appreciate innovations in research

PSO5: Understand the importance and judicious use of mathematical software's for the sustainable growth of mankind

EXAMINATION AND ASSESSMENTS (Theory)

EXAMINATION AND ASSESSMENTS (Practicals)

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

Course Description: This course is intended to assist students in acquiring necessary skills on the use of research methodology  in Mathematics. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project and also to the preparation of a research article.

Course Objectives: This course will help the learner to

COBJ 1: Know the general research methods

COBJ 2: Get hands on experience in methods of research that can be employed for research in mathematics

Introduction to research and research methodology, Scientific methods, Choice of research problem, Literature survey and statement of research problem, Reporting of results, Roles and responsibilities of research student and guide.

Introducing mathematics Journals, Reading a Journal article, Mathematics writing skills. - Standard Notations and Symbols, Using Symbols and Words, Organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail, Tools for checking Grammar and Plagiarism.

Package for Mathematics Typing, MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, producing various types of documents using TeX.

Course Description: This course will help students to understand the concepts of functions of single and several variables. This course includes such concepts as Riemann-Stieltjes integral, Sequences and series of functions, Special Functions, and the Implicit Function Theorem.

Course objectives​: This course will help the learner to

COBJ1. Develop in a rigorous and self-contained manner the elements of real variable functions.

COBJ2. Integrate functions of a real variable in the sense of Riemann – Stieltjes.

COBJ3. Classify sequences and series of functions which are pointwise convergent and uniform Convergent.

COBJ4. Explore the properties of special functions.

COBJ5. Understand and apply the functions of several variables.

Definition and existence of Riemann-Stieltjes integral, Linearity properties of Riemann-Stieltjes integral, Riemann-Stieltjes integral as the limit of sums, Integration and differentiation, Integration of vector-valued functions, Rectifiable curves.

Pointwise and uniform convergence, Uniform convergence: Continuity, Integration and differentiation, Equicontinuous families of functions, The Stone-Weierstrass theorem

Power Series, The Exponential and Logarithmic functions, The Trigonometric functions, The Algebraic completeness of the complex field, Fourier series, The Gamma function.

Linear transformations, Differentiation, The Contraction principle, The Inverse function theorem, The Implicit function theorem.

Course Description: This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory.

Course objectives​: This course will help the learner to

COBJ1. Enhance the knowledge of advanced level algebra.

COBJ2. Understand the proof techniques for the theorems on advanced group theory, rings and Galois theory.

Automorphisms, Cayley’s theorem, Cauchy’s theorem, permutation groups, symmetric groups, alternating groups, simple groups, conjugate elements and class equations of finite groups, Sylow theorems, direct products, finite Abelian groups, solvable groups.

Euclidean Ring, polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings.

Extension fields, roots of polynomials, construction with straightedge and compass, more about roots.

The elements of Galois theory, solvability by radicals, Galois group over the rationals, finite fields.

Course description : This helps students understand the beauty of the important branch of mathematics, namely, differential equations. This course includes a study of second order linear differential equations, adjoint and self-adjoint equations, existence and uniqueness of solutions, Eigenvalues and Eigenvectors of the equations, power series method for solving differential equations. Non-linear autonomous system of equations.

Course Objectives: This course will help the learner to

COBJ 1: Solve adjoint differential equations and understand the zeros of solutions

COBJ 2:Understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems.

COBJ 3:Solve the differential equations by power series method and also hypergeometric equations.

COBJ 4:Understand and solve the non-linear autonomous system of equations.

Linear differential equations, fundamental sets of solutions, Wronskian, Liouville’s theorem, adjoint and self-adjoint equations, Lagrange identity, Green’s formula, zeros of solutions, comparison and separation theorems.

Fundamental existence and uniqueness theorem, Dependence of solutions on initial conditions, existence and uniqueness theorem for higher order and system of differential equations, Eigenvalue Problems, Strum-Liouville problems, Orthogonality of eigenfunctions.

Ordinary and singular points of the differential equations, Classification of singular points, Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, Hermite, Laguerre, Chebyshev and Hypergeometric differential equation and its polynomial solutions, standard properties.

Linear system of homogeneous and non-homogeneous equations, Non-linear autonomous system of equations, Phase plane, Critical points, Stability, Liapunov direct method, limit cycle and periodic solutions, Bifurcation of plane autonomous systems.

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.

Course Objectives: This course will help the learner to

COBJ 1: Have thorough understanding of Linear transformations and its properties.

COBJ 2: Understand and apply the elementary canonical forms, rational and Jordan forms in real life problems.

COBJ 3: Gain knowledge on Inner product space and the orthogonalisation process.

COBJ 4: Explore different types of bilinear forms and their properties.

Linear transformations, algebra of linear transformations, isomorphism, representation of transformation by matrices, linear functionals, the transpose of a linear transformation, determinants: commutative rings, determinant functions, permutation and the uniqueness of determinants, additional properties of determinants.

Elementary canonical forms: characteristic values, annihilating polynomials, invariant subspaces, simultaneous triangulation and diagonalization, direct sum decomposition, invariant dual sums, the primary decomposition theorem. the rational and Jordan forms: cyclic subspaces and annihilators, cyclic decompositions and the rational form, the Jordan form, computation of invariant factors, semi-simple operators.

Inner products, Inner product spaces, Linear functionals and adjoints, Unitary operators – Normal operators, Forms on Inner product spaces, Positive forms, Spectral theory, Properties of Normal operators.

Bilinear forms, Symmetric Bilinear forms, Skew-Symmetric Bilinear forms, Groups preserving Bilinear forms.

Course Description: This course deals with the fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration, recurrence relations and analysis of algorithms.

Course Objectives: The course will help the learner to

COBJ 1: develop mathematical foundations to understand and create mathematical arguments.

COBJ 2: demonstrate logical reasoning to solve a variety of practical problems.

COBJ 3: implement enumeration techniques in a variety of real-life problems.

COBJ 4: develop efficient algorithms and determine their efficiency.

COBJ 5: communicate the basic and advanced concepts of the subject precisely and effectively.

Sets: Cardinality and countability, recursively defined sets, relations, equivalence relations and equivalence classes, partial and total ordering, representation of relations, closure of relations, functions, bijection, inverse functions.

Logic: Propositions, logical equivalences, normal forms, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies.

Fundamental principles, pigeon-hole principle, permutations – with and without repetitions, combinations- with and without repetitions, binomial theorem, binomial coefficients, the principle of inclusion and exclusion, derangements, arrangements with forbidden positions, rook polynomial.

Ordinary and exponential generating functions, recurrence relations, first-order linear recurrence relations, higher-order linear homogeneous recurrence relations, non-homogeneous recurrence relations, solving recurrence relations using generating functions.

Real-valued functions, big-O, big-Omega and big-Theta notations, orders of power functions, orders of polynomial functions, analysis of algorithm efficiency, the sequential search algorithm, exponential and logarithmic orders, binary search algorithm.

1. R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction,5th ed., New Delhi: Pearson, 2014.
2. S. S. Epp, Discrete Mathematics with Applications, Boston:Cengage Learning, 2019 (for Unit-4).

1. C. L. Liu, Introduction to Combinatorial Mathematics, New York: McGraw-Hill Book Co., 1968.
2. K. Erciyes, Discrete Mathematics and Graph Theory, New York: Springer, 2021.
3. K.H. Rosen, Discrete Mathematics with Applications, New York: McGraw-Hill Higher Education, 2019.
4. B. Kolman, R. C. Busby and S. C. Ross, Discrete Mathematical Structures, 6th ed., New Jersey: Pearson Education, 2013.
5. J. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Application to Computer Science, Noida: Tata McGraw Hill Education, 2008.
6. S. Lipschutz and M. Lipson, Discrete Mathematics, New Delhi: Tata McGraw-Hill, 2013.
7. N. L. Biggs, Discrete Mathematics, 2nd ed., New Delhi: Oxford University Press, 2014.

Course description: This course aims at introducing the programming language Python and its uses in solving problems on discrete mathematics and differential equations.

Course objectives: This course will help the learner to

COBJ1. Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ2. Gain proficiency in using Python to solve problems on Differential equations. 

COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph.

Installation, IDE, variables, built-in functions, input and output, modules and packages, data types and data structures, use of mathematical operators and mathematical functions, programming structures (conditional structure, the for loop, the while loop, nested statements)

Use of Sympy package, Symbols, Calculus, Differential Equations, Series expressions, Linear and non-linear equations, List, Tuples and Arrays.

Standard plots (2D, 3D), Scatter plots, Slope fields, Vector fields, Contour plots, stream lines, Manipulating and data visualizing data with Pandas, Mini Project

Course Description: This course aims at introducing the elementary notions on Machining learning and focuses on some simple application of machine-learning, algorithms on supervised machine learning and unsupervised learning.

Course Objective: This course will help the learner to:

COBJ1. Be proficient on the idea of machine learning

COBJ2. Implement Supervised Machine Learning Algorithms

COBJ3. Handle computational skills related to unsupervised learning and preprocessing

Introduction - simple machine learning applications: classifying iris species: meet the data, training and testing data, pair plot of iris dataset - k-nearest neighbours model, evaluating model.

Classification and regression - generalization, overfitting and underfitting - relation of model complexity to dataset size, supervised machine learning algorithms: k-nearest neighbour algorithm: k-neighbors classification, k-neighbors regression, strengths, weakness and parameters of k-NN algorithm, linear models: linear models for regression, linear models for classification, linear models for multiclass classification, strengths, weakness and parameters of linear models, decision trees: building decision trees, controlling complexity of decision trees, analyzing decision trees, strengths, weakness and parameters of decision trees.

Types of unsupervised learning, challenges in unsupervised learning, preprocessing and scaling: different kinds of preprocessing, applying data transformations, scaling training and test data, principal component analysis, non-negative matrix factorization.

Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.

Course objectives​: This course will help the learner to:

COBJ1. Provide precise definitions and  appropriate examples and counter-examples of  fundamental concepts in general topology.

COBJ2. Acquire knowledge about a generalisation of the concept of continuity and related properties.

COBJ3. Appreciate the beauty of deep mathematical results such as  Uryzohn’s lemma and understand and apply various proof techniques.

Elements of topological spaces, basis for a topology, the order topology, the product topology on X x Y, the subspace topology, Closed sets and limit points.

Continuous functions, the product topology, metric topology.

Connected spaces, connected subspaces of the Real line, components and local connectedness, compact spaces, Compact Subspaces of the Real line, limit point compactness, local compactness.

The countability axioms, the separation axioms, normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension theorem.

Course Description: This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 3-circles theorem.

Course objectives​: This course will help the learner to

COBJ1. enhance the understanding the advanced concepts in complex Analysis.

COBJ2. acquire problem solving skills in complex Analysis.

Power series, radius and circle of convergence, power series and analytic functions, Line and contour integration, Cauchy’s theorem, Cauchy integral formula, Cauchy integral formula for derivatives, Cauchy integral formula for multiply connected domains, Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus principles.

Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles.

Rational functions, behavior of functions in the neighborhood of an essential singularity, Cauchy’s residue theorem, contour integration problems, Mobius transformation, conformal mappings.

Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.

Course Description: This helps students understand the beauty of the important branch of mathematics, namely, partial differential equations. This course includes a study of first and second order linear and non-linear partial differential equations, existence and uniqueness of solutions to various boundary conditions, classification of second order partial differential equations, wave equation, heat equation, Laplace equations and their solutions by Eigenfunction method and Integral Transform Method.

Course Objectives: This course will help the learner to

COBJ 1: Understand the occurrence of partial differential equations in physics and its applications.

COBJ 2: Solve partial differential equation of the type heat equation, wave equation and Laplace equations.

COBJ 3: Also solving initial boundary value problems.

Formation of PDE, initial value problems (IVP), boundary value problems (BVP) and IBVP, solutions of first, methods of characteristics for first order PDE, linear and quasi, linear, method of characteristics for one-dimensional wave equations and other hyperbolic equations.

Origin of second order PDE, Classification of second order PDE, Initial value problems (IVP), Boundary value problems (BVP) and IBVP, Mathematical models representing stretched string, vibrating membrane, heat conduction in solids, second-order equations in two independent variables. Cauchy’s problem for second order PDE, Canonical forms, General solutions.

Occurrence of heat equation in Physics, resolution of boundary value problem, elementary solutions, method of separation of variables, method of eigen function expansion, Integral transforms method, Green’s function.

Occurrence of wave and Laplace equations in Physics, Jury problems, elementary solutions of wave and Laplace equations, methods of separation of variables,, the theory of Green’s function for wave and Laplace equations.

Course Description: This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.

Course objectives: This course will help the learner to

COBJ 1: Know the history and development of Graph Theory

COBJ 2: Understand all the elementary concepts and results

COBJ 3: Learn proof techniques and algorithms in Graph Theory

Graphs as models, degree sequences, classes of graphs, matrices, isomorphism, distances in graphs, connectivity, Eulerian and Hamiltonian graphs, Chinese postman problems, travelling salesman problem and Dijkstra’s algorithm.

Properties of trees, rooted trees, spanning trees, algorithms on trees- Prufer’s code, Huffmans coding, searching and sorting algorithms, spanning tree algorithms.

Graphical embedding, Euler’s formula, platonic bodies, homeomorphic graphs, Kuratowski’s theorem, geometric duality.

Vertex and edge coloring, chromatic polynomial and index, matching, decomposition, independent sets and cliques, vertex and edge covers, clique covers, digraphs and networks.

Course Description: This course aims at introducing the fundamental aspects of fluid mechanics. They will have a deep insight and general comprehension on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of non-Newtonian fluids.

Course Objectives: This course will help the learner to

COBJ1: Understand the basic concept of tensors and their representatives.

COBJ2: Derive and understand the governing equations in fluid mechanics.

COBJ3: Familiarize with two- or three-dimensional incompressible flows.

COBJ4: Determine properties of inviscid and viscous fluids.

COBJ5: Describe and construct analytically standard two- or three-dimensional viscous flows.

Cartesian tensors: Cartesian tensors, basic properties, transpose, symmetric and skew symmetric tensors, gradient, divergence and curl in tensor calculus, integral theorems. Continuum hypothesis: deformation gradient, strain tensors, infinitesimal strain, compatibility relations, principal strains, material and local time derivatives, transport formulas, stream lines, path lines.

Stress and Strain: stress components and stress tensor, normal and shear stresses, principal stresses, transformation of the rate of strain and stress, relation between stress and rate of strain. Fundamental basic physical laws: The equation of continuity, conservation of mass, equation of motion (Navier-Stokes equations), conservation of momentum, the energy equation, conservation of energy.   

Bernoulli equation, applications of Bernoulli equation, circulation theorems, circulation concept, Kelvin’s theorem, constancy of circulation, Laplace equations, stream functions in two- and three-dimensional motion. Two-dimensional flow: Rectilinear flow, source and sink, the theorem of Blasius.

Flow between parallel flat plates, Couette flow, plane Poiseuille flow, the Hagen- Poiseuille flow, flow between two concentric rotating cylinders.

Course Description: This course aimsto solve mathematical models using differential equations, linear algebra and fluid mechanics using Python libraries.

Course objectives​: This course will help the learner to

COBJ1: Acquire skill in using suitable libraries of Python to solve real-world problems giving rise to differential equations

COBJ2: Gain proficiency in using Python to solve problems on linear algebra.

COBJ3: Build user-defined functions to deal with the problem on fluid mechanics.

1. Linear and non-linear model growth and decay Newton’s law of cooling/warming, mixtures, computational models with quadratic growth, concentration of a nutrients.
2. Lotka-Volterra predator-prey model, competition models.
3. Spring/mass systems: Free undamped motion, free damped motion, driven motion, nonlinear springs, simple pendulum, projectile, double pendulum.
4. Matrices and determinants, operations on matrices, powers and inverse of a matrix.
5. Rank, solving systems of linear equations (Gauss-Elimination method, Gauss-Jordan method, LU decomposition method), eigenvalues, eigenvectors, and diagonalization.
6. Linear combinations, linearly independence and dependence, basis and dimension, linear transformation, orthogonal set, orthogonal projection of a vector, orthonormal, Gram Schmidt process.
7. Stream lines, path lines, vortex lines and their plots.
8. Calculating Rayleigh number for Rayleigh-Bénard convection with external constraints magnetic field, rotation, non-uniform temperature gradients.
9. Solution of Lorenz equations, Nusselt number.

Course Description: This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching, they are exposed to the principles, procedures and techniques of planning and implementing teaching techniques. Through service learning they will apply the knowledge in real-world situations and benefit the community.

Course objectives: This course will help the learner to

COBJ 1: Understand the pedagogy of teaching.

COBJ 2: Able to use various ICT tools for effective teaching.

COBJ 3: Apply the knowledge in real-world situation.

COBJ 4: Enhances academic comprehension through experiential learning.

Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (Audio-Visual), Teaching aids (projected and non-projected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching.

Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community.

A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives.

 < marks to be converted to credits >

Course description: The Course covers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue.

Course objectives: This course will help the learner to 

COBJ1. Enhance the understanding of the advanced notions from Mathematical Analysis 

COBJ2. Know more about the Measure theory and Lebesgue Integration

Lebesgue Outer Measure, The s-Algebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Additivity, Continuity and the Borel-Cantelli Lemma, Nonmeasurable Sets, The Cantor Set and the Canton-Lebesgue Function, Sums, Products and Compositions of Lebesgue Measurable Functions, Sequential Pointwise Limits and Simple Approximation, Littlewood’s three principles, Egoroff’s Theorem and Lusin’s Theorem.

The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure, The Lebesgue Integral of a Measurable Nonnegative Function; The General Lebesgue Integral; Countable Additivity and Continuity of Integration, Uniform Integrability, Uniform Integrability and Tightness, Convergence in measure, Characterizations of Riemann and Lebesgue Integrability.

Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives.

Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The L^p spaces, Approximation and Separability, The Riesz Representation for the Dual of L^p, Weak Sequential Convergence in L^p, Weak Sequential Compactness, The Minimization of Convex Functionals.