Syllabus for
Master of Science (Mathematics)
Academic Year  (2023)

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 / Numerical Linear Algebra using MATLAB respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning (during 2024-2025 for 2023-2024 batch) 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

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.

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. Demonstrate the ability to manipulate and use of special functions

COBJ5. Use and operate functions of several variables.

Definition and Existence of Riemann-Stieltjes Integral, Linearity Properties of Riemann-Stieltjes Integral, The Riemann-Stieltjes Integral as the Limit of Sums, Integration and Differentiation, Integration of Vector-valued Functions, Rectifiable Curves.

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.

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.

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

Course Objectives: The course will help the learner to

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

COBJ 2: implement enumeration techniques in a variety of real-life problems.

COBJ 3: analyze the order and efficiency of algorithms.

COBJ 4: communicate the basic and advanced concepts of the topic precisely and effectively.

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: gain proficiency in using Python for programming.
COBJ2: acquire skills in usage of suitable functions/packages of Python to solve mathematical problems.
COBJ3: acquaint with Sympy and Numpy packages for solving concepts of calculus, linear algebra and Differential equations.
COBJ4: illustrates use of built-in functions of Pandas and Matplotlib packages for visualizing of data and plotting of graphs.

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 situations.

COBJ 4: Enhances academic comprehension through experiential learning.

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.

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 and theorems on meromorphic functions.

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.

Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus theorems.  Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles.

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.

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

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 representations.
COBJ2: applies the Physics and Mathematics concepts for derivations and interpretations of concepts of fluid mechanics.
COBJ3: familiarize with two- or three-dimensional incompressible flows and viscous flows.
COBJ4: classify Newtonian and non-Newtonian Fluids.

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.

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 situations.

COBJ 4: Enhances academic comprehension through experiential learning.

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

Course Description: This course deals with the theory and application of various advanced methods of numerical approximation. These methods or techniques help us to approximate the solutions of problems that arise in science and engineering. The emphasis of the course will be the thorough study of numerical algorithms to understand the guaranteed accuracy that various methods provide, the efficiency and scalability for large scale systems and issues of stability.

Course Objectives: This course will help the learner

COBJ1. To develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use.

COBJ2. To become familiar with the methods which will help to obtain solution of algebraic and transcendental equations, linear system of equations, finite differences, interpolation numerical integration and differentiation, numerical solution of differential equations and boundary value problems.

COBJ3. Understand accuracy, consistency, stability and convergence of numerical methods.

Fixed point iterative method, convergence criterion, Aitken’s -process, Sturm sequence method to identify the number of real roots, Newton-Raphson methods (includes the convergence criterion for simple roots), Graeffe’s root squaring method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Cholesky method), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm.

Course Description: This course helps learners to acquire active knowledge and understanding of the basic concepts and properties of the geometry of curves and surfaces in Euclidean space. Also, this course aims at connecting geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus.

Course Objectives: This course will help the learner to

COBJ 1: understand the calculus on the Euclidean geometry of E3

COBJ 2: implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms.

COBJ 3: derive and understand about the intrinsic geometry of the surfaces and curves on surfaces.

COBJ 4: interpret the structure of surfaces using the first and second fundamental forms.

Course Description: This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems. This course also includes the mathematical modelling of nano-liquids

Course objectives: This course will help the learner to

COBJ 1: Understand the different modes of heat transfer and their applications.

COBJ 2: Understand the importance of doing the non-dimensionalization of basic equations.

COBJ 3: Understand the boundary layer flows.

COBJ 4: Familiarity with porous medium and non-Newtonian fluids

Course description: Theory of intersection graphs, perfect graphs, chromatic graph theory and eigenvalues of graphs are dealt with in this course.

Course objectives: This course will help the learner to

COBJ 1: understand and apply the fundamental and advanced concepts in distance-related graph coloring problems.

COBJ 2: understand and apply the fundamental and advanced concepts in color connections and disconnections.

COBJ 3: understand and apply the fundamental and advanced concepts in spectral properties of graphs.

COBJ 4: enhance the skill of proof-writing techniques.

Data Science is an interdisciplinary, problem-solving oriented subject that learns to apply scientific techniques to practical problems. This course provides strong foundation for data science and application area related to information technology and understand the underlying core concepts and emerging technologies in data science.

Course Description: This course aims at introducing the computational aspects of Linear Algebra.

Course Objectives​:  This course will help the learner to

COBJ1. Demonstrate the computational ability in handling matrices, norms and method of least squares.

COBJ2.   Solve systems of equations using various methods of numerical linear algebra..

Course Description: This course provides the fundamentals of magnetohydrodynamics, which include theory of Maxwell’s equations, basic equations, exact solutions and applications of classical MHD.

Course objectives​: This course shall help the students to

COBJ1: understand mathematical form of Gauss’s law, Faraday’s law and Ampere’s law and their corresponding boundary conditions.

COBJ2: derive the basic governing equations and boundary conditions of MHD flows.

COBJ3: finding the exact solutions of MHD governing equations.

COBJ4: understand the Alfven waves and derive their corresponding equations.

Course Description: This course covers a large area of domination in graphs. This course discusses different types of dominations with their applications in real-life situations, the relation of domination related parameters with other graph parameters such as vertex degrees, chromatic number, independence number, packing number, matching number etc.

Course objectives: This course will help the learner to
COBJ 1: Understand the advanced topics of domination in graphs.
COBJ 2: Understand different types of dominations related to various real-life situations
COBJ 3: Enhance the understanding of techniques of writing proofs for advanced topics in domination theory.

The main aim of this course is to provide fundamental knowledge of neural networks and deep learning. On successful completion of the course, students will acquire fundamental knowledge of neural networks and deep learning, such as Basics of neural networks, shallow neural networks, deep neural networks, forward & backward propagation process and build various research projects.

Course Description: The main aim of this course is to provide fundamental knowledge of fractional calculus. This course includes a study of special functions and different fractional differential and integral operators and their properties. This course also helps to know how to solve fractional differential equations using different methods.

Course objectives: This course will help the learner to

COBJ1.fundamentals and properties of special functions and their properties.

COBJ2.familiarize with the difference between different fractional derivatives.

COBJ3. analyze and develop problem-solving skills for fractional differential equations by various methods.

Course description: In this course programming Numerical Methods in Python will be focused. How to program the numerical methods step by step to create the most basic lines of code that run on the computer efficiently and output the solution at the required degree of accuracy.

Course objectives: This course will help the learner to

COBJ1. Program the numerical methods to create simple and efficient Python codes that output the numerical solutions at the required degree of accuracy.

COBJ2. Use the plotting functions of matplotlib to visualize the results graphically.

COBJ3. Acquire skill in usage of suitable functions/packages of Python to solve initial value problems numerically.

The objective of this course is to provide the students an opportunity to gain work experience in the relevant institution, connected to their subject of study. The experienced gained in the workplace will give the students a competetive edge in their career.

This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment.

·       Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester.

·       Each student shall be under the supervision of a faculty mentor/guide.

·       The 15 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers.

·       A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide.

·       Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters.

Course Description: This course deals with some of the key ideas of classical mechanics like generalized coordinates, Lagrange's equations and Hamilton's equations. Also, this course aims at introducing the Lagrangian Mechanics and Hamiltonian mechanics on Manifolds.

Course Objectives: This course will help the learner to

COBJ 1: derive necessary equations of motions based on the chosen configuration space.

COBJ 2: gain sufficient skills in using the derived equations in solving the applied problems in Classical Mechanics.

COBJ 3: effectively use Lagrangian mechanics on the manifolds.

COBJ 4: deal with the Hamiltonian mechanics on the manifolds.

Course Description: This abstract course imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics.

Course Objective. This course will help learner to

COBJ1: know the notions behind functional analysis

COBJ2. enhance the problem solving ability in functional analysis