Curriculum Information at IIT Delhi

Department of Mathematics

MTD6010: M.Sc. Project Part I

4.00 credits (0.0-0.0-8.0)
It is an independent research project engaging students in systematic research themes, strong foundation and analysis, experimental/computational work, data analysis, developing critical problem-solving skills, and advanced research and technical knowledge in their specialized project domain.

On successful completion of the course, a student will be able to:
  1. Formulate and execute research projects with systematic investigation, research survey, building understanding, analysis and implementation.

MTD6020: M.Sc. Project Part II

6.00 credits (0.0-0.0-12.0)
Pre-requisites: MTD6010
This is Part II of the M.Sc. Project, serving as a continuation of Part I under specified academic guidelines and evaluation criteria. In this course, students pursue an independent research project within a defined research theme, building upon the work initiated in Part I. The emphasis is on deepening theoretical foundations, conducting experimental and/or computational investigations, performing systematic data analysis, and further developing advanced critical thinking, problem-solving skills, and technical expertise in their specialized domain.

On successful completion of the course, a student will be able to:
  1. Formulate and execute research projects with systematic investigation, research survey, building understanding, analysis and implementation.

MTD6030: Cornerstone Project

3.00 credits (0.0-0.0-6.0)
This course is designed for M.Sc. Mathematics students to foster collaborative and interdisciplinary learning through team-based projects. Students will engage in identifying mathematical and scientific problems or themes, particularly by connecting advanced concepts from senior secondary levels (Classes XI–XII) with foundational ideas from Classes VI–X, or by developing content based on NCERT textbook activities. The course emphasizes concept mapping, the design of mathematical or experimental models, and the creation of educational resources such as demonstrations and explanatory videos. Students will also develop skills in mathematical communication, documentation, and presentation, while gaining exposure to interdisciplinary approaches in mathematics education and outreach.

On successful completion of the course, a student will be able to:
  1. Upon successful completion of the course, students will be able to work effectively in teams to address scientific and educational challenges, and demonstrate the ability to identify and integrate concepts across different levels of learning. They will be capable of designing and developing experimental models or educational content, while applying interdisciplinary perspectives. Additionally, students will develop proficiency in communicating scientific ideas through written, visual, and oral formats, and will exhibit enhanced critical thinking, creativity, and problem-solving skills in project-based environments.

MTL1001: Mathematics I Calculus

4.00 credits (3.0-1.0-0.0)
Sequences of real numbers: real number system and its completeness property, Archimedean property, definitions of sequence and convergence, Cauchy sequence, definition of limit superior and inferior. Infinite series: series of real numbers, absolute and conditional convergence, alternating series, test for convergence: comparison, Cauchy condensation, ratio, root, Leibniz’s tests. Differential calculus: limits, continuity, differentiability, mean value theorems, Taylor’s theorem, Taylor’s series, power series, maxima and minima. Integral calculus: definition of Riemann integral, properties, fundamental theorems, improper integrals of first and second kind, beta and gamma functions. Multivariable differential calculus: functions of several variables, limits, continuity, partial derivatives, directional derivatives, differentiability, chain rule, gradient, Taylor’s theorem, maxima and minima in two variables, method of Lagrange multipliers. Multivariable integral calculus: double and triple integrals, Jacobian and change of variables, polar, spherical, and cylindrical coordinate systems, surface integrals, applications to area, volume and surface.

On successful completion of the course, a student will be able to:
  1. Appreciate the need for precision, logical reasoning and mathematical rigour by analyzing various functions with diverse behaviours.
  2. Explore the role of completeness in the real number system and appreciate its significance in the convergence of sequences and series.
  3. Develop problem-solving skills using limits, derivatives, and integrals to solve problems in single and multivariable calculus.
  4. Apply calculus techniques to analyze and model real-world problems, including optimization, area/volume computations, and coordinate transformations.
  5. Recognize how the concept of convergence facilitates the approximation of solutions and ensures their validity, especially in cases where exact solutions are impractical or unavailable.

MTL1002: Mathematics II Linear Algebra and Differential Equations

4.00 credits (3.0-1.0-0.0)
Systems of linear (homogeneous and non-homogeneous) equations, matrices and Gauss elimination, elementary row operations, row reduced echelon form, row space, column space, null space, and rank of a matrix. Vector spaces over Q, R, and C, subspaces, linear independence, linear span of a set of vectors, basis and dimension of a vector space, sum and direct sum. Introduction to inner product and norm (definitions and examples), linear transformation, rank-nullity theorem and its applications, matrix representation of a linear transformation, change of basis, and similarity. Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem (without proof) and applications. Review of first-order differential equations, existence and uniqueness of solutions of an initial value problem, Picard’s theorem. Linear independence of solutions of linear differential equations, dimensionality of the space of solutions, and Wronskian. Linear ODE with constant coefficients of second and higher order, Cauchy-Euler equations, method of undetermined coefficients and method of variation of parameters. System of linear differential equations with constant coefficients, fundamental matrix, matrix methods. Taylor series method, Laplace transform and its application to solve ODE.

On successful completion of the course, a student will be able to:
  1. Analyze and communicate mathematical ideas with logic, rigor and draw logical conclusions
  2. Use the concept of row reduction and echelon forms to find inverse of a square matrix, rank of a matrix and to solve system of linear equations which arise in many real-world problems.
  3. Analyze vector spaces, linear transformations, evaluate eigenvalues and eigenvectors, diagonalize linear transformations, and apply these concepts to various problems across various disciplines.
  4. Apply the inner product and norm space concepts to different applications.
  5. Demonstrate how to model physical phenomenon in the mathematical way via differential equations and apply linear algebra concepts along with various methods (techniques) to solve various differential equations.

MTL1080: Discrete Mathematical Structures

4.00 credits (3.0-1.0-0.0)
Preclusions: COL1002
Logic: propositional logic: language of propositional logic, truth table, natural deduction, predicate logic: language of predicate logic, logical inference with quantifiers. Proof techniques: introduction to different standard proof techniques. Set theory: review of basic set operations, cardinality of a set. Relations: types of relations, operations of relations and applications, posets, topological ordering; congruence arithmetic. Combinatorics: counting techniques: pigeon hole principle, inclusion exclusion principle, recurrence relation and generating function. Graph theory: graph as a discrete structure, modeling applications using graphs, Hamiltonian graphs, planar graphs, graph coloring, matching.

On successful completion of the course, a student will be able to:
  1. construct and analyze rigorous mathematical arguments through formal logic.
  2. apply posets and topological ordering to order theory and algorithm design
  3. apply counting techniques, including the Pigeonhole Principle, Inclusion-Exclusion Principle, recurrence relations, and generating functions to essential problems in discrete mathematics, probability, and algorithm analysis.
  4. model real-world problems in scheduling, optimization, and network design, reinforcing their ability to apply mathematical reasoning to computational and practical challenges.

MTL2003: Optimization Methods and Applications

4.00 credits (3.0-1.0-0.0)
Preclusions: MEL2004, MTL5008, COL7156, MEL7101, CVL7055
Linear programming - formulation through examples from decision making problems, preliminary theory and geometry of linear programs, basic feasible solution, simplex method, variants of simplex method. Duality and its principles, dual simplex method. Linear integer programming, applications in real decision making problems, Branch- Bound and Cutting plane methods, Nonlinear programming, gradient descent, stochastic gradient, and Newton’s Method for unconstrained nonlinear problems, introduction to these algorithms and their convergence, nonlinear constrained optimization problem, KKT optimality conditions, sufficiency of KKT for convex programs, second order conditions, examples of convex programs least norm problems, statistics parameter estimation problems, applications of convex programs to support vector classification, regression, and clustering problems.

On successful completion of the course, a student will be able to:
  1. Understanding the real life decision making problem formulation as an optimization problem.
  2. Understanding the various methods and algorithms to solve linear programming and linear integer programming problems.
  3. Understanding basic methods and algorithms to solve nonlinear unconstrained and constrained optimization problems
  4. Application of convex optimization problems in support vector classification, regression and clustering problems.

MTL2004: Advanced Linear Algebra

3.00 credits (3.0-0.0-0.0)
Preclusions: MTL7860, MTL5002, ELL7100, COL7126, MTL1002
Pre-requisites: (MTL1002)
Review the basis and dimension of a vector space, linear transformations, eigenvalues, and eigenvectors of an operator. Dual and double dual of a vector space and transpose of a linear transformation. Diagonalizability of linear operators of finite-dimensional vector spaces, triangularization, simultaneous triangulization and simultaneous diagonalization. The primary decomposition theorem - diagonal and nilpotent parts. Minimal polynomial, Jordan canonical form and its applications. Inner product spaces, Gram-Schmidt orthogonalization, Best approximation of a vector by a vector belonging to a given subspace, and application to least square problems. Adjoint of an operator, Hermitian, unitary and normal operators. Introduction of bilinear and quadratic forms, positive definite matrices. Spectral decomposition, polar decomposition, singular value decomposition and its applications.

On successful completion of the course, a student will be able to:
  1. Analyze and communicate linear algebra concepts with logic, rigour and draw logical conclusions to solve algebraic and applied problems.
  2. Students will rigorously analyze the duality of vector spaces—including dual, double duals, and their relation to the fundamental subspaces through the lens of linear transformations. The study will also include the construction of Jordan canonical forms of the linear operators. Theoretical insights will be applied to problems in optimization and computational mathematics.
  3. Students will characterize invariant subspaces and minimum polynomials of linear operators, leveraging their interplay to determine when a linear operator admits a triangular or diagonal matrix representation. The students will also be able to identify commuting operators, their nature of simultaneously diagonalizable and simultaneously triangularizable.
  4. Students will develop a comprehensive understanding of inner product spaces and their fundamental properties, and their geometric implications. They will explore the construction of orthonormal bases through the Gram-Schmidt process and examine the theoretical foundations of orthogonal complements and projections. Building on these concepts, students will apply orthogonal projections to solve least squares approximation problems, connecting abstract theory to practical applications in data fitting and linear regression. Through proofs and computational exercises, students will develop confidence in these concepts and their relevance to both theoretical and real-world problem-solving.
  5. Students will explore key concepts such as self-adjoint and normal operators, the spectral theorem, isometries, and unitary operators. They will learn to apply matrix factorization techniques such as QR decomposition and singular value decomposition (SVD), examining their consequences—including polar decomposition and the approximation of linear maps. Additionally, students will investigate how singular values relate to operator norms and volume calculations, connecting theory to applications in other branches of their respective study.

MTL2005: Algebra

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL5001
Group Theory: Equivalence relations and partitions, maximal elements, minimal elements, least upper bound, greatest lower bound, lattices, with examples. Permutations, cyclic permutation, transpositions, and the sign of a permutation. Binary operations, formal definition of group, subgroup, basic examples of groups, symmetric group(permutation group), matrix group, group of rigid motions of the plane, and finite group of motions (Dihedral group). Finite group, cyclic group, generators and relations, cosets and Lagrange's theorem, group homomorphisms, Cayley's theorem, normal subgroup, quotient group and isomorphism theorems. Group action, Class equation, Cauchy's theorems, Sylow's theorems, direct products and structure theorem for finite abelian groups (without proof) and their applications. Ring Theory: Definition and examples, ring homomorphism, ideals and quotient rings, Chinese Remainder Theorem, integral domain and quotient field, ring of Gaussian integers, Unique factorisation domain, Principal ideal domain, Euclidean domain, Gauss' lemma (statement only), polynomial ring, irreducibility of polynomials and Eisenstein's criterion. Field Theory: Fields, extension of a field, degree of an extension, finite extension, algebraic extension, Splitting field, Finite field and their structures.

On successful completion of the course, a student will be able to:
  1. Basics of Group Theory: Introduction to the fundamental concepts of group theory, including the definition and properties of groups, types of groups, and basic examples, in order to build a foundational understanding of group structures.
  2. Some advance topics in Group Theory: These topics are foundational for the development of group theory and have various applications in physics, cryptography, and geometry.
  3. Ring Theory : This introductory module provides the basics of ring theory which has applications in many areas of mathematics such as Number Theory and Algebraic Geometry.
  4. Field Theory : In this module, we study the basics of field theory, focusing on the fundamental properties and structure of fields. Special attention is given to finite fields which plays a crucial role in modern applications such as cryptography, coding theory, and error correction.

MTL2006: Introduction to Probability and Stochastic Processes

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL2008, MEL7102, MTL7725, ELL7400, MTL5005
Random Experiments, Probability Definition, Properties of Probability Function, Bayes Theorem, Random Variables, Distribution Function, Probability Mass and Density Functions, Moments, Generating Functions, Moment Inequalities, Standard Probability Distributions, Two and Higher Dimensional Random Variables, Independent Random Variables, Covariance and Correlation, Conditional Distribution, Conditional Expectation, Law of Large Numbers, Central Limit Theorem, Stochastic Processes and their Classification, Markov and Stationary properties, Discrete and Continuous Time Markov Chains, Poisson and Birth-Death Processes, Simple Markov Queuing Systems, Applications in Engineering.

On successful completion of the course, a student will be able to:
  1. Understanding probability concepts, how the world around us is inherently random, and how probabilistic models can be used to model real-life scenarios.
  2. Developing analytical skills to model uncertainty using probability distributions for real-world problem-solving in engineering and science.
  3. Model real-world systems using appropriate stochastic process types, such as Markov chains, Poisson processes, and birth and death processes.
  4. Solve practical problems using Markov chain models, such as modeling queuing systems or predicting future states in a system.

MTL2007: Numerical Analysis and Computation

4.00 credits (3.0-1.0-0.0)
Preclusions: AML3000, MTL5009, MTL8251, PYL8000, PYL3025
Floating point representation, floating point arithmetic, and roundoff errors. Roots of a nonlinear equation in one variable. Direct and iterative methods for linear system of equations, Condition number and ill-conditioned systems. Solutions of nonlinear system of equations. Interpolation: Lagrange, Newton’s interpolations, and errors in interpolation. Approximation: least square approximations, QR Decomposition, Continuous least square, Legendre, Chebyshev’s polynomials, Fourier Series. Numerical differentiation: Finite difference formulas and errors. Numerical integration: Newton-Cotes formulas, Gauss quadrature rules. Numerical methods for Initial value problem for ODEs

On successful completion of the course, a student will be able to:
  1. Develop ability to implement basic numerical methods
  2. Ability to analyze numerical algorithms by understanding there convergence, stability and efficiency
  3. Demonstrate ability to analyze computational error and its origin in the computational results.

MTL2008: Introduction to Probability and Statistics

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL2006
Random experiments, probability definition, properties of probability function, Bayes theorem, discrete and continuous random variables, distribution function, probability mass and density functions, moments, generating functions, moment inequalities, standard probability distributions, two and higher dimensional random variables, independent random variables, joint distributions, marginal and conditional distributions, covariance and correlation, conditional expectation, law of large numbers, central limit theorem, descriptive statistics, graphical representation of data, sampling distributions, point estimation, properties of estimators, methods of estimation, interval estimation, hypothesis testing: one and two-sample tests, goodness of fit.

On successful completion of the course, a student will be able to:
  1. Understanding probability concepts, how the world around us is inherently random, and how probabilistic models can be used to model real-life scenarios.
  2. Developing analytical skills to model uncertainty using probability distributions for real world problem-solving in engineering and science.
  3. Develop understanding of interpreting data using exploratory data analysis and descriptive statistics.
  4. Making statistical inferences, and applying statistical methods for decision-making in engineering and scientific contexts.

MTL2022: Introduction to Analysis

4.00 credits (3.0-1.0-0.0)
Metric space: Limits in metric spaces, Open sets, Closed sets, Relative metric, Completeness, Compact sets, Connected sets, Separability Continuity: Continuous functions on metric spaces, Homeomorphisms, Continuity and compactness, Continuity and connectedness, Uniform continuity Sequences and series of functions: Pointwise and uniform convergence, The Weierstrass M-test, Uniform convergence and continuity, Uniform convergence and integration, Uniform convergence and differentiation, The Weierstrass theorem, equicontinuity and Ascoli-Arzela theorem. Lebesgue measure: Outer measure, Lebesgue measure, Measurable sets, A Nonmeasurable set, Measurable functions Lebesgue Integration: Simple functions, Integration of non-negative measurable functions, Monotone convergence theorem, Fatou’s lemma, Integration of absolutely integrable functions, Lebesgue dominated convergence theorem, Comparison with the Riemann integral

On successful completion of the course, a student will be able to:
  1. Enable students to Communicate mathematical arguments clearly and precisely, Construct formal mathematical proofs in analysis, apply analytical reasoning to solve complex mathematical problems and extend their knowledge to advanced topics; Bulid a strong foundation in analysis and applications in mathematics, statistics, and applied sciences.

MTL2042: Analysis and Design of Algorithms

4.00 credits (3.0-1.0-0.0)
Preclusions: COL2051, ELL7281, AIL7002
Asymptotic notation; DFS, BFS, and some applications; greedy algorithms: interval scheduling, optimal caching, minimum spanning trees (Prim’s and Kruskal’s ), shortest path, etc.; divide and conquer: sorting, matrix multiplication, etc.; dynamic programming: scheduling, subset sum, etc.; branch and bound, backtracking techniques; network flow algorithms and applications; NP-completeness and reductions; introduction to approximation algorithms; recent trends in algorithms.

On successful completion of the course, a student will be able to:
  1. Apply fundamental algorithm design strategies, including divide-and-conquer, dynamic programming, greedy methods to develop efficient solutions for computational problems.
  2. Formally verify the correctness of algorithms by constructing rigorous proofs using mathematical invariants and inductive reasoning.
  3. Examine the efficiency of algorithms by assessing their computational resource requirements, such as time complexity and memory usage, to determine their scalability and feasibility.
  4. Identify, design, and implement appropriate data structures that enhance the performance of algorithms and contribute to more efficient problem-solving approaches. Learn recent advances in Algorithms for solving hard problems.
  5. Explore the concept of computational hardness by understanding complexity classes and demonstrating problem hardness through reduction techniques.

MTL3107: Topology

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL6507
Topological spaces: definitions and examples, basis and subbasis for a topology, limit points, closure, interior; continuous functions, homeomorphisms; subspace topology, metric topology, product and box topology, order topology; quotient spaces. Connectedness and compactness: connectedness, path connectedness; connected subspaces of the real line; components and local connectedness; compact spaces, limit point compactness, sequential compactness; local compactness, one point compactification; Tychonoff theorem (without proof), characterizations of compact metric spaces. Countability axioms: first countable spaces, second countable spaces, separable spaces, Lindeloff spaces. Separation axioms: Hausdorff, regular and normal spaces; Urysohn’s lemma (without proof), Urysohn’s metrization theorem (without proof), tietze extension theorem (without proof), and their applications. Completely metrizable spaces, Baire’s category theorem.

On successful completion of the course, a student will be able to:
  1. Understand the definitions and examples of topological spaces, basis and subbasis, limit points, closure, interior, and their role in topology. Develop proficiency in continuous functions, homeomorphisms, and different types of topologies, including subspace, metric, product, box, order, and quotient topologies.
  2. Analyze various forms of connectedness, including path connectedness and local connectedness. Understand compactness, limit point compactness, sequential compactness, and local compactness. Explore the concept of one-point compactification and apply Tychonoff’s theorem and characterizations of compact metric spaces.
  3. Develop an understanding of first-countable, second-countable, separable, and Lindelöf spaces, and their implications in topology.
  4. Gain knowledge of Hausdorff, regular, and normal spaces. Understand key theorems such as Urysohn’s lemma, Urysohn’s metrization theorem, and Tietze’s extension theorem, along with their applications in topology.
  5. Study completely metrizable spaces and apply Baire’s category theorem to analyze the structure of topological and metric spaces.

MTL3111: Functional Analysis and Applications

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL5007
Review of Basic Concepts: Metric spaces and basic concepts. Normed and Inner Product Spaces: Normed linear spaces, Inner product spaces, Banach spaces, Hilbert spaces, Properties and examples, Geometric concepts in inner product spaces., Schauder basis and separability. Bounded Operators: Bounded linear operators, Examples and properties, Bounded linear functionals, Dual spaces of some concrete Banach spaces, Weak convergence. Closed and Compact Operators: Closed operators, Compact operators, Examples, Some simple properties and applications. Fundamental Theorems in Functional Analysis: Hahn-Banach extension theorem, Applications of Hahn-Banach extension theorem, Uniform boundedness principle, Closed graph theorem, Open mapping theorem. Hilbert Space Theory: Riesz representation theorem and some applications, Orthogonal complements, Projection theorem, Compact self-adjoint operators, Spectral theorem for compact self-adjoint operators.

On successful completion of the course, a student will be able to:
  1. Understand fundamental concepts in metric spaces, normed linear spaces and Banach spaces, including their topological and geometric properties.
  2. Apply the principles of linear operators, duality, and weak convergence to solve problems in functional analysis and related fields.
  3. Appreciate the significance of key theorems such as the Hahn-Banach theorem, Uniform Boundedness Principle, Open Mapping theorem and Closed Graph Theorem in analysing the structure of infinite-dimensional spaces.
  4. Develop proficiency in working with compact and closed operators, including their continuity properties and applications in functional analysis.
  5. Utilize the concepts of Hilbert spaces, orthogonality, and the projection theorem to solve approximation and optimization problems.
  6. Explore the spectral theorem for compact self-adjoint operators and its applications to solving differential equations and functional equations.
  7. Recognize the relevance of functional analysis techniques in broader mathematical and applied contexts, including physics, engineering, and data science.
  8. Solve problems applying abstract functional analysis concepts to concrete examples and real-world scenarios.

MTL3176: Graph Algorithms

4.00 credits (3.0-1.0-0.0)
Pre-requisites: MTL2042/COL2051
Introduction to Graphs: Definition and basic concepts, efficient representations of Graphs, revision of graph searching and different MST algorithms, Enumeration of all spanning trees of a graph. Graph Connectivity algorithms, Hamiltonian Graphs: sufficient conditions for Hamiltonian graphs, checking Hamiltonicity: Bellman-Held-Karp Algorithm, Characterization of Eulerian graphs, The Chinese Postman problem, Traveling Salesman Problem, Planar Graphs: properties of planar graphs, a planarity testing algorithms; Graph Coloring: vertex coloring, edge coloring, planar graph coloring; Matching: maximum matching in bipartite graphs, maximum matching in general graphs; Finding Perfect Matchings, NP-completeness of graph problems, Exact Exponential Algorithms for NP-complete graph problems, Approximation algorithms for some NP-Hard graph problems; Algorithms for some NP-Hard graph problems on special graph classes; Treewidth: Definition, Dynamic Programming on Bounded Treewidth Graphs, Approximating Treewidth.

On successful completion of the course, a student will be able to:
  1. Apply fundamental graph algorithms to analyze connectivity, spanning trees, matchings, planarity, and coloring in graphs
  2. Design and analyze algorithms for classical and NP-hard graph problems using exact, approximation, and structure-based techniques
  3. Assess computational complexity and exploit graph structure (planarity, bounded treewidth) to solve hard cases efficiently

MTL3190: Statistical Inference

4.00 credits (3.0-1.0-0.0)
Pre-requisites: MTL1108/MTL1106
Basic concepts and data visualization: measures of central tendency, dispersion, skewness, kurtosis, data representation using histogram, pie chart, boxplot, biplot; revision of probability distribution: some standard discrete and continuous distributions; some standard probability inequalities; order statistics: different order statistics and their single and joint distribution; mode of convergence, CLT; random number generation: generation of random numbers following certain distributions; sampling distribution of mean, variance; emphasis on normal, chi-square, student's t, F distributions; theory of estimation: properties of an estimator, sufficient, completeness, minimal sufficient, ancillarity, unbiased estimator, MVUE, BLUE, Fisher information, Cramer-Rao inequality, Rao-Blackwell theorem, Lehmann-Scheffé theorem; methods of finding estimators: MoM, MLE; testing of hypothesis: mean and variance, Neyman-Pearson lemma, MLR, Karlin-Rubin theorem, LRT; non-parametric methods run tests, rank tests, signed rank tests, Kolmogorov-Smirnov test, chi-squared goodness of fit test, test of homogeneity, test of independence, etc.; confidence interval estimation: inversion of a test procedure, pivotal approach; methods of evaluating interval estimation.

On successful completion of the course, a student will be able to:
  1. Understanding the basic concepts of Probability, different types of distributions, mode of convergence of Random Variable, and Random Number Generation, briefly.
  2. Understanding the basic concepts of Sampling Distribution and Theory of Estimation
  3. Understanding the basic concepts of Testing of Hypothesis
  4. Understanding the basic concepts of Confidence Interval

MTL5001: Algebra

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL2005
Definition and some examples of groups, Subgroups, Lagrange's theorem, Normal subgroups, Quotient groups, Cyclic groups, Permutation and Alternating groups, Direct product of groups, Isomorphism theorems, Class equation and its applications, Automorphism group, Cayley's theorem, Group actions, Sylow theorems and its applications, Simple groups, Simplicity of alternating groups, Structure theorem of finite abelian groups, Free abelian groups, Structure of finitely generated abelian groups (statement only), Semi-direct product of groups, Solvable and nilpotent groups, Composition series, Jordan-Holder theorem. Definition of some examples of commutative and non-commutative rings, Zero divisors, Idempotent and nilpotent elements, Subrings, Polynomial rings, Formal power series rings, Matrix rings, Boolean rings, Rings of continuous functions, Ring of Gaussian integers, Product of rings, Ideals, Algebra of ideals, Quotient rings, Isomorphism theorems, Chinese remainder theorem, Prime ideals, Maximal ideals, Local rings, Definition of fields and some examples, Integral domains, Construction of field of fractions, Irreducible and prime elements in a commutative ring with unity, Principal ideal domains, Unique factorisation domains, Primitive polynomials over UFDs, Gauss's lemma and Gauss's theorem on UFDs, Eisenstein's criterion, Euclidean domains.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to understand and work with the basic concepts of groups, subgroups, normal subgroups and various isomorphism theorems
  2. On successful completion of this course, a student should be able to understand and apply concepts of Sylow theorems and understand advanced concepts in group theory
  3. On successful completion of this course, a student should be able to understand and work with basic concepts of rings, subrings, ideals and various isomorphism theorems
  4. On successful completion of this course, a student should be able to understand basic concept of factorization rings and apply some irreducibility criterion to study various polynomial rings

MTL5002: Linear Algebra

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL7860, MTL2004, MTL1002, ELL7100, ELL7120
Review system of equations, row reduced echelon form, vector spaces, subspaces, basis and dimension of a vector space, row equivalence. In addition, the course also reviews linear transformations, Isomorphisms, null space and range, linear functionals, and representation of linear transformations by matrices. Dual and double dual of a vector space and transpose of a linear transformation, Invariant subspaces, quotient spaces, eigenvalue and eigenvector of an operator, Minimal polynomial and characteristic polynomial. Diagonalizability of linear operators of finite-dimensional vector spaces, triangularization, simultaneous triangulization and simultaneous diagonlization. The primary decomposition theorem - diagonal and nilpotent parts. Jordan canonical form and its applications. Inner product spaces, Gram-Schmidt orthogonalization, Best approximation of a vector by a vector belonging to a given subspace, and application to least square problems. Adjoint of an operator, Hermitian, unitary and normal operators. Introduction of bilinear and quadratic forms, positive definite matrices. Spectral decomposition, polar decomposition, singular value decomposition and its applications.

On successful completion of the course, a student will be able to:
  1. Ability to analyze and effectively communicate linear algebra concepts with clarity and logical rigor, and apply these concepts to solve the theoretical and real -world problems.
  2. Students successfully completing this course will be able to determine the row equivalence of matrices, comprehend and apply the ideas of vector spaces, subspaces, basis, and dimension. They will be able to analyze and solve systems of linear equations using row reduced echelon form. Additionally, students will be able to compute null spaces and ranges, identify isomorphisms, interpret and construct linear transformations, work with linear functionals, and represent linear transformations using matrices.
  3. Students will be able to compute the transpose of linear transformations, analyze and construct a vector space's dual and double dual, and investigate the structure of vector spaces using quotient spaces and invariant subspaces. Along with applying the ideas of minimal and characteristic polynomials to the study of the algebraic properties of linear transformations, students will also be able to calculate the eigenvalues and eigenvectors of linear operators.
  4. Students will be able to calculate and interpret linear operators' minimal and characteristic polynomials, ascertain whether operators on finite-dimensional vector spaces are diagonalizable and triangulable, and evaluate the requirements for simultaneous triangularization and diagonalization. The students will also be able to identify commuting operators, its nature of simultaneously diagonalizable and simultaneously triangularization. Additionally, students will use the Jordan Canonical Form to analyze linear transformations and solve related theoretical and computational problems, as well as apply the Primary Decomposition Theorem to decompose operators into diagonal and nilpotent parts.
  5. A comprehensive understanding of inner product spaces, including their fundamental properties and geometric interpretations will be studied by the student. They will explore the construction of orthonormal bases using the Gram-Schmidt process and examine the theoretical foundations of orthogonal complements and projections. Building on these concepts, students will apply orthogonal projections to solve least squares approximation problems, effectively bridging abstract linear algebra theory with practical applications such as data fitting and linear regression.
  6. With a focus on their algebraic and geometric characteristics, students will gain a firm grasp of the adjoint of a linear operator as well as the distinction between Hermitian, unitary, and normal operators. Bilinear and quadratic forms, as well as the description of positive definite matrices, will be presented to them.Additionally, students will explore key matrix factorizations such as spectral decomposition, polar decomposition, and singular value decomposition (SVD), and apply these tools to a range of theoretical and practical problems.

MTL5003: Real Analysis

4.00 credits (3.0-1.0-0.0)
Review: Elementary set theory, countable and uncountable sets, the real number system and its order completeness. Metric spaces, open sets and closed sets, limit and continuity, separability and homeomorphism. Complete metric space, contraction mappings, Baire category theorem, and totally bounded sets. Connected metric spaces, compact metric spaces, compactness and continuous spaces, characterisation of compactness, Arzela-Ascoli theorem. Sequence of functions, uniform convergence, Dini's theorem, Weierstrass approximation theorem Directional derivative, total derivative, mean-value theorem, higher-order derivatives, inverse function theorem and implicit function theorem.

On successful completion of the course, a student will be able to:
  1. Enable students to Communicate mathematical arguments clearly and precisely, Construct formal mathematical proofs in analysis, apply analytical reasoning to solve complex mathematical problems and extend their knowledge to advanced topics; Bulid a strong foundation in analysis and applications in mathematics, statistics, and applied sciences.

MTL5004: Introduction to Computer Programming

4.00 credits (3.0-0.0-2.0)
Introduction to python, basic concepts and data organization, data types, data structures, operators, built-in functions, strings, conditional and loop based control structures, if/else statements, for/while loops, iterating, functions, modules, importing, lists and tuples, slicing techniques, dictionaries and classes, mathematical operations, object-oriented programming, recursions, inheritance, files, handling exceptions, operations on arrays, operations on singly and doubly linked structures, searching and sorting algorithms, linking python with data science, NumPy, pandas, data visualization, exploratory data analysis (python will be used for all concepts and modules).

On successful completion of the course, a student will be able to:
  1. Explain the fundamental concepts of programming using Python syntax.
  2. Define writing, debugging, and formulate Python programs for basic computational problems.
  3. Explain the effective use of functions, control structures, and data structures such as dictionaries and lists.
  4. Solve the real-data problems using Python programming

MTL5005: Introduction to Probability

4.00 credits (3.0-1.0-0.0)
Preclusions: ELL7400, MEL7102
Random experiments and events, axiomatic definition of probability, properties of probability functions, conditional probability, Bayes’ theorem, discrete and continuous random variables, distribution functions, probability mass and density functions, expectations and moments, moment-generating functions, moment inequalities, standard probability distributions, joint, marginal, and conditional distributions, independence, covariance and correlation, conditional expectation, law of large numbers, and central limit theorem.

On successful completion of the course, a student will be able to:
  1. Understanding probability concepts, how the world around us is inherently random, and how probabilistic models can be used to model real-life scenarios.
  2. Analyze and compute properties of random variables using distributions, expectations, moment generating functions, and inequalities.
  3. Understand independent random variables, convergence concepts, apply limit theorems and their interpretations.

MTL5006: Ordinary Differential Equations

4.00 credits (3.0-1.0-0.0)
Review of the method of variation of parameter, method of separation of variables, exact equation, integrating factor for 1st order ODE, Bernoulli equation, first order scalar and vector valued ODE, Cauchy-Peano existence theorem, Picard-Lindelof theorem, maximal interval of existence, continuous dependence, Gronwall's inequalities, higher order linear ODE with constant coefficients, Sturm's theory on separation and comparison properties of solutions for second order ODE, two point boundary value problems, Sturm-Liouville problems, Green function, maximum principle and its application, power series method for some special second order ODE, Legendre and Bassel equations, linear system with constant coefficients, fundamental matrix, linear systems with periodic coefficients, Boundedness and asymptotic behaviour of linear systems, critical point for nonlinear systems and their stability, phase plane analysis.

On successful completion of the course, a student will be able to:
  1. Demonstrate how to model physical phenomenon in the mathematical way via ordinary differential equations; apply various methods (techniques) to solve various ODEs (with logic and rigor) and to analyze the behavior of solutions of ODEs.
  2. Enable to apply these in-depth knowledge to various problems across various disciplines.

MTL5007: Functional Analysis

4.00 credits (3.0-1.0-0.0)
Normed linear spaces: Normed linear spaces and their examples; Banach spaces and their examples; quotient spaces; finite-dimensional Banach spaces and properties. Bounded linear operators: definition and examples of bounded linear operators; properties of bounded linear operators. Dual and transpose: definition and examples of dual spaces; characterization of the dual of certain Banach spaces; definition and properties of the transpose of a bounded linear operator. Fundamental theorems of functional analysis: Hahn–Banach theorems; uniform boundedness principle; open mapping theorem; closed graph theorem. Geometry of Hilbert spaces: inner product spaces and their properties; Hilbert spaces and examples; best approximation in Hilbert spaces; orthogonal complements; orthonormal bases; dual of a Hilbert space. Basic operator theory in Hilbert spaces: Adjoint of an operator; self-adjoint operators; normal and unitary operators; projections. Compact operators: definition, examples, and basic properties of compact operators; spectral theorem for compact self-adjoint operators.

On successful completion of the course, a student will be able to:
  1. Demonstrate a thorough understanding of the structure of normed linear and Banach spaces — including quotient spaces, finite-dimensional cases, and dual spaces — and effectively communicate the underlying mathematical reasoning.
  2. Understand and employ fundamental theorems of functional analysis — including the Hahn–Banach theorem, uniform boundedness principle, open mapping theorem, and closed graph theorem — in order to formulate, analyse, and solve problems involving operators in Banach and Hilbert spaces.
  3. Analyse the geometry of Hilbert spaces and apply operator theory, including the study of adjoint, self-adjoint, normal, and unitary operators, projections, and compact operators, and employ the spectral theorem for compact self-adjoint operators in problem-solving.

MTL5008: Mathematical Programming

4.00 credits (3.0-1.0-0.0)
Linear programming - formulation through examples from decision making problems, preliminary theory and geometry of linear programs, basic feasible solution, simplex method, variants of simplex method. Duality and its principles, dual simplex method, primal dual method. Linear integer programming, applications in real decision making problems, Branch- Bound and Cutting plane methods, dynamic programming, Nonlinear programming, gradient descent, stochastic gradient, and Newton’s Method for unconstrained nonlinear problems, introduction to these algorithms and their convergence, nonlinear constrained optimization problem, KKT optimality conditions, sufficiency of KKT for convex programs, second order conditions, examples of convex programs least norm problems, statistics parameter estimation problems, applications of convex programs to support vector classification, regression, and clustering problems.

On successful completion of the course, a student will be able to:
  1. Understanding the real life decision making problem formulation as an optimization problem.
  2. Develop understanding of the various methods and algorithms to solve linear programming and linear integer programming problems.
  3. Understanding basic methods and algorithms to solve nonlinear unconstrained and constrained optimization problems
  4. Application of convex optimization problems in support vector classification, regression and clustering problems.

MTL5009: Numerical Analysis

4.00 credits (3.0-1.0-0.0)
Numerical algorithms and errors, floating point systems, roundoff error accumulations. Solution of nonlinear equations: fixed point iteration, bisection, secant, regula-falsi, Newton-Raphson methods. Interpolation: Lagrange interpolation, Newton’s divided difference interpolation, finite differences, Hermite interpolation, cubic splines. Numerical differentiation. Numerical integration: Newton cotes formulas, Gaussian quadrature, composite quadrature formulas approximation: least squares approximation, minimum maximum error techniques. Legendre and Chebyshev polynomials. Solution of linear systems: direct methods, Gauss elimination, LU and Cholesky factorizations. Iterative methods – Jacobi, Gauss- Seidel and SOR methods. System of nonlinear equations, eigen-value problems: power and inverse power method. Numerical solution of IVPS for ODEs: Taylor series, Euler and Runge-Kutta methods.

On successful completion of the course, a student will be able to:
  1. Students will be able to develop numerical methods for the underlying problems
  2. Students will be able to analyze convergence, stability and efficiency of numerical algorithms
  3. Students will be able to develop ability to implement basic numerical methods and analyze computational error and its presence in the numerical results

MTL5010: Measure and Integration

4.00 credits (3.0-1.0-0.0)
Riemann Integral: A brief review of the Riemann integral and its limitations; Lebesgue’s method for extending the Riemann integral. The Lebesgue Measure on the Real Line: Outer measure induced by the length function; Carathéodory’s criterion for measurability; Lebesgue measurable sets in R; existence of non-measurable sets in R. Abstract Measure Spaces: General construction of measures and outer measures; extension of a pre-measure to a measure; uniqueness of the extension; completion of a measure space; measurable functions; convergence of measurable functions (almost everywhere, in measure, in mean, and almost uniformly) with respect to a measure; Fatou’s Lemma, the monotone convergence theorem, and dominated convergence theorem; L^p spaces. Measure and Integration in Product Spaces: Construction of product measure spaces; integration over product spaces; Fubini’s and Tonelli’s theorems. Fundamental Theorem of Calculus for the Lebesgue Integral: Absolutely continuous functions; differentiability of monotone functions; the fundamental theorem of calculus for the Lebesgue integral and its applications.

On successful completion of the course, a student will be able to:
  1. Explain the limitations of the Riemann integral and describe the construction of the Lebesgue measure on the real line.
  2. Identify and verify measurable sets and functions; distinguish between different modes of convergence with illustrative examples.
  3. Apply key theorems of Lebesgue integration—such as Fatou’s Lemma, Monotone and Dominated Convergence Theorems, and Fubini’s Theorem— to solve problems in integration and convergence.
  4. Demonstrate foundational understanding of abstract measure spaces and the Fundamental Theorem of Calculus for the Lebesgue integral, preparing for further study or research in analysis.

MTL5011: Statistical Inference

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL3190, MTL2008, MEL7102, HSL5013
Pre-requisites: MTL5005
Basic concepts and data visualization: measures of central tendency, dispersion, skewness, kurtosis, data representation using histogram, pie chart, boxplot, biplot; revision of probability distribution: some standard discrete and continuous distributions; some standard probability inequalities; order statistics: different order statistics and their single and joint distribution; mode of convergence, CLT; random number generation: generation of random numbers following certain distributions; sampling distribution of mean, variance; emphasis on normal, chi-square, student's t, F distributions; theory of estimation: properties of an estimator, sufficient, completeness, minimal sufficient, ancillarity, unbiased estimator, MVUE, BLUE, Fisher information, Cramer-Rao inequality, Rao-Blackwell theorem, Lehmann-Scheffé theorem; methods of finding estimators: MoM, MLE; testing of hypothesis: mean and variance, Neyman-Pearson lemma, MLR, Karlin-Rubin theorem, LRT; non-parametric methods run tests, rank tests, signed rank tests, Kolmogorov-Smirnov test, chi-squared goodness of fit test, test of homogeneity, test of independence, etc.; confidence interval estimation: inversion of a test procedure, pivotal approach; methods of evaluating interval estimation.

On successful completion of the course, a student will be able to:
  1. Understanding the basic concepts of Probability, different types of distributions, mode of convergence of Random Variable, and Random Number Generation, briefly.
  2. Understanding the basic concepts of Sampling Distribution and Theory of Estimation
  3. Understanding the basic concepts of Testing of Hypothesis
  4. Understanding the basic concepts of Confidence Interval

MTL5021: Professional Ethics in Mathematics

1.00 credits (1.0-0.0-0.0)
Identify ethical issues in mathematical modeling, data handling, statistics, algorithmic decisions, and AI, ChatGPT, internet etc, analyse real-world ethical dilemmas using mathematical reasoning. Apply ethical frameworks to research publishing, proof validation, and plagiarism. Demonstrate responsible conduct in teaching, assessment, and supervision.

On successful completion of the course, a student will be able to:
  1. Describe key ethical concepts and professional codes of conduct relevant to mathematics.
  2. Analyzing real-world mathematics problems to identify ethical dilemmas and apply structured problem solving approaches.

MTL6001: Complex Analysis

4.00 credits (3.0-1.0-0.0)
Field of complex numbers, Complex plane, Polar representation, Stereographic projection, Holomorphic functions, Cauchy-Riemann equation and applications, Harmonic conjugates, Power series, Exponential function, Branch of logarithm, Contour integration, ML-inequality, Rectangle/Triangle theorem, Integral theorem, Cauchy-Goursat theorem, Cauchy integral formula, Power series representation of a holomorphic function, Morera’s theorem, Liouville’s theorem, Fundamental theorem of Algebra, Identity theorem, Maximum modulus principle, Minimum modulus principle, Open mapping theorem, Inverse function theorem, Schwarz’s Lemma, Möbius transformations, Isolated singularities, Classification of singularities, Casaroti-Weierstrass theorem, Laurent Series expansion, Cauchy residue theorem and application, Meromorphic functions, Argument principle, Rouche’s theorem and applications, Applications of contour integral techniques and residue theorem: Calculating real improper integrals, evaluating infinte series, Introduction to Fourier series, Fourier transform and calculating inverse Fourier transform.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to understand the complex numbers, their properties, holomorphic functions, properties of holomorphic functions and basic examples through power series, contour integration techniques and their applications in studying holomorphic functions via Cauchy integral theorem, moreover they would be able to write power series expanison of holomorphic functions.
  2. On successful completion of this course, a student should be able to understand key properties of holomorphic and entire functions, like the Liouville's theorem, Identity theorem, Maximum modulus principle, Open mapping theorem, Inverse function theorem.
  3. On successful completion of this course, a student should be able to understand key points about different type of isolated singularities, behaviour of the functions around the singularities, Laurent expansion around them, integration of functions with singularity via Cauchy residue theorem and several apllications including Argument principle, Rouche's theorem.
  4. On successful completion of this course, a student should be able to apply countour integration techniques and Cauchy resideue theorem to evaluate improper integrals, study Fourier transform and inverse Fourier transform, moreover calculate inverse Laplace transform.

MTL6011: Discrete Mathematical Structures

4.00 credits (3.0-1.0-0.0)
Logic: propositional logic: language of propositional logic, truth table, natural deduction, predicate logic: language of predicate logic, logical inference with quantifiers. Proof techniques: introduction to different standard proof techniques. Review on basic set theory and relation: basic set operations, cardinality of a set, types of relations, operations of relations and applications, posets, topological ordering; congruence arithmetic. Combinatorics: counting techniques: pigeon hole principle, inclusion exclusion principle, recurrence relation and generating function. Graph theory: graph as a discrete structure, modeling applications using graphs, reachability, connectivity, cycles, forests, trees, Euler tours, Hamiltonian graphs, planar graphs, graph coloring, matching.

On successful completion of the course, a student will be able to:
  1. construct and analyze rigorous mathematical arguments through formal logic.
  2. apply counting techniques, including the Pigeonhole Principle, Inclusion-Exclusion Principle, recurrence relations, and generating functions to essential problems in discrete mathematics, probability, and algorithm analysis.
  3. model real-world problems in scheduling, optimization, and network design, reinforcing their ability to apply mathematical reasoning to computational and practical challenges.

MTL6507: Topology

4.00 credits (3.0-1.0-0.0)
Topological spaces: definitions and examples, basis and subbasis for a topology, limit points, closure, interior; continuous functions, homeomorphisms; subspace topology, metric topology, product and box topology, order topology; quotient spaces. Connectedness and compactness: connectedness, path connectedness; connected subspaces of the real line; components and local connectedness; compact spaces, limit point compactness, sequential compactness; local compactness, one point compactification; Tychonoff theorem (without proof), characterizations of compact metric spaces. Countability axioms: first countable spaces, second countable spaces, separable spaces, Lindeloff spaces. Separation axioms: Hausdorff, regular and normal spaces; Urysohn’s lemma (without proof), Urysohn’s metrization theorem (without proof), tietze extension theorem (without proof), and their applications. Completely metrizable spaces, Baire’s category theorem.

On successful completion of the course, a student will be able to:
  1. Understand the definitions and examples of topological spaces, basis and subbasis, limit points, closure, interior, and their role in topology. Develop proficiency in continuous functions, homeomorphisms, and different types of topologies, including subspace, metric, product, box, order, and quotient topologies.
  2. Analyze various forms of connectedness, including path connectedness and local connectedness. Understand compactness, limit point compactness, sequential compactness, and local compactness. Explore the concept of one-point compactification and apply Tychonoff’s theorem and characterizations of compact metric spaces.
  3. Develop an understanding of first-countable, second-countable, separable, and Lindelöf spaces, and their implications in topology.
  4. Gain knowledge of Hausdorff, regular, and normal spaces. Understand key theorems such as Urysohn’s lemma, Urysohn’s metrization theorem, and Tietze’s extension theorem, along with their applications in topology.
  5. Study completely metrizable spaces and apply Baire’s category theorem to analyze the structure of topological and metric spaces.

MTL6632: Nonparametric Statistical Methods

3.00 credits (2.0-0.0-2.0)
Pre-requisites: MTL5005&MTL5011
Order statistics, their distributions, and associated statistical properties; Introduction to nonparametric tests; Comparison of parametric and nonparametric tests; Tests for randomness: run test; Goodness of fit tests: chi-square test, one-sample Kolmogorov-Smirnov test; One-sample and paired-sample tests: sign test, Wilcoxon signed-rank test; Rank-order statistics; Two-sample tests: Wald-Wolfowitz run test, Kolmogorov-Smirnov two-sample test, median test, Mann-Whitney 'U' test; Linear rank tests for location and scale problems, k-sample problems; Measures of association; Asymptotic relative efficiency; Bootstrap and Jackknife resampling methods; Nonparametric regression.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to gain a thorough understanding of the theory and applications of order statistics.
  2. On successful completion of this course, a student should be able to draw valid statistical inferences in distribution-free scenarios.
  3. On successful completion of this course, a student should be able to explore real-world examples and understand the advantages of nonparametric methods over parametric techniques by utilizing modern statistical tools.
  4. On successful completion of this course, a student should be able to understand the concepts of resampling methods and nonparametric regression techniques.

MTL6777: Foundations of Statistical Learning

3.00 credits (2.0-0.0-2.0)
Preclusions: CML7211, COL7341, CRL7005
Pre-requisites: MTL5011
Summary measures (mean, variance) and statistical inference (confidence intervals, hypothesis testing). A review of limit theorems (central limit theorem, law of large numbers) and exploratory data analysis, linear regression and least square estimation, generalized linear models and logistic regression, classification problem, Fisher’s discriminant analysis and nearest neighbor methods. Model assessment and cross-validation. Data dimension reduction, principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE), model selection, ridge and LASSO regressions. Data-depth, multivariate ordering of data and classification based on data-depth.

On successful completion of the course, a student will be able to:
  1. Apply statistical inference, hypothesis testing, and confidence intervals to analyze real-life data for informed decision-making in various domains.
  2. Understand and implement regression models, classification techniques, and model assessment strategies to address real-life challenges in healthcare, finance, and technology.
  3. Understand and implement dimensionality reduction methods like PCA and t-SNE to extract meaningful insights from large-scale real-life data, improving efficiency and interpretation.
  4. Learn to implement Ridge and LASSO regression, data-depth techniques, and cross-validation to ensure reliable, scalable, and interpretable machine learning solutions for real-life applications.

MTL7012: Computational Methods for Differential Equations

4.00 credits (3.0-0.0-2.0)
Preclusions: CHL2001, AML3000, MEL7403
Pre-requisites: (MTL2007)
Numerical methods for solving IVPs for ODEs: Single step method: Taylor series method, convergence, order. Multistep methods: Development of linear multistep method using interpolation and undetermined parameter approach, consistency, zero stability, convergence, order. Absolute stability for linear multistep methods: motivation, Routh-Hurwitz criterion, boundary locus method etc., absolute stability for systems, stiff systems. Linear multistep methods for nonlinear system: Predictor Corrector methods. Numerical methods for solving BVPs for ODEs: Finite difference methods for second order ODEs. Numerical methods for solving PDEs: Finite difference methods for PDEs on real line, Consistency, stability (von neumann analysis, energy method) and convergence. Finite difference methods for PDE on interval: incorporation of Boundary Conditions, Consistency, stability (matrix method, energy method) and convergence

On successful completion of the course, a student will be able to:
  1. Development of numerical method
  2. Demonstrate how to solve physical phenomenon via solving differential equations numerically, and analyze numerical methods properties like convergence, absolsute stability etc..
  3. Enable to apply these in-depth knowledge to various problems across various disciplines.

MTL7030: Fundamentals of Cryptography

4.00 credits (3.0-0.0-2.0)
Preclusions: SIL7159, ELL7408, COL7159
Pre-requisites: MTL2005/MTL5001
Classical cryptosystems (substitution ciphers, Vigene’re ciphers, matrix ciphers, transposition ciphers, ADFGX cipher, ADFGVX cipher, permutation cipher, playfair cipher and Enigma machine cipher) and their cryptanalysis. Preview from number theory, Congruences and residue class rings, quadratic residue and Legendre symbol. Public Key Cryptosystems of RSA, Rabin, etc. their security and cryptanalysis. Primality tests (Fermat PT, Miller-Rabin PT, AKS PT), factorization and quadratic sieve, efficiency of other factoring algorithms (Fermat’s, Pollard’s (p-1), Monte Carlo factorization, Factor base, quadratic sieve and continued fraction). Finite fields: Construction and examples, polynomials on finite fields and their factorization/ irreducibility. Key-Exchange protocols Diffie-Hellman key exchange, Massey Omura cryptosystems and ElGamal cryptosystem. Discrete logarithm problems in general and on finite fields. Algorithms for finding discrete logarithms (Shank Baby step Giant step algorithm, Pollard’s ρ algorithm, index calculus algorithm in Z_p and Pohlig-Hellman algorithm) and use them for cryptanalysis. Elliptic curves, public key cryptosystems particularly on elliptic curves. Problems of key exchange, discrete logarithms and the elliptic curve logarithm problem. Implementation of elliptic curve cryptosystems. Counting of points on Elliptic Curves over Galois Fields of order 2^m. Cryptographic hash functions. Authentication, Digital Signatures, Identification, certification infrastructure and other applied aspects. DES- security and generalizations.

On successful completion of the course, a student will be able to:
  1. Understanding the theory of encryption and decryption for secured communication
  2. Analyzing various crypto systems
  3. Designing cryptographic techniques and systems

MTL7146: Methods of Applied Mathematics

3.00 credits (3.0-0.0-0.0)
Preclusions: MEL7403, AML8410, PYL1012
Sturm-Liouville eigenvalue problems, Expansion in Eigen functions, Fourier series and Fourier Integral, orthogonal expansion, mean square approximation, completeness, orthogonal polynomials and their properties. Integral transform and their applications. General variation of a functional, direct variation, variational problems. Integral equations of voltera and Fredhlom type, seperable and symmetric kernels, Hilbert-Schmidth theory, Singular integral equations, approximation methods of solving integral equations.

On successful completion of the course, a student will be able to:
  1. Analyze and communicate mathematical ideas with logic, rigor and draw logical conclusion
  2. Fourier series, Generalized Fourier series, Different integral transform and their appplications in PDEs, variational calculus and their applications, Theory and applications of integral equations
  3. Enable to apply these in-depth knowledge to various problems across various disciplines.

MTL7237: Introduction to Differential Geometry

3.00 credits (3.0-0.0-0.0)
Curves in the plane and space, arc-length, reparametrization, curvature of a plane curve, curvature and torsion of a space curve, simple closed curves, isoperimetric inequality, Four-Vertex Theorem, surfaces, smooth surfaces and examples, level surfaces, quadric surfaces, surfaces of revolution, ruled surfaces, smooth maps, tangent space, derivatives, orientability of surfaces, first fundamental form, lengths of curves on surfaces, isometries, conformal mappings, equiareal maps, second fundamental form, Gauss and Weingarten maps, normal and geodesic curvatures, Gaussian and mean curvatures, principal curvatures, surfaces of constant Gaussian curvature, surfaces of constant mean curvature, flat surfaces, parallel transport, geodesics and their examples, geodesic equations, geodesics as shortest paths, Gauss and Codazzi-Mainardi equations, Theorema Egregium, Gauss–Bonnet Theorem, introduction to hyperbolic and spherical geometry.

On successful completion of the course, a student will be able to:
  1. Analyze and compute geometric invariants (curvature, torsion, etc.) of curves in Euclidean space using Frenet–Serret formulas.
  2. Identify and describe properties of surfaces in 3D space, including tangent planes and normal vectors using parametrizations.
  3. Compute and interpret the first and second fundamental forms, Gaussian and mean curvature of surfaces.
  4. Investigate intrinsic properties of surfaces such as geodesics and isometries, and understand their implications through theorems like Theorema Egregium and Gauss–Bonnet.

MTL7239: Representation Theory of Finite Groups

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2005/MTL5001
Revision of basic group theory. Definition and examples of representation. Sub-representation, sum and tensor product of representations, irreducible representations; Character Theory: Character of a representation, Schur’s Lemma, Maschke’s theorem, Orthogonality relations for characters, decomposition of regular representation, number of irreducible representations of a group; Representation of subgroups and product of groups, induced representations; Group Algebra: Representations and C[G]-modules; Decomposition of complex algebra C[G], Integrability properties of characters, Burnside’s pq theorem. Induced representations, restriction to subgroups, Reciprocity formula, Mackey’s irreducibility criterion; Irreducible representations of symmertric groups (S_n) and alternating groups (A_n).

On successful completion of the course, a student will be able to:
  1. Understanding the foundational concepts of group representations
  2. Theory of characters
  3. classification of irreducible representations up to isomorphism
  4. Analyzing representations of specific finite groups

MTL7244: Mathematical Theory for Error Correcting Codes

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2005/MTL5001
The notion of error detection and correction, communication channels, maximum likelihood and nearest neighborhood decoding principles, Hamming distance, codes and their parameters, ISBN Code. Structure of finite fields, Linear codes and their duals, generator and parity-check matrices of linear codes, equivalence of linear codes, encoding and decoding of a linear code, coset and syndrome decoding using coset leader. Hamming codes. Hadamard matrix codes, Golay codes, Codes and Latin squares. Non-linear codes, Nordstrom-Robinson code, Kerdock codes, Preparata codes. Bounds in coding theory: Sphere covering bound, Hamming bound and perfect codes, Singleton bound and MDS codes, Gilbert-Varshamov bound, Plotkin bound, Griesmer bound. Weight enumerators, MacWilliams Identity. Construction of new codes: Propagation rules, Reed-Muller codes, subfield sub-codes, trace codes. The structure of cyclic codes, encoding and decoding with a cyclic code, minimal codes, burst errors, burst error-correcting codes, decoding of cyclic burst-error-correcting codes. Some special cyclic codes including BCH codes and their decoding algorithm, Reed-Solomon codes, quadratic residue codes. Generalized Reed-Solomon codes, Alternant codes, Goppa codes, Sudan decoding for generalized RS codes

On successful completion of the course, a student will be able to:
  1. understanding how mathematical theory works in error detection and correction in digital communication and data storing
  2. characterization of useful codes, classical examples of such codes and construction of new codes
  3. encoding and decoding methods for various classical codes

MTL7246: Combinatorics

3.00 credits (3.0-0.0-0.0)
Basic counting techniques; The Pigeonhole Principle; the Principle of Inclusion–Exclusion; recurrence relations and generating functions; Catalan Numbers, and its; Difference Sequences and Stirling Numbers; Lattice Paths and Schroder Numbers; Systems of Distinct Representatives; Stable Marriages; Hall’s Theorem; Burnside's lemma and Pólya enumeration theorem; Latin squares; combinatorial designs and Steiner Triple Systems; Ramsey theory; Mobius Inversion; Extremal set theory.

On successful completion of the course, a student will be able to:
  1. Apply basic counting techniques, the Pigeonhole Principle, and the Principle of Inclusion–Exclusion, along with recurrence relations and generating functions, to solve combinatorial problems
  2. Analyze and compute key combinatorial structures including Catalan numbers, difference sequences, Stirling numbers, lattice paths, and Schröder numbers, along with Burnside’s Lemma and the Pólya Enumeration Theorem. Apply these concepts to count graphs and related combinatorial configurations under symmetry.
  3. Formulate and solve matching and design problems using Systems of Distinct Representatives, Hall’s Theorem, stable marriage algorithms, Latin squares, and combinatorial designs including Steiner Triple Systems.
  4. Apply fundamental results from Ramsey theory to understand unavoidable patterns and structures in large or complex systems.
  5. Utilize the Möbius inversion method and techniques from extremal set theory to derive bounds and solve advanced problems in discrete mathematics.

MTL7255: Algebraic Curves

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2005&MTL3107
Plane curves: historical context and classical examples; Affine space and projective space, Zariski topology, coordinate rings, projective space, homogenization; Morphisms between varieties, rational maps, birational equivalence; Function fields, birational isomorphism; Derivations, Jacobians, smooth points, singular points; Local rings, completions, formal local parameters; Multiplicities, intersections of plane curves, Bézout's theorem; Divisors, principal divisors, linear equivalence, degree; Linear systems, dimension of divisor spaces; Statement of Riemann-Roch. Canonical divisors, genus, Elliptic curve, applications of Riemann-Roch.

On successful completion of the course, a student will be able to:
  1. Understand the Basic Concepts of Algebraic Curves: Students will be able to define and work with affine and projective algebraic curves, understand the role of polynomial rings and coordinate rings, and interpret algebraic curves as zero loci of polynomial equations.
  2. Develop Geometric Intuition through Algebraic Tools: Students will gain proficiency in using algebraic tools—such as elimination theory, and local rings—to study geometric properties like intersection multiplicity, tangents, and singularities of curves.
  3. Apply the Theory of Function Fields and Divisors: Students will understand the function field of an algebraic curve, the concept of divisors, and their relation to rational maps, leading to foundational knowledge of the Riemann-Roch theorem in a concrete setting.
  4. Bridge Classical and Modern Perspectives: Students will appreciate classical algebraic geometry through examples and problems, and will be able to transition to more abstract modern concepts such as schemes and sheaves, thereby preparing for further studies in algebraic geometry.

MTL7257: Introduction to Algebraic Topology

3.00 credits (3.0-0.0-0.0)
Paths and homotopy; the fundamental group of the circle; induced homomorphisms; free products of groups; the van Kampen theorem and its applications; applications to cell complexes; lifting properties; the classification of covering spaces; deck transformations and group actions. Delta complexes; simplicial homology; singular homology; homotopy invariance; exact sequences and excision; the equivalence of simplicial and singular homology; degree; cellular homology; Mayer–Vietoris sequence and its applications.

On successful completion of the course, a student will be able to:
  1. Students will understand paths and homotopy, compute the fundamental group of the circle, and analyze induced homomorphisms. They will study free products of groups and apply the van Kampen theorem to compute fundamental groups of complex spaces, enhancing their ability to connect algebraic structures with topological properties through rigorous proofs.
  2. Students will apply algebraic topology to cell complexes, understand lifting properties of maps, and classify covering spaces using the fundamental group. They will analyze deck transformations and group actions, gaining tools to study symmetries and structure in topological spaces through covering space theory and its connection to algebraic invariants.
  3. Students will understand delta complexes and compute simplicial and singular homology groups. They will explore the relationship between these theories and apply them to analyze topological spaces. Emphasis will be placed on homotopy invariance, enabling students to use homology as a tool to distinguish and classify spaces up to homotopy equivalence.
  4. Students will understand exact sequences, excision, and the equivalence of simplicial and singular homology. They will compute the degree of maps, apply cellular homology to cell complexes, and use the Mayer-Vietoris sequence to analyze complex spaces, developing strong skills in homological methods and their topological applications.

MTL7258: Introduction to Differential Topology

3.00 credits (3.0-0.0-0.0)
Smooth manifolds and smooth maps, tangent spaces and derivatives, manifolds with boundary, proof of Sard's theorem, immersions and submersions, one-manifolds and some consequences, transversality, intersection theory mod 2, winding numbers, the Jordan-Brouwer separation theorem, the Borsuk-Ulam theorem and its applications, orientation, oriented intersection number, Lefschetz fixed-point theory, vector fields, the Poincare-Hopf theorem, the Hopf degree theorem, the Euler characteristic and triangulations, integration on manifolds, differential forms.

On successful completion of the course, a student will be able to:
  1. By the end of the course, students will understand the foundational concepts of smooth manifolds and smooth maps, compute tangent spaces and derivatives, and analyze manifolds with boundary. They will grasp the statement and proof of Sard’s Theorem and apply this understanding to study immersions, submersions, and their geometric implications.
  2. By the end of the course, students will understand the classification of one-manifolds and key consequences, apply transversality and intersection theory mod 2, and compute winding numbers. They will also grasp and utilize the Jordan-Brouwer Separation Theorem and the Borsuk-Ulam Theorem, exploring their topological significance and diverse applications.
  3. By the end of the course, students will understand the concept of orientation on manifolds and compute oriented intersection numbers. They will explore Lefschetz fixed-point theory, analyze vector fields on manifolds, and apply the Poincaré-Hopf Theorem to relate vector field behavior to topological invariants, deepening their understanding of global manifold structure.
  4. By the end of the course, students will understand the Hopf Degree Theorem and its implications, compute the Euler characteristic via triangulations, and perform integration on manifolds. They will gain proficiency in using differential forms to study global properties of manifolds, linking topology, geometry, and analysis in a unified framework.

MTL7281: Finite Element Theory and Applications

3.00 credits (3.0-0.0-0.0)
Preclusions: MEL7206, CVL8451, AML7070, AML3060, CVL8451, ESL3873
Pre-requisites: (MTL5007&MTL5009) /(MTL2007&MTL3111)
Sobolev Spaces, Variational formulation of elliptic boundary value problems; Lax Milgram lemma; existence and uniqueness of weak solutions; equivalence of Galerkin and Ritz variational formulations; triangulation of domains; finite element approximation and discrete formulation; conforming and non-conforming finite element methods, Cea’s lemma, interpolation on simplexes in Rn, different Lagrange and Hermite finite elements, affine, isoparametric finite elements; triangulation using isoparametric mapping; numerical integration; construction of element stiffness matrices and assembly into global stiffness matrix; skyline method of solution of finite element equations; solution of model problems and computer implementation procedures; asymptotic error estimate results; eigenvalue problems of Laplace operator.

On successful completion of the course, a student will be able to:
  1. Develop understanding of fundamental concepts of the theory of finite element methods.
  2. Develop ability to implement finite element algorithms, focusing on aspects such as mesh generation, boundary conditions, and post-processing of results.

MTL7455: Partial Differential Equations

3.00 credits (3.0-0.0-0.0)
First order PDEs: classification and examples of PDEs, method of characteristic, complete and general solutions. Second order PDEs in two independent variables: classification, examples and canonical form. Cauchy-Kowalewsky theorem for quasilinear equations, Holmgren’s uniqueness theorem for linear equations. Theory of distribution: test functions and distributions, some operations with distributions, convolution of distributions, fundamental solutions, Schwartz space, tempered distributions. Weak/generalized solution of PDE in divergence form: scalar conservation laws, Rankine-Hugoniot condition, entropy condition, Riemann's problem. Laplace equation: fundamental solution , Poisson's equation, harmonic functions and their properties, maximum principle, Green's function, Dirichlet's principle and method of separation of variables. Heat equation: fundamental solution, mean value theorem and maximum principle, uniqueness results and method of separation of variables. Wave equation: D' Alembert's formula, weak solution, method of separation of variables, method of spherical means (n=2,3), Duhamel`s principle and energy method for uniqueness, the domain of dependence, Range of influence.

On successful completion of the course, a student will be able to:
  1. Demonstrate the ability to analyze mathematical ideas rigorously, communicate them logically, and draw valid conclusions
  2. Develop mathematical models of physical phenomena via partial differential equations, and analyze solution concepts, well-posedness, and qualitative behavior of solutions for some fundamental PDEs using a variety of methods
  3. Enable to apply these in-depth knowledge to various problems across various disciplines.

MTL7717: Fuzzy Sets and Applications

4.00 credits (3.0-0.0-2.0)
Fuzzy sets, fuzzy relations, matrix representation of fuzzy relations, fuzzy numbers, fuzzy arithmetic, Zadeh’s extension principle, ordering fuzzy numbers, fuzzy aggregation, t-norm, t-conorm, fuzzy negation, OWA operators, fuzzy relational equations (FRE), algorithms to solve systems of FRE, fuzzy logic and approximate reasoning, fuzzy inference, Mamdani model, Larsen model, TSK model, SAM model, fuzzy measures, belief and plausibility, necessity and possibility, applications of fuzzy sets and logics in areas of clustering, pattern recognition, image processing, controller design, optimization, neuro-fuzzy, AI, and similar domains in project component.

On successful completion of the course, a student will be able to:
  1. Developing an understanding of mathematical modeling or ordinal and qualitative variables through fuzzy sets; to get familiar with fuzzy representation and operations on fuzzy sets.
  2. To get familiar with fuzzy operators, fuzzy numbers, and underline solution structure of fuzzy relational equations and algorithms to solve them
  3. To get to know basic fuzzy propositional logic, fuzzy if then rules, and fuzzy logic modeling via foundational logic models.
  4. Implementation and group project on state of the art applications of fuzzy inference and fuzzy sets in engineering, health science, decision-making, and applications to real-situations; to encourage group tasks based projects.

MTL7725: Stochastic Processes and its Applications

3.00 credits (3.0-0.0-0.0)
Pre-requisites: (MTL5005) /(MTL2006) /(MTL2008)
Stochastic processes and their classifications, stationary, memoryless and martingales properties, simple stochastic processes, discrete and continuous time Markov chains, Poisson process, birth-death process, simple queueing models, solution techniques, point process, renewal process, Monte-Carlo simulation, random number generation, simulation of stochastic processes, modeling on system reliability, block diagram, fault tree and Markov methods, simulation methods.

On successful completion of the course, a student will be able to:
  1. Understanding stochastic process concepts
  2. Developing analytical skills to model using stochastic process
  3. Model real-world systems using appropriate stochastic process
  4. Solve practical problems using stochastic modeling

MTL7732: Financial Mathematics

4.00 credits (3.0-0.0-2.0)
Preclusions: MTL7725
Financial markets, Interest computation, growth and discount factors, present and future values, no arbitrage, forwards and futures, forward rates agreements, derivative products, basic option theory: single- and multi-period binomial pricing models, Cox-Ross-Rubinstein (CRR) model, volatility, Black-Scholes formula for option pricing as a limit of CRR model, Black-Scholes PDE, Greeks and hedging, delta and omega hedging, mean-Variance portfolio theory: Markowitz model, Capital Asset Pricing Model (CAPM), factor models, Fama and French higher factor model for asset pricing, zero coupon and coupon bonds, forward and short interest rates, bond yield, Macaulay duration of bond, interest rate option derivatives, Binomial tree models for interest rates options, Black-Derman Toy model, Cox-Ingersoll-Ross model

On successful completion of the course, a student will be able to:
  1. To learn the basic framework and techniques necessary for financial engineering.
  2. To learn valuation, with discrete case, of financial assets and more specifically the derivative products
  3. Introduce the concept of risk and relation between risk and return for optimal investment planning
  4. To learn and understand interest rates models, bonds valuations, interest rate options.

MTL7733: Stochastic of Finance

3.00 credits (3.0-0.0-0.0)
Pre-requisites: (MTL2006) /(MTL2008) /(MTL5005)
Basic stochastic process involved in the study of mathematics of finance, martingales, Brownian motion, geometric Brownian motion, Itô calculus, Itô formula, stochastic differential equations, strong solution, weak solution, derivative security pricing, change of measure, martingale representation theorem, Girsanov’s theorem, option pricing, Feymann-Kac theorem, Black-Scholes model, forward contracts and bond pricing, Vasicek model, CIR model, Hull-White model, Monte-Carlo simulation, random number generation, simulation of Black-Scholes model, simulation of stochastic differential equation, simulation models for option pricing.

On successful completion of the course, a student will be able to:
  1. Understanding finance and stochastic processes concepts
  2. Developing analytical skills to model finance problems using stochastic process
  3. Model finance problems using appropriate stochastic calculus
  4. Solve practical problems in finance using stochastic modeling

MTL7735: Introduction to Number Theory

4.00 credits (3.0-1.0-0.0)
Preclusions: MTL3145
Divisibility, Prime numbers, Infinitude of primes, Fundamental theorem of arithmetic, Euclidean algorithm, Euler totient function, Möbius function, Divisor function, von Mangoldt function, Chebyshev's functions, Big-O and Little-o notions, Möbius inversion formula, Abel's partial summation formula and applications, Estimates of prime numbers (statements only), Complete and reduced residue systems, Euler-Fermat theorem, Wilson's theorem, Linear congruences and Chinese remainder theorem, Hensel's lemma and applications, Quadratic residues and non-residues, Legendre symbol, Quadratic reciprocity law and applications, Jacobi symbol, Representation of integers as sum of squares, Existence of primitive roots, Index computation and its applications in solving polynomial congruences, Dirichlet's theorem, Algebraic and irrational numbers, Liouville's theorem and applications, Roth's theorem (statement only), Continued fraction expansions, Properties of convergents and partial quotients, Improvements of Dirichlet's theorem, Characterisation of quadratic irrationals, Brahmagupta-Pell equation, Pythagorean triples, Fermat equations, Tangent-chord method, p-adic norm, p-adic numbers, Hensel's lemma revisited, Product formula, Ostrowski's theorem, Binary quadratic forms, Reduction and equivalence of binary quadratic forms, Class number problem (statements only), General quadratic forms, Statement of Hasse-Minkowski theorem, Partition function, Statements of Ramanujan congruences, Roger-Ramanujan continued fractions, Primality and factorisation.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to understand the growth of various arithmetic functions and their implications in the theory of distribution of prime numbers.
  2. On successful completion of this course, a student should be able to solve various types of congruences, like linear congruences, quadratic congruences, apply quadratic reciprocity law, work with primitive roots for index computations and solving higher order congruences.
  3. On successful completion of this course, a student should be able to understand how closely one can approximate real numbers by rational numbers, and understand problems related to arithmetic nature of real numbers, moreover they should be able to solve some Diophantine equations using these techniques.
  4. On successful completion of this course, a student should be able to apply tools developed in this course to factorize and comment about primality of reasonably big numbers, and they would also get exposed to some of Ramanujan's fundamental works.

MTL7736: Analytic Number Theory

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL3122/MTL6001
Fundamental Theorem of Arithmetic, Arithmetic functions, Order and average order of arithmetic functions, Partial summation formula, Dirichlet’s hyperbola method, Dirichlet convolution, Partial sums of Dirichlet convolution, Statement of Prime Number Theorem and its equivalent formulations, Chebyshev’s Theorem, Bertrand’s postulate. Dirichlet series, Half plane of convergence and absolute convergence, Euler product, Riemann zeta function and its analytic continuation via Riemann’s functional equation, Non-vanishing of Riemann zeta function on the line Re(s)=1, Proof of Prime Number Theorem, Dirichlet L-function and its analytic continuation, Non-vanishing at s=1, Proof of Dirichlet’s theorem for primes in arithmetic progression. Modular group and its action on the upper half plane, Modular forms, Eisenstein series, Cusp forms, Valence formula, j-function, Hecke operators and their eigenvalues.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to understand the growth of various arithmetic functions and their implications in the theory of distribution of prime numbers.
  2. On successful completion of this course, a student should be able to understand the theory of Dirichlet series, its key importance and and apply it to study various number theoretic problems using analytic tools
  3. On successful completion of this course, a student should be able to understand the importance of modular forms and it's rich structural properties to aim to understand deeper number theoretic questions connecting elliptic curves, Diophantine equations etc.

MTL7737: Algebraic Number Theory

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2005/MTL5001
Algebraic numbers, transcendental numbers, minimal polynomial, conjugates. Algebraic number fields, primitive element, real and complex embeddings, norm, trace and discriminant. Algebraic integers, ring of integers in a number field, integral basis. Examples of quadratic fields, cubic fields, cyclotomic fields. Dedekind domain, ideal factorization, fractional ideal, ideal class group. Lattices, Minkowski's theory, finiteness of class group, computation of class group of number fields. Dirichlet Unit Theorem, fundamental units, units in quadratic fields, solution of Pell's equation. Extensions of Dedekind domains, Galois extensions, decomposition and inertia groups, ramification Dedekind zeta function and analytic class number formula (proof of the quadratic class number formula).

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to understand the concepts of algebraic number, algebraic integer, Dedekind domains, in elementary cases calculate the norm and trace of an algebraic number.
  2. On successful completion of this course, a student should be able to understand the concepts of fractional ideal, ideal class, ideal class group, the geometry of numbers, Minkowski's theory, in elementary cases apply Minkowski's bound for computation of the ideal class groups
  3. On successful completion of this course, a student should be able to understand the concepts of units in a number field, Dirichlet unit theorem and learn its applications, also understand the concepts of extensions of Dedekind domains, ramification, Dedekind zeta function and analytic class number formula.

MTL7738: Commutative Algebra

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2005/MTL5002/MTL2004/MTL5001/
Revision of rings and ideals: prime and maximal ideals, chinese remainder theorem, nilradical, Jacobson radical, operations on ideals, extension and contraction. Basics of module: module, submodule, quotient module, sums and products of modules, Nakayama’s lemma, module homomorphism, kernel, cokernel, direct sum, direct product, universal properties, free module, exact sequences, projective and injective modules, tensor product and flat modules. Rings and modules of fractions. Localisation of rings and modules and the functorial properties of localisation. Primary decomposition: primary ideals, primary decomposition, uniqueness theorems and associated primes. Integral dependence: integral dependence and independence, Noether normalisation lemma, going-up and going down theorems, Nullstellensatz (weak form). Chain conditions: Noetherian rings, Artinian rings, discrete valuation rings and Dedekind domains. Krull dimension, and completion.

On successful completion of the course, a student will be able to:
  1. On completion of this section students will develop a solid understanding of ring and module theory, including prime and maximal ideals, the Chinese Remainder Theorem, nilradical, Jacobson radical, and ideal operations. They will explore modules, submodules, quotient modules, module homomorphisms, and exact sequences. Key concepts such as Nakayama’s Lemma, projective, injective, free, and flat modules, tensor products, and universal properties will be introduced and applied.
  2. Students will understand the construction and properties of rings and modules of fractions, localization techniques, and their functorial behavior. They will explore primary ideals, primary decomposition, associated primes, and related uniqueness theorems, gaining the ability to analyze algebraic structures through localization.
  3. Students will learn concepts of integral dependence and independence, the Noether Normalization Lemma, and understand the Going-Up and Going-Down theorems and the weak form of Hilbert’s Nullstellensatz. They will also study chain conditions and properties of Noetherian and Artinian rings.
  4. Students will understand the structure and properties of discrete valuation rings and Dedekind domains, including fractional ideals. They will study the Krull dimension of rings and completion.

MTL7741: Fractal Geometry

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2022/MTL5003
Basic measure theory, Hausdorff measure, Hausdorff dimension, Examples, Packing measure and dimensions, Box-counting dimensions, Modified box-counting dimensions, Relationship between various dimensions, Mass distribution principle, Iterated function systems, self-similar sets, Hausdorff metric, Continued fractions, Dimensions of graphs of functions, Fractal interpolation functions, Julia sets and Mandelbrot sets.

On successful completion of the course, a student will be able to:
  1. Explain the fundamental concepts of measure theory, Hausdorff and packing measures, and the major notions of fractal dimension used in fractal geometry.
  2. Compute and compare Hausdorff, packing, box-counting, and modified box-counting dimensions for classical fractal sets, self-similar sets, and number-theoretic fractals.
  3. Apply dimension estimation techniques, including the mass distribution principle and similarity methods, to determine the geometric properties of fractals.
  4. Construct and analyze self-similar sets and fractal interpolation functions using iterated function systems, the Hausdorff metric, and related fixed point methods.
  5. Investigate and interpret fractals arising from dynamical systems, such as Julia sets and the Mandelbrot set, and describe their structural and dimensional features.

MTL7743: Fourier Analysis

3.00 credits (3.0-0.0-0.0)
Fourier series - definition, uniqueness, convolution, summability, convergence of Fourier series, Fourier series for square integrable functions, Plancheral theorem, Riesz-Fischer theorem, Dini’s condition, divergence of Fourier series. Fourier transform – Schwartz space on R^n, Fourier transform on the Schwartz space, Fourier transform of L1 functions, Schwartz class, Fourier transform for L2 functions, Plancherel theorem. Tempered distributions- topology on the Schwartz space, tempered distributions and its properties, Fourier transform of tempered distributions. Hardy-Littlewood maximal function- approximations of the identity, weak-type in-equality, Marcinkiewicz interpolation theorem, Hardy-Littlewood maximal function and its properties, Calderon-Zygmund theorem, Riesz-Thorin interpolation theorem. The Hilbert transform: conjugate Poisson kernel, principal value distribution, Riesz and Kolmogorov theorems, multipliers.

On successful completion of the course, a student will be able to:
  1. Analyze Fourier series and their convergence Understand the definition and uniqueness of Fourier series. Apply convolution techniques in the context of Fourier series. Evaluate different types of convergence and summability of Fourier series. Extend Fourier series concepts to square integrable functions. Apply the Plancherel Theorem and Riesz-Fischer Theorem. Understand Dini’s Condition and its implications for convergence. Recognize cases of divergence in Fourier series.
  2. Understand the theory and applications of the Fourier transform: Define the Schwartz space on R^n imes R^n and understand its structure. Apply the Fourier transform on Schwartz space functions and L^1 functions. Extend the Fourier transform to L^ 2 functions and proving the Plancherel Theorem.
  3. Study tempered distributions and their Fourier transforms : Understand the topology of the Schwartz space and its significance in distribution theory. Define and analyze tempered distributions and their properties. Apply the Fourier transform to tempered distributions and understand its consequences.
  4. Apply maximal function techniques in harmonic analysis: Understand approximations of the identity and weak-type inequalities. Apply the Hardy-Littlewood maximal function and analyze its properties. Explore the dyadic maximal function and related maximal inequalities. Use the Marcinkiewicz and Riesz-Thorin interpolation theorems. Understand and apply the Calderón-Zygmund theorem in analysis.
  5. Analyze the Hilbert transform and its properties Understand the conjugate Poisson kernel and principal value distributions. Apply the Hilbert transform and study its implications in harmonic analysis: Explore key results like the Riesz and Kolmogorov theorems. Identify and apply multiplier theorems related to the Hilbert transform.

MTL7744: Applied Harmonic Analysis

3.00 credits (3.0-0.0-0.0)
Basics of Fourier series and Fourier transform, Poisson summation formula, Shannon-sampling theorem, Heisenberg uncertainty principle. Short-time Fourier transform(STFT), orthogonality relations and inversion formula of STFT, Lieb's uncertainty principle, Bargmann transform. Bessel sequences and Riesz bases in Hilbert spaces, frames in a Hilbert space, frame operator, characterizations of frames, frame series and its convergence. Frames of translates and its characterization, Gabor frames, existence of Gabor frames, Zak transform, Gabor systems at the critical density, Balian-low theorem, continuous wavelet transfrom and inversion, wavelet frames.

On successful completion of the course, a student will be able to:
  1. Enable students to understand signal representation/decomposition using small atoms and analyze signals/images in transformed domains.

MTL7750: Abstract Harmonic Analysis

3.00 credits (3.0-0.0-0.0)
Fourier transform of L1 functions on Rn, Fourier- Stieltjes transform, Plancherel theorem, convolution on L1 on Rn, Fourier series of L1 functions on Tn, Fourier- Stieltjes series, Plancherel theorem, convolution on L1 on Tn. Banach algebras, spectrum of an element and its properties, spectral radius, ideals in a commutative Banach algebra, spectrum of a commutative Banach algebra, Gelfand transform, Gelfand-Naimark theorem. Topological groups and its properties, Haar measure on a locally compact group, existence and uniqueness, modulator function, convolution, homogeneous spaces, existence of an invariant measure, weil’s formula. Unitary representation of locally compact groups, Schur’s lemma, relation between unitary representation of groups and the *-representations of the group algebra, positive definite functions, Gelfand-Raikov theorem. Harmonic analysis of LCA groups: dual group of locally compact abelian group, dual group as a locally compact abelian group, Pontriagin duality theorem, spectrum of the L1 space with respect to the convolution, Fourier analysis on locally compact abelian groups, Plancherel theorem, Bochner’s theorem. Harmonic analysis on compact groups: finite dimensionality of the irreducible unitary representations of a compact group, Schur’s orthogonality relations, Peter-Weyl thereom, Fourier analysis on compact groups, Plancherel theorem, representations of SU(2) and SO(3).

On successful completion of the course, a student will be able to:
  1. Representation of groups and its properties are one of the important properties of abstract harmonic analysis
  2. After representation theory, it natural to expect what are known as Fourier analysis
  3. Before all these, some kind of abstract things are needed in order to understand.

MTL7761: Basic Ergodic Theory

3.00 credits (3.0-0.0-0.0)
Pre-requisites: (MTL2022)(MTL3107)(MTL6507)
Measure spaces, Haar measure, Poincare recurrence theorem, Hopf's maximal ergodic theorem, Birkhoff's ergodic theorem, von Neumann's ergodic theorem, isomorphism, conjugacy and spectral isomorphism, entropy, topological pressure and its relationship with invariant measures.

On successful completion of the course, a student will be able to:
  1. On completion of this course a student will learn about basics of measurable spaces..
  2. On successful completion of this course a student will learn about definition and properties of ergodic systems
  3. On successful completion of this course a student will learn about spectral properties and invariant measures.

MTL7763: Introduction to Game Theory

3.00 credits (3.0-0.0-0.0)
Preclusions: COL7155, HSL6011
Optimal strategies and value of two player zero-sum games, existence of saddle point equilibrium using Brouwer’s fixed point theorem, graphical method and Linear programming formulations for two player zero-sum game, Existence of Nash equilibrium for n-player games using Kakutani fixed point theorem, Pareto efficiency, symmetric games, Algorithm to compute Nash equilibrium: i) Equilibria by support enumeration, ii) Equilibria by labelled polytopes, iii) Equilibria by vertex enumeration, iv) Lemke-Howson algorithm. Bayesian Games, Strategic games with continuous strategy sets, Applications in market competition, Auctions, Electricity Market. Sequential games, sub-game perfect Nash equilibrium. Inefficiency of Equilibria: i) Price of Anarchy, ii) Price of Stability. Selfish routing games, Network Design Games. Network formation games: i) pairwise stability ii) Nash stability iii) strong stability, iv) efficiency. Some famous examples: Co-author example, connection models.

On successful completion of the course, a student will be able to:
  1. Understanding the rational decision-making, game formulation, and various equilibrium concept like Nash equilibrium, Subgame perfect Nash equilibrium
  2. Understanding the various methods and algorithms to compute Nash equilibrium and subgame perfect Nash equilibrium of game problem
  3. Application of game theory in Market competition and auction theory
  4. Understanding the various types of Network games and underlying equilibrium and stability notions

MTL7766: Multivariate Statistical Analysis

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2006/MTL2008/MTL5011/AIL7001/HSL5013
Multivariate distributions, sample geometry, expectations of sample mean and covariance, multivariate normal distribution, sampling and maximum likelihood estimation, sampling distributions and asymptotics, Wishart and Hotelling distributions, likelihood ratio tests, confidence regions, hypothesis testing for mean vectors, comparisons across populations, simultaneous confidence intervals, two-way MANOVA, multiple and multivariate regression, least squares estimation, inference on models and parameters, linear discriminant analysis, classification methods, principal component analysis, graphical representation of components, factor analysis including orthogonal models, estimation, rotation, and scores, and canonical correlation analysis for exploring relationships between multivariate variable sets.

On successful completion of the course, a student will be able to:
  1. Understand and apply multivariate probability distributions to analyze relationships among multiple variables and derive expected values, sampling distributions, and estimators.
  2. Perform inference on multivariate data using techniques such as hypothesis testing, confidence regions, MANOVA, and likelihood ratio tests for mean vector comparisons.
  3. Develop and assess multivariate models including regression, discriminant analysis, and classification for prediction and decision-making.
  4. Apply dimension reduction and correlation techniques such as PCA, factor analysis, and canonical correlation analysis to simplify, visualize, and interpret high-dimensional datasets.

MTL7768: Graph theory

3.00 credits (3.0-0.0-0.0)
Preclusions: MTL3176
Introduction to graphs; Trees and distance: properties, spanning trees and enumeration etc.; Matchings and covers; Connectivity: cuts, vertex/edge connectivity, Menger's theorem, etc.; Network flow; Graph coloring: vertex coloring, Brooks' theorem, color critical graphs etc.; Planar graphs: Embeddings, Euler Formula, Kuratowski's theorem, coloring of planar graphs etc.; Line graphs and edge coloring; Hamiltonian cycles; Special classes of graphs: chordal graphs, interval graphs, perfect graphs etc.

On successful completion of the course, a student will be able to:
  1. Explain fundamental concepts and properties of graphs, trees, connectivity, and network flows, and illustrate their relationships through examples.
  2. Apply graph-theoretic techniques to analyze and solve problems related to spanning trees, matchings, cuts, and flows in networks.
  3. Analyze structural properties of graphs involving connectivity, colorability, and planarity using key theorems such as Menger’s, Brooks’, and Kuratowski’s.
  4. Evaluate graph classes (e.g., chordal, interval, and perfect graphs) and their applications to optimization and real-world problems.
  5. Design and justify graph-based models and algorithms for real-world scenarios involving routing, scheduling, or resource allocation.

MTL7770: Combinatorial Optimization

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL2003/MTL5008
Optimization problems, Convex sets and convex functions, Important combinatorial optimization problems, Convex hulls, Polytopes, Facets, Background on Polytope and Linear programming; Integral polytopes, Total Unimodularity, Total dual integrality, Matroids, Greedy algorithm, Properties, Axioms and constructions of matroids, Matroid Intersection problem, Applications of matroid intersection, Weighted matroid intersection, Matroid Union Approximation algorithms to well-known combinatorial optimization problems: TSP, Knapsack.

On successful completion of the course, a student will be able to:
  1. On successful completetion of this course, a student should be able to formulate real-world decision-making problems as combinatorial optimization models
  2. Learning how to analyze the polyhedral structure of combinatorial problems and derive valid inequalities
  3. Describe and analyze the complexity and correctness of classical algorithms
  4. Apply optimization techniques to well-known problems such as the Traveling Salesman Problem, Knapsack Problem, Assignment Problem, Matching

MTL7777: Introduction to Statistical Learning

3.00 credits (2.0-0.0-2.0)
Preclusions: CML7211, COL7341, CRL7005
Pre-requisites: MTL3190/HSL5013/MCL7102
Summary measures (mean, variance) and statistical inference (confidence intervals, hypothesis testing). A review of limit theorems (central limit theorem, law of large numbers) and exploratory data analysis, linear regression and least square estimation, generalized linear models and logistic regression, classification problem, Fisher’s discriminant analysis and nearest neighbor methods. Model assessment and cross-validation. Data dimension reduction, principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE), model selection, ridge and LASSO regressions. Data-depth, multivariate ordering of data and classification based on data-depth.

On successful completion of the course, a student will be able to:
  1. Apply statistical inference, hypothesis testing, and confidence intervals to analyze real-life data for informed decision-making in various domains.
  2. Understand and implement regression models, classification techniques, and model assessment strategies to address real-life challenges in healthcare, finance, and technology.
  3. Understand and implement dimensionality reduction methods like PCA and t-SNE to extract meaningful insights from large-scale real-life data, improving efficiency and interpretation.
  4. Learn to implement Ridge and LASSO regression, data-depth techniques, and cross-validation to ensure reliable, scalable, and interpretable machine learning solutions for real-life applications.

MTL7778: Bayesian Statistics and Modelling

3.00 credits (3.0-0.0-0.0)
Pre-requisites: MTL3190/AIL7051/HSL5013/MEL7102/MTL5011/ESL3172
Bayes’ theorem, prior and posterior distributions, conjugate priors, Bayesian decision theory, loss functions, posterior predictive distributions, maximum a posteriori estimation, Markov Chain Monte Carlo (MCMC), Gibbs sampling, Metropolis-Hastings algorithm, Bayesian model comparison (Bayes factors, DIC, WAIC), generalized linear models in Bayesian framework, hierarchical Bayesian models, empirical Bayes methods, nonparametric Bayesian methods (e.g., Dirichlet process), Bayesian regression, Bayesian classification, model diagnostics and convergence, prior sensitivity analysis, real-world applications in science and engineering, and case studies using Bayesian data analysis tools in R.

On successful completion of the course, a student will be able to:
  1. Apply Bayes’ theorem to derive posterior distributions and make probabilistic statements about parameters.
  2. Use MCMC methods to estimate Bayesian models and perform inference.
  3. Develop and interpret hierarchical Bayesian models for structured data.
  4. Critically assess and compare models using Bayesian criteria and diagnostics.
  5. Apply Bayesian methods to real-world data and interpret the results in context.

MTL7783: Theory of Computation

4.00 credits (3.0-1.0-0.0)
Preclusions: COL3152, MTL3183
Introduction to the Theory of Computation: Proof Techniques, Basic concepts of language, Grammar and Automata; Chomsky Hierarchy, Regular Languages, Finite automata, Equivalence, DFA and NFA, Minimization, Myhill-Nerode Theorem; Context Free Grammar, Pushdown Automata, their equivalence and Application, Properties of Context-Free Languages; Turing Machine, Recursive and Recursively Enumerable Languages, Undecidability, Rice's Theorem, Post's Correspondence Problem, Complexity Theory, Intractable Problems.

On successful completion of the course, a student will be able to:
  1. Use formal mathematical language to precisely describe the structure and functioning of various abstract machines used in computation.
  2. Evaluate and compare different computational models to determine whether they can simulate each other or differ in expressive capabilities.
  3. Identify the most basic computational model required to solve a given problem and classify problems accordingly.
  4. Develop tailored versions of computational models to execute specific tasks and justify their correctness through logical reasoning.
  5. Explore the inherent limitations of computational systems and rigorously prove when certain tasks cannot be performed by a given model.

MTL7799: Mathematical Analysis in Learning Theory

3.00 credits (3.0-0.0-0.0)
Review: Hilbert space, orthonormal basis, projection theorem, Riesz representation theorem, compact operators, spectral theorem for self-adjoint compact operators. Reproducing kernel Hilbert space (RKHS), positive definite functions, feature maps, Gaussian kernel and their RKHSs, Mercer’s theorem, the elements of statistical learning theory, probabilistic inequalities, Tikhonov-type regularization, general regularization in Hilbert space, representer theorem, convergence analysis of empirical risk minimization, optimal convergence rates for kernel methods, adaptive regularization parameter choice rules. Approximation by trigonometric polynomials, neural networks, universal approximation, ReLU neural networks, convergence rates for holder continuous functions, analysis of deep versus shallow, universal approximation of deep convolution neural networks.

On successful completion of the course, a student will be able to:
  1. Enable students to analyze and justify the convergence behavior of learning algorithms, as well as develop stable prediction/decision-making algorithms using regularization techniques

MTL7848: Pseudodifferential Operators with Application in PDE

3.00 credits (3.0-0.0-0.0)
Fourier transform on Rn, Schwartz space and tempered distributions, Convolution and Fourier multipliers, Sobolev spaces, Symbol classes, Definition of pseudodifferential operators, Examples and basic properties , Symbolic calculus, Composition of operators, Adjoints and asymptotic expansions, Pseudolocal property, Boundedness on 𝐿^p, Action on Sobolev spaces, elliptic operators, Construction of parametrices, Elliptic equations and regularity, Propagation of singularities, Garding's Inequality , Weyl Law

On successful completion of the course, a student will be able to:
  1. Understand the theory of pseudodifferential operators, including symbol classes and their calculus, and analyze properties such as boundedness, adjoints, and composition on Sobolev spaces. Apply microlocal analysis techniques to study linear partial differential equations.

MTL7860: Linear Algebra II

3.00 credits (3.0-0.0-0.0)
Review of vector space, linear transformations, dual and double dual of a vector space and transpose of a linear transformation. Eigenvalue and eigenvector of an operator, characteristic and minimal polynomials. Diagonalizability of linear operators, triangulation, simultaneous triangulation and simultaneous diagonalization. The primary decomposition theorem, cyclic decomposition theorem and the rational and Jordan canonical forms, computation of invariant factors. Inner product spaces, adjoint of an operator, Hermitian, unitary and normal operators, Spectral theorem for normal operators, polar decomposition, Spectral decomposition, singular value decomposition and its applications. Bilinear and quadratic forms, Symmetric and Skew-symmetric bilinear forms, positive definite matrices. Non-negative matrices, Perron-Frobenius theory and applications, generalized inverse of a matrix.

On successful completion of the course, a student will be able to:
  1. On successful completion of this course, a student should be able to apply concepts of vector spaces and linear transformation to solve the theoretical and real-world problems, by employing ideas of eigen values and eigen vectors.
  2. On successful completion of this course, a student should be able to understand and apply concepts of diagonalisability and triangulability in solve important mathematical problems by employing the decomposition theorems and Jordan cannonical forms.
  3. On successful completion of this course, a student should be able to understand the concepts of inner product spaces and operators on them and thereby study spectral properties to analyze and employ important decomposition ideas to solve real life problems.
  4. On successful completion of this course, a student should be able to understand the concepts of quadratic forms, positive definite matrices, non-negative matrices and their usage by applying the Perron-Frobenius theory.

MTL7890: Optimization Algorithms

4.00 credits (3.0-0.0-2.0)
Preclusions: AIL7011, MEL7116, AML7020, ELL7124, AIL7001
Convex function, quasiconvex function, log convex function, strongly convex functions, supporting and separating hyperplanes, separation theorems, convex optimization, first and second order necessary and sufficient conditions, Lagrange dual, duality theorems, algorithm features, how quickly they work, one-dimensional line search and approximate line search methods, Armijo conditions, gradient descent method, stochastic gradient method, batch optimization, gradient method with momentum, accelerated gradient method, splitting methods and ADMM, Newton method, modified Newton method, conjugate gradient method, quasi-Newton method, stochastic quasi-Newton method, constrained optimization, projection method, penalty method, barrier method, sequential quadratic method, solution methods for quadratic problems; descriptions, features, and applications of these methods into practice

On successful completion of the course, a student will be able to:
  1. Foundation of optimization enabling in designing and understanding of algorithms encountered in ML, AI, and Engineering
  2. Gradient methods and its various variants including stochastic and batch gradient methods extensively used in large-scale machine learning
  3. Improved rate of convergence methods, Newton, Conjugate gradient, quasi Newton methods and their theoretical foundation, convergence analysis, and advantages with limitations
  4. Solve constrained optimization problems by projection, penalty, barrier, and sequential quadratic methods and realize their applications to real problems

MTL8880: Parameterized Algorithms for NP-hard Problems

3.00 credits (3.0-0.0-0.0)
Review of NP-completeness and reductions. Introduction to parameterized complexity, parameterized algorithms through bounded search trees, iterative compression, randomized methods in parameterized algorithms, dynamic programming over subsets, treewidth and parameterized algorithms for bounded treewidth graphs, algebraic techniques in parameterized complexity, Matroids and representative sets, lower bounds for parameterized algorithms based on Exponential Time Hypothesis and Strong Exponential Time Hypothesis.

On successful completion of the course, a student will be able to:
  1. Explain fundamental concepts of parameterized complexity, including FPT, kernelization, the W-hierarchy, and the limitations of classical NP-completeness.
  2. Design and analyze parameterized algorithms using major techniques such as bounded search trees, iterative compression, randomized methods, dynamic programming, and treewidth-based methods.
  3. Apply algebraic techniques, matroids, and representative sets to develop advanced FPT algorithms and evaluate their correctness and complexity.
  4. Assess parameterized lower bounds using ETH/SETH and justify optimality or hardness for key computational problems.

MTN8001: Introduction to Research Methodology and Communication in Mathematics

1.00 credits (0.0-0.0-2.0)
Unit 1: Research Foundations and Ethics Definition, concepts, and overview of research; approaches to research including basic, applied, interdisciplinary, and multidisciplinary research. Ethical principles in research: ethical reasoning in scientific practice, scientific conduct, intellectual honesty, responsible authorship, and proper citation and attribution. Research integrity: plagiarism, fabrication, falsification, and redundant publications, including duplication, overlapping publications, and salami slicing. Good research practices: data management, transparency, reproducibility, and conflicts of interest. Unit 2: Foundations of Mathematical and Statistical Reasoning Reasoning and Proof Techniques: Mathematical statements and logic; necessary and sufficient conditions; contrapositive, converse, and negation; direct and indirect proofs; proof by contradiction; principle of mathematical induction; counterexamples; Axiom of Choice and Zorn’s Lemma; inductive and deductive reasoning. Mathematical Modelling and Numerical Analysis: Mathematical modelling and problem formulation; analytical and numerical solutions; approximations and error analysis; interpretation of analytical and computational results. Statistical Reasoning and Data Analysis: Data collection and sampling methods; data analysis and visualization; statistical hypothesis testing; introduction to statistical software, including SPSS, R, and Python-based tools. Unit 3: Research Databases, Metrics, and Scientific Writing Research Databases and Indexing Platforms: Indexing and citation databases, including Web of Science, Scopus, Google Scholar, MathSciNet, and zbMATH Open; search strategies; citation networks; and understanding indexed literature. Research Metrics and Evaluation Indicators: Journal-level metrics such as Impact Factor, CiteScore, SNIP, SJR, and IPP; author-level metrics including h-index, g-index, and i10-index; article-level metrics and altmetrics; scope and limitations of research metrics. Scientific Writing, Documentation, and Publication Ethics: LaTeX for documentation and Beamer presentations; reference management tools such as Mendeley and JabRef; bibliographic styles and citation formats; plagiarism and similarity checks; intellectual property rights and copyrights; preprints (arXiv), open access, and publication models. Responsible Use of AI Tools in Mathematical Research: Introduction to AI-assisted search, literature exploration, and scientific writing; appropriate use of AI for summarisation, LaTeX assistance, and bibliographic support, with emphasis on verification of mathematical content, academic integrity, and ethical considerations.

On successful completion of the course, a student will be able to:
  1. Understand foundational concepts of research, major research typologies, and principles of scientific integrity and research ethics.
  2. Develop core competencies in rigorous mathematical reasoning, proof techniques, modelling, statistical interpretation, and algorithmic analysis.
  3. Acquire systematic understanding of the research process—from formulation of problems to publication of scholarly work.
  4. Gain working familiarity with essential open-source tools, programming environments, and software for mathematical research and documentation.