Course Details:

The Real number system, Extended real number system, Complex numbers, Euclidean spaces. Finite, countable and uncountable sets. Numerical sequences and series : Convergent sequences, subsequences, Elementary properties. Cauchy sequences, uUpper and lower limits, some special sequences. Infinite series. Series of non negative terms, absolute convergence. Power series, properties and applications. Limits, continuity of functions, Discontinuities, monotonic functions. Differentiability, Mean value theorems, L’Hospital rule, Continuity of derivatives. Higher order derivatives, Taylor’s theorem, Differentiation of vector valued functions. Introduction to Riemann integration. Fundamental theorem of integral calculus and other properties, applications. Improper integrals. Applications of integral calculus to length, Area and volume. Surface area and volume of solid of revolution; other applications.

Course References:

Text Books:
1. Tom Apostol, Calculus Vol I, Second Edition, Wiley, 2025.

Reference Books:
1. S. R. Ghorpade and B. V. Limaye, A Course in and Real Analysis, Springer India, 2006.
2. M.D. Weir, J. Hass, C. Heil and P. Bogacki, Thomas' Calculus, 15th Edition, 2024.
3. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, Wiley India, 4th Edition, 2014.

Course Details:

Introduction to MATLAB environment, workspace, exact vs floating-point arithmetic. Vectors, matrices, array indexing, and basic matrix operations.
Relational and logical operators, conditional statements (if-elseif-else), loops (for, while). Writing scripts and creating user-defined functions.
2D and 3D plotting (plot, surf, scatter). Solving basic linear systems of equations, polynomial roots, and basic numerical methods.
Notebook interface, syntax rules (capitalization, brackets). Exact arithmetic, symbolic vs. numeric computation, lists, and basic matrix operations.
Symbolic algebra (Expand, Factor, Simplify). Calculus operations (Limit, D, Integrate). Solving algebraic and ordinary differential equations (Solve, DSolve).
Functional programming concepts (Map, Apply). 2D and 3D graphics (Plot, Plot3D). Creating interactive mathematical models using Manipulate.
Introduction to Python interpreter, variables, dynamic typing. Basic data types (integers, floats, booleans, strings). Introduction to lists and tuples.
Control structures (if-elif-else, for loops, while loops). Defining functions (def), local vs global scope, basic error handling, and importing standard modules.
Introduction to NumPy arrays and vectorization (bridging concept from MATLAB). Basic mathematical plotting using Matplotlib (pyplot).
Introduction to R and RStudio interface. Basic data structures in R: vectors, matrices, factors, and an introduction to data frames.
Data manipulation, indexing, reading/writing basic datasets (CSV). Writing custom functions and basic control flow in R.
Summary statistics (mean, variance). Generating random numbers from standard distributions (runif, rnorm, rbinom). Basic data visualization using base R graphics.
Solving a singular mathematical problem (e.g., a Monte Carlo simulation or finding roots of a complex system) across all 4 environments to compare syntax, speed, and approach.
Presentations of mini-projects utilizing one or more of the learned software tools to solve a mathematical or statistical problem.

Course References:

Text Books:
1. Amos Gilat, MATLAB: An Introduction with Applications, 6th Edition, Wiley, 2016.
2. John V. Guttag, Introduction to Computation and Programming Using Python, 3rd Edition, MIT Press, 2021.
Reference Books:
1. Stephen Wolfram, An Elementary Introduction to the Wolfram Language, 2nd Edition, Wolfram Media, 2017.
2. Norman Matloff, The Art of R Programming: A Tour of Statistical Software Design, No Starch Press, 2011.
3. Cleve B. Moler, Numerical Computing with MATLAB, SIAM, 2004.

Course Details:

Matrix operations: Basic operations, Transpose and adjoint, Elementary row operations, RREF, Determinant, Computing matrix inverse.
Systems of linear equations: Linear independence, Rank, Solvability, Gauss-Jordan elimination and Gauss elimination.
Matrix as a linear map: subspace and span, Basis & dimension, Linear transformation, Coordinate vectors, Coordinate matrices, Change of basis matrix, Equivalence and similarity.
Orthogonality: Inner products, Gram-Schmidt process, QR-factorization, Orthogonal projection, Best approximation, Least squares solutions.
Eigenvalues and Eigenvectors: The characteristic polynomial, spectrum, Special types of matrices.
Canonical forms: Schur triangularization, Annihilating polynomials, Diagonalizability, Jordan form, Singular value decomposition.

Course References:

Text Books:
A. Singh, Introduction to Matrix Theory, Springer, 2021.
Reference Books:
G. Strang, Introduction to Linear Algebra, 6th Ed., Wellesley-Cambridge Press, 2023.

Course Details:

Formal Logic: Atomic and compound statements, implications, and logical equivalences. Translating between formal logic and plain English. Quantifiers in Linear Algebra. Writing Focus: Translating Calculus definitions (like the ϵ-δ limit) into formal logic. Negating nested quantifiers. Precise definitions.
Proof Methods: The "Mathematical Etiquette" of starting a proof. Direct proof. Proof by contrapositive. Writing Focus: The structural difference between "Suppose P" and "Suppose not Q." Avoiding circular reasoning. How to structure a proof. Ex: Proving that if A is invertible, then Ax=0 has only the trivial solution.
Contradictions and disproofs: The logic of P∧¬Q⟹falsehood. Disproof and counterexamples. Writing Focus: How to construct a clear counterexample. Proving the irrationality of \sqrt(p) for any prime p. Connect it back to incompleteness of rationals. Proving the linear independence of eigenvectors corresponding to distinct eigenvalues by contradiction.
Set Operations: Cartesian products and Boolean algebra of sets. Writing Focus: Proving A⊆B vs A=B. The notation of indexed families of sets ⋃_i∈I A_i. Set theory as the language of Subspaces. Intersection of subspaces vs. Union (and why the union isn't usually a subspace).
Functions: Injections, surjections, and bijections. Composition and well-definedness. Writing Focus: Proving a function is a bijection by finding an explicit inverse vs. proving it is injective and surjective separately.
Relations and Partitions: Reflexivity, symmetry, transitivity. Equivalence classes and the Fundamental Theorem of Equivalence Relations. Writing Focus: Proving that the set of equivalence classes forms a partition.
Mathematical Induction: Principle of Mathematical Induction (weak induction). The Well-Ordering Principle of N. Strong induction and its uses (recursive structures). Induction Traps. Writing Focus: The formal "template" for an inductive proof (Base Case, Inductive Hypothesis, Inductive Step). Proof of division algorithms for integers / Prove that the determinant of an n×n upper triangular matrix is the product of its diagonal entries. Proof of Fundamental Theorem of Arithmetic using strong induction.
Modular Arithmetic: Defining Z_n as a set of equivalence classes. Addition and multiplication in Z_n. Writing Focus: Proving operations are well-defined (independent of the choice of representative).
Cardinality, Countability: The Cantor-Schroeder-Bernstein Theorem. Writing Focus: Writing a bijective proof. Constructing a bijection between (0,1) and [0,1] or other non-trivial sets.
Posets and Zonr’s Lemma: Partial orders, chains, upper bounds, and maximal elements.
Writing Focus: The Blueprint for using Zorn's Lemma (Define a poset, show every chain has an upper bound, invoke the Lemma).
Axiom of Choice: The Axiom of Choice. Discussion on the equivalence of AC WOP, ZL (brief overview) OR Construction of Real numbers (LUB, Dedekind cuts, R completes Q. Define R as set of Cauchy sequences mod eq relation of seq converting to 0. Writing Focus: Every vector space has a basis. Recall that RREF works for finite dim cases. Show that for an infinite-dimensional space (like the space of all continuous functions C[0,1]), you must invoke AC (via Zorn's) because you cannot "construct" the basis manually. OR Build R and prove that it satisfies the expected axioms (Field axioms + Order axioms). Well definedness: If we define R via equivalence classes, write proofs showing that if (a_n)∼(a_n′) and (b_n)∼(b_n′), then (a_n+b_n)∼(a_n′+b_n′). Gap-filling: Write arguments for why Q is insufficient and how the new construction fills the gaps.
Combinatorial Reasoning: Combinatorial "rules of thumb" (Sum Rule, Product Rule). Selections (permutations, combinations), and Binomial theorem. Writing Focus: How to write a "combinatorial proof" (counting in two ways).
Polynomials: Definitions, algebra of $F[x]$, the Division Algorithm for polynomials. Remainder and Factor theorems. Writing Focus: Division Algorithms/ Comparing the structural similarity between Z and F[x].
Zeros and Polynomials: The Rational Zeros Theorem (proof combining number theory and polynomials). Irreducible polynomials. Writing Focus: Rational Zeros Theorem.

Course References:

Text Books:
1. Mathematical thinking: Problem-Solving and Proofs – John P. D’Angelo and Douglas West, Pearson, 2nd edition, 2019.
2. A foundation course in Mathematics, Ajit Kumar, S Kumaresan, BK Sharma. Narosa, 2018.
3. A primer of mathematical writing, Steven G Krantz, arxiv 2016.
Reference Books:
1. Mathematical Proofs: A transition to Advanced Mathematics – Chatrand, Polimeni, Zhang- Pearson.
2. Mathematical Writing by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts. (online).
3. Naive Set Theory by Paul Halmos. Benediction Classics/ Springer.

Course Details:

Ordinary Differential Equations:Geometrical meaning of a first order ODE, First order linear and exact equations, integrating factors, homogeneous linear equations with constant coefficients of higher order, non-homogeneous linear equations with constant coefficients of higher order, linear independence of solutions and Wronskian, complex roots and repeated roots of characteristic equation, solution of non- homogeneous equations using Method of variation of parameters.
Series solution of ODE: Power series method, Legendre's equation, Orthogonality of Legendre polynomials, Frobenius method, Bessel's equation, Orthogonality of Bessel functions.
Partial differential equations:
First order equations using Lagrange’s method, classification of second order equations, D'Alembert's solution of wave equation, solutions of heat equation, wave equation and Laplace equation (Cartesian coordinates) using separation of variables.

Course References:

Text Books:
E. Kreyszig, Advanced Engineering Mathematics, 10th Ed., John Willey & Sons, 2010.
Reference Books:
1. W.E. Boyce and R.C. DiPrima, Elementary Differential Equations, 7th Ed., John Wiely& Sons, 2002.
2. S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, 2006.
3. N. Piskunov, Differential and Integral Calculus Vol. I & II, Mir Publishers, 1974.

Course Details:

Principle of Mathematical Induction; basic properties of integers; divisibility language and examples. The Division Algorithm; Euclidean Algorithm; gcd computation and applications.
Infinitude of primes; Euclid’s proof; prime factorization viewpoint and consequences. The Sieve of Eratosthenes; elementary prime counting ideas and computational aspects. Congruences; modular arithmetic; linear congruences and solvability criteria. The Chinese Remainder Theorem; constructive solutions; applications to computations. Fermat’s Little Theorem; introduction to primality testing (basic tests and motivation). Euler’s Totient Function; properties, computations, and examples. Euler’s theorem (generalization of Fermat); applications to modular exponentiation. Arithmetical functions; multiplicative functions (including φ, divisor functions). Möbius function; Möbius inversion formula; elementary applications. Quadratic residues; Legendre symbol; basic properties and computations. Quadratic Reciprocity Law; statement, examples, and standard applications. Introduction to cryptography and RSA; continued fractions; Pell’s equation (overview and key methods).

Course References:

Text Books
1. David M. Burton, Elementary Number Theory, 7th Edition, McGraw-Hill (2017).
Reference Books
1. Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer-Verlag.
2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford University Press (2008).

Course Details:

Intro to Experimental Math & Tools: Familiarization with SageMath, Python, SymPy. Plot functions, derivatives, integrals. Symbolic math.
Surfaces & Geometry: 3D paper/card models of paraboloids, cones. Build polyhedra skeletons with straws/balls, compute Euler characteristic V−E+F. Tangent plane demo with transparent sheet.
Calculus in Practice I, II & III: Visualize Riemann sums, trapezoidal rule, Simpson’s rule. Approximate areas/volumes by cutting/rearranging paper models. Taylor series approximations: visualize convergence, compare with actual function using plots and small code experiments. Numerical differentiation & integration: implement finite differences, Riemann sums, trapezoid, Simpson’s rule; compare errors.
Linear Algebra I: Code Gaussian elimination, solve systems of equations. Collaborative “human” solving of equations.
Probability I: Coin toss experiments (physical + simulation). Compare empirical vs. theoretical frequencies. Monty Hall, Catan (Dice probabilities, Longest Road / Largest Army Probabilities, Monte Carlo Game Simulation) Problem. Real + MC simulations
Logic & Puzzles I: Truth-teller/ liar puzzles. Logic grid puzzles. Students act as “human logic gates” constructing truth tables.
Differential Equations I & II: Code Euler’s method vs. Runge–Kutta. Compare with analytic solutions. Build dice-based population growth simulation. Predator–prey game: role-play rules to mimic ODEs. Pendulum/spring–mass demos. Then code the ODE numerically.
Basic Algebra I: Vector Space Machine – Use rods and markers to explore spans, linear combinations, and basis changes.
Math Writing - LaTeX I & II: Write a two-page report on one of the projects did before in Linear Algebra/Differential Equations. Submit a PDF assignment solution for one of your classes.
Mini Projects & Presentations: Small groups present projects

Course References:

Text Books
TBU
Reference Books
TBU

Course Details:

Vector Spaces- definition of field and vector space, examples (both finite and infinite dimensional), subspaces , span, linear dependence and independence (of infinte sets). Basis and Dimension - Definition of Basis (without co-ordinate), Dimension and invariance of dimension, Existence of basis (both finite and infinite (discuss Axiom of Choice)). Linear Transformation - Linear maps between vector spaces, Image, Kernel, Rank-Nullity Theorem.
Isomorphism and Quotient spaces- Isomorphisms and classification, Quotient spaces and induced maps, examples. Operators vs Matrices - Linear operators and invariant subspaces, Operators vs matrices (conceptual distinction), Similarity as change of basis. Determinants- permutations, signature of permutation, alternating multilinear map, determinant (of a square matrix) as alternating multilinear map, existence and uniqueness of determinant. Determinants- Geometric meaning of determinant, properties of determinant. Determinant of linear operator (on finite dimensional space). Non-existence of determinant in infinite dimension. Determinant and Invertibility. Eigenvalues and Eigenvectors- Eigenvalue and eigenvector of operators, Eigenspaces and invariant subspaces, characteristic polynomial. Minimal Polynomial and Diagonalization - Minimal polynomial, polynomial function as operator, Diagonalizable operator. Triangulation and Simultaneous Triangulation - Existence of triangulation, commuting family of operators, Simultaneous Triangulation. Cayley-Hamilton Theorem- statement, proof and applications. Inner Product Spaces- Inner products and norms, Orthogonality, Orthonormal sets (in R^n and in infinite dimensions). Projection and Approximation - Orthogonal complement, Orthogonal Projection, Best approximation and motivation for completeness. Dual spaces and Reflexivity- Linear functional, dual spaces and annihilator, Double dual, finite dimensional reflexivity and failure in infinite dimension.

Course References:

Text Books
1. Hoffman, K. & Kunze, R., Linear Algebra, 2nd Edition
Reference Books
1. Sheldon Axler, Linear Algebra Done Right
2. P.R. Halmos, Finite Dimensional Vector Spaces

Course Details:

Metric spaces: Motivation via real numbers, definition, examples, open sets, closed sets, Relation of open sets to unions and intersections. Metric spaces: interior points, limit points, closure, Relative openness, Product spaces. Compactness, relative compactness, Limit point compactness, Nested intersections theorem. Metric spaces: Completeness, the Heine-Borel Theorem, the Baire category theorem. Metric spaces: Contraction mapping theorem, Connectedness, continuous functions. Relation of continuous functions with compactness, connectedness and products, Succinct review of Riemann integration. Riemann-Stieltjes integral as a generalization of the Riemann integral, Integrals of continuous and monotone functions, Functions of bounded variation as the difference of monotone functions and its integral. Properties of the Riemann-Stieltjes integral, Fundamental theorems of calculus, Integration by parts, Change of variables formula. Succinct review of numerical sequences and series, Sequences and series of functions, pointwise and uniform convergence. Weierstrass’s M-test, Uniform convergence and continuity, The metric of uniform convergence. Completeness w.r.t. the metric of uniform convergence, Uniform convergence: integration and differentiation. A continuous nowhere differentiable function, Weierstrass’ approximation theorem. Equicontinuous families of functions and the Arzela-Ascoli theorem. Equicontinuous families of functions and the Arzela-Ascoli theorem.

Course References:

Text Books
1. W. Rudin - Principles of mathematical analysis, Mcgraw-Hill 1976.
2. Terence Tao - Analysis I and II, with Chapter 11 from Volume I and Chapters 1,2 and 3 from Volume II, TRIM series 37 and 38, Hindustan Book Agency.
Reference Books
1. C. C. Pugh, Real Mathematical Analysis, Springer, 2002.
2. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.
3. G. F. Simmons, Topology and Modern Analysis, Kreiger, 2003.
4. Richard R. Goldberg, Methods of Real Analysis, John Wiley and Sons, 1976.
5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2000.

Course Details:

Functions of several variables, graphs, level sets, limits and continuity. Partial derivatives, directional derivatives, higher order derivatives. Differentiability, Derivative of function a linear map. Gradient vector, relation between directional derivatives and gradient vector, tangent planes, applications. Chain rule; Jacobian matrix and applications. Implicit Function Theorem. Inverse Function Theorem. Critical points; extrema; Hessian matrix; Lagrange multipliers. Double integrals; Fubini’s theorem; integration over regions, double integration using polar coordinates. Change of variables; Jacobian; triple integrals in coordinate systems: cartesian, cylindrical and spherical. Vector fields; line integrals; conservative fields. Green’s Theorem and applications. Surface integrals; flux integrals. Stokes’ Theorem; Gauss Divergence Theorem.

Course References:

Text Books:
1. Tom M. Apostol, Calculus, Volume II, 2nd Edition, Wiley India, 2022.
Reference Books:
1. S. R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis, Undergraduate Texts in Mathematics, Springer, 1st edition (corrected publication 2019; originally published 2010),.

Course Details:

Method of Successive, approximations, Lipschitz condition, convergence of the successive approximations. Non-local existence of solutions, approximations to and uniqueness solutions, continuous dependence on initial conditions. Second-order differential equations: General solution of homogeneous equations, Non-homogeneous equations, Wronskian, Method of variation of parameters, Euler’s equation. Solution in Series: Convergence Theorems. Oscillatatory equations; Sturm comparison theorem; Sturm separation theorem. Boundary value problems; Green’s function, Non-homogeneous boundary conditions. Sturm–Liouville system, Eigenfunction expansions, Singular Sturm–Liouville system. Linear systems; Fundamental matrix solution; System with costant coefficient (eigenvalues and eigenvectors method), Periodic linear systems; Floquet’s Theorem. Autonomous systems, Stability for linear systtem with constant coefficients (node, saddle, degenarate case, center, spiral). Perturbed Systems, Method of Lyapunov for Nonlinear Systems. Limit cycles Poincare-Bendixson theorem.

Course References:

Text Books
Tyn Myint U, Ordinary Differential Equations, Elsevier North- Holland, 1978.
Reference Books
1. E.A. Coddington, An Introduction to Ordinary Differential Equations, PHI Learning 1999.
2. R.P. Agarwal and D. O'Regan, An Introduction to Ordinary Differential Equations, Springer- Verlag, 2008
3. G.F. Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed., McGraw-Hill, 1991.

Course Details:

Review of basic properties of integers, division algorithm, infinitudes of primes, integers modulo n, congruence class modulo n, addition and multiplication modulo n. Permutations - definition, cycles, product of cycles, permutation as product of disjoint cycles, order of cycles, transposition, even and odd transpositions. Binary operations on a set, Isomorphism of binary structures, Definition of Groups and Examples (integers, integers modulo n, rationals, reals, complex numbers, S1, permutations, set of all real valued functions on a domain, matrices under addition, GL2(C), GL3(C), SL2(C), SL3(C)), Some examples of sets with binary opertions which are not groups (Z, Q, R, C with multiplication, matrices with multiplication, set of all functions from X to X with composition as operation). Finite groups and infinite groups, Finite group tables, Basic properties of groups (cancellation laws, uniqueness of identity and inverse).
Subgroups, Examples of subgroups, Subgroups of cyclic groups (finite and infinite), Cayley’s Theorem, Even and odd permutations, An, groups of symmetries of a regular n-gon, group of isometries of the plane. Cosets via relations, Lagrange’s theorem and applications, Intersection of subgroups, Product (sum) of sugroups. Generators of a group, Direct sum and direct product, Structure of finitely generated abelian group. Homomorphism, Kernel, Image, Properties of homomorphisms. One-one, onto in terms of kernel and image, Normal subgroup, Quotient subgroups. Examples of normal subgroups and quotient groups of Zn, Sn and other groups, 1st isomorphism theorem and applications, Normal subgroups under homomorphisms, Simple groups, center and commutator subgroups, An is simple for n > 4. Definition of field, examples, Some basic properties of fields. Examples of finite fields, some examples of field extensions over Q, motivate algebraic closure, motivate “algebraic closure exists”.

Course References:

Text Books:
1. J. B. Fraleigh, A First Course in Abstract Algebra, Pearson Education India, 7th Edition, 2013
Reference Books:
1. J. A. Gallian, Contemporary Abstract Algebra, Cengage Learning India, 9th Edition, 2019
2. M. Artin, Algebra, Pearson Publication 2nd Edition, 2024.
3. I. N. Herstein, Topics in Algebra, Wiley Student Edition, 2016.

Course Details:

Revision of Riemann Integration, Function theoretic view of Riemann integral. Outer measure induced by the length, the Caratheodory condition, Construction of the Lebesgue measure in R, Non-measurable sets in R. Abstract Measurable space, Borel σ-algebra. Measure, Continuity properties of a measure, Monotone class theorem. Uniqueness of the extension, Completion of a measure space, Completeness of Lebesgue σ-algebra. Measurable functions, Convergence of measurable functions. Egorov’s theorem, Lusin’s theorem. Integral of nonnegative measurable function, Monotone convergence theorem. Fatou’s lemma, Lebesgue Integrable functions. Dominated convergence theorem, Generalized dominated convergence theorem. Completeness of L^1(μ), L^1[a, b] as the completion of R[a,b], Fundamental theorem for Lebesgue integrable functions. Product of measure spaces, Product measure. Integration on product spaces, Fubini’s theorem.

Course References:

Text Books
1. G. de Barra, Measure and Integration, Wiley Eastern, 1981.
2. I. K. Rana, An Introduction to Measure and Integration, Second Edition, Narosa, 2005.
Reference Books
1. W. Rudin - Principles of Mathematical Analysis, McGraw-Hill, 1976.
2. H. L. Royden, Real Analysis, Third edition, Prentice-Hall of India, 1995.
3. Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, AMS, 2011
4. G. Folland, Real Analysis: Modern Techniques and Their Applications.
5. W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, International Editions, 1987

Course Details:

Complex numbers, Topology of the Complex plane, Stereographic Projection, Extended Complex plane, Complex functions, Limits. Continuity, Introduction to the Complex exponential and the Complex logarithm. Complex trigonometric functions, Branch of the logarithm (Intro). Linear fractional transformations. Complex differentiability, Cauchy-Riemann equations, Sufficient conditions for differentiability. Holomorphic functions, Harmonic functions, Harmonic conjugates, Branch of the logarithm. Basics of Series of Complex numbers, Power series, Taylor series in particular, Radius of convergence, Differentiation of Power series. Complex integration - Contour Integrals, Cauchy’s theorem. Morera’s theorem, Cauchy integral formula, Liouville’s theorem. Fundamental theorem of algebra, Maximum modulus principle, Schwarz lemma. Singularities, Laurent series, Classification of singularities. Classification ctd., Behavior near singularities, Residue theorem. Evalution of Real integrals. Argument principle, Rouche’s theorem, Open mapping theorem.

Course References:

Text Books
1. S. Ponnusamy and H. Silverman, Complex Variables with Applications, Birkhauser, Boston, 2006.
2. D. Sarason, Complex Function Theory, Hindustan Book Agency, 2008.
Reference Books
1. L. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.
2.T.W. Gamelin, Complex Analysis, Springer-Verlag, 2001.
3. J.B. Conway, Functions of one Complex Variable, 2nd edition, Springer-Verlag, 1978.
4. R.E. Greene and S.G. Krantz, Function theory of One Complex variable, AMS, 2006.
5. J.W. Brown and R.V. Churchill, Complex Variables and Applications, McGraw Hill, 2008.

Course Details:

Introduction and motivation; sets, functions, equivalence relations; metric spaces; open and closed sets. Topological spaces; definition and examples; basis and subbasis; neighborhoods. Continuous functions; homeomorphisms; topological invariance. Subspace topology; product topology; projection maps. Quotient topology; identification spaces; examples. Connected and path-connected spaces; components. Compactness; open cover definition; continuous images of compact spaces. Countability axioms; separation axioms; Hausdorff, regular, and normal spaces. Urysohn’s Lemma; normal spaces; applications. Tietze Extension Theorem; statement, proof outline, applications. Product spaces and compactness; Tychonoff’s Theorem. Local compactness; local connectedness; metrizable spaces. One-point compactification; function spaces; classical examples. Introduction to homotopy; homotopy of maps; homotopy equivalence; contractible spaces.

Course References:

Text Books
1. M. A. Armstrong, Basic Topology, Springer, 2004.
Reference Books
1. J. R. Munkres, Topology, Pearson, 2021.
2. Vikram Aithal and S. Kumaresan, Topology – A Core Course, Techno World, 2023

Course Details:

Basic Principles: The Art of Counting Without Counting. Permutations & Combinations: Order and Selection. Binomial & Multinomial Identities: The Bridge to Algebra. The Twelve-fold Way: The Grand Classification. Setting up & Solving Recurrences: Thinking Recursively. Advanced Recurrences: Complexity and Inhomogeneity. Ordinary Generating Functions : The Clothesline Strategy. Generating Function Method: Manipulating the Clothesline. Exponential Generating Functions: When Labels Matter. Principle of Inclusion-Exclusion : When we over count. Classic Applications of PIE: Chaos and Derangements. Introduction to Symmetry. Burnside’s Lemma and applications. Polyas’ theorm and applications.

Course References:

Text Books
TBU
Reference Books
TBU

Course Details:

Classical Probability: Sample space, events, probability axioms, some properties, equally likely outcomes, Axioms of Probability. Conditional probability and independence: Conditional probability, total probability theorem. Bayes’s formula, independent events. Discrete random variables: Discrete random variables, expectation, variance, the Bernoulli, binomial, and Poisson random variables. Other discrete probability distributions, expectation of sums of random variables, properties of cumulative distribution function. Continuous random variables: Definition, expectation, variance, uniform, Normal, and exponential random variables. Other continuous distributions, distribution of a function of a random variable. Jointly distributed random variables: joint distribution functions, independent random variables and their sums. Conditional distribution in discrete and continuous cases, joint probability distribution of functions of random variables. Properties of expectation: Expectation and variance of sums of random variables, covariance and correlations, conditional expectation and prediction. Moment generating functions, properties of Normal random variables, general definition of expectation. Limit theorems: Chebyshev’s inequality, weak law of large numbers, central limit theorem. Strong law of large numbers, other inequalities. Simulation: General techniques for simulating continuous random variables, simulation from discrete distributions, variance reduction techniques.

Course References:

Text Books
1. Sheldon M Ross, A First Course in Probability, Pearson Education, 10th edition, 2022.
Reference Books
1. William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edition, Wiley, 2025.
2. William Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edition, Wiley, 2024.
3. Kai Lai Chung and Farid AitSahlia, Elementary Probability Theory With Stochastic Processes and an Introduction to Mathematical Finance, 4th edition, Springer, 2003.
4. C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2006.
5. P. . G. Hoel, S.C. Port and C.J. Stone, Introduction to Probability Theory, Houghton Mifflin, 1972.

Course Details:

Multivariable Calculus I: Gradient Field Map – Plot a 2D contour map of a function and overlay gradient vectors; identify local maxima/minima. Optimization – Implement hill-climbing in Python to find the maximum of a multivariable function. Elementary Number Theory I: Toy Cryptography – Implement a simple RSA encryption/decryption with small primes in Python/Sage. Differential Geometry I: Build cylinders, cones, and saddle surfaces; classify curvature qualitatively. Geodesic Paths – Explore shortest paths on surfaces using string or paper models; optionally simulate in code. Logic & Puzzles I: Algorithmic puzzles (Towers of Hanoi, coin-choosing game). Relate to recursion and induction. Probability & Statistics I&II: Monte Carlo estimation of π. Dice sum histograms, and CLT demonstration, Experimental Design: Which paper-airplane design flies farthest? Intro to hypothesis testing and inference. Linear Algebra I & II: Eigenvalues/ eigenvectors (toy PageRank example). SVD-based image compression. Use transparent sheets for rotations/reflections/shears. Graphs & Networks I & II: Clock arithmetic – cyclic group, symmetry group of squares/triangle using cutouts. Grids, chessboard, small graphs. Rubik’s Cube/permutation puzzles as group actions. Other similar puzzles. Model friendship networks with students. Introduce graph invariants (degree, connectivity). Explore social network examples. Activity: Use colored beads or tiles to explore Burnside’s Lemma (count colorings). Make human Cayley graphs. Build Cayley graphs with students as nodes (each move represents a generator). Dynamical Systems I & II: Bifurcation, nonlinear dynamics, plotting trajectories, chaos, orbits. Art of Writing Proofs I & II: See some examples here: https://matthbeck.github.io/papers/aop.noprint.pdf
Mini Projects: Small groups present projects.

Course References:

Text Books
TBU
Reference Books
TBU

Course Details:

Recall: The Euclidean algorithm for integers, polynomials and the Euclidean algorithm, vector spaces and elementary linear algebra, matrices and elementary operations, permutations and their structure, Zorn's lemma Subgroups, Group homomorphisms, Isomorphisms, Normal subgroups, Cosets, Kernels, of homomorphisms, Quotient groups, Lagrange's theorem, Cyclic (sub)groups, The first isomorphism theorem, The second isomorphism theorem, The third isomorphism theorem. Cayley's theorem, group action, Orbits, Stabilizers, fixed points and the counting lemma,
Example : Counting inequivalent colourings, Analysing groups with order power of a prime, Free groups and viewing groups via presentations (generators modulo relations),
Example : The tile puzzle (optional). The class equation and the first Sylow theorem (existence of a p-Sylow subgroup).
More Sylow theory : Normalizers, conjugacy of p-Sylow subgroups, arithmetic of the number of p-Sylow subgroups and the existence of a p-Sylow subgroup containing a given subgroup of order a power of p. Application in classifying groups of a specified order. Semi-direct products. More applications in classifying groups of a specified order using semi-direct products. Basic definitions and examples of rings.
Special classes : unital, commutative rings, integral domains, fields, division rings. Basic properties of rings. Definition of a subring. Examples. Ring homomorphisms and left/right/two-sided ideal. Some properties and examples. Quotient rings. Prime ideals and maximal ideals. Examples involving the evaluation map for rings of functions. Existence of maximal ideals. The first, second and third isomorphism theorems for ideals. The fraction field of an integral domain. The Chinese remainder theorem. Factorization in integral domains. Definitions of UFDs, PIDs and EDs. Examples. ED is a PID, PID is a UFD. UFDs need not be PIDs : ℤ[x]. PIDs need not be EDs :ℤ[(1+√−19)/2]. Construction of finite fields.
Basics of modules : definitions, abelian groups as modules over ℤ, submodules, quotients, homomorphisms, direct sums. Linear algebra over rings : generating sets, linear independence, free modules, bases, rank, homomorphisms are matrices. Torsion elements, torsion modules, cyclic modules. The Smith normal form. The structure theorem for finitely generated modules over a p.i.d. Jordan canonical form.

Course References:

Text Books:
1. Anthony W. Knapp, Basic Algebra, Downloadable From Here.
Reference Books:
1. Micheal Artin, Algebra, Pearson Publication 2nd Edition, 2024.
2. David S. Dummit and Richard M. Foote, Abstract Algebra, Third Edition, John Wiley & Sons, 2004.

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Review of vector-valued functions, parametrized curves, regular curves, arc length, unit-speed parametrization. Tangent, normal, binormal vectors; curvature; examples of plane and space curves. Torsion, Frenet–Serret formulas, fundamental theorem of space curves. Parametrized surfaces, regular surfaces, tangent planes, normal vectors. First fundamental form, metric on surfaces, lengths and angles. First fundamental form, metric on surfaces, lengths and angles. Second fundamental form, shape operator, principal curvatures. Gaussian curvature, mean curvature, examples. Gauss map, Theorem of Egregium, intrinsic nature of curvature. Geodesics, geodesic curvature, examples on standard surfaces. Parallel transport, geodesic coordinates, local geometry. Euler characteristic, triangulations, curvature integrals. Gauss–Bonnet Theorem: statement, proof for surfaces with boundary. Gauss–Bonnet for closed surfaces, applications, review.

Course References:

Text Books
1. A. Pressley, Elementary Differential Geometry, Springer.
Reference Books
1. J.A. Thorpe Elementary Topics in Differential Geometry
2. Manfredo Perdigão do Carmo Differential geometry of curves and surfaces, Dover, 2016.

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Data and Statistical Questions: Types of data; populations vs. samples; observational studies vs. experiments. Descriptive Statistics & Distributions: Summary statistics; histograms and boxplots; Binomial, Normal, and Poisson distributions. Probability, Sampling Variability & Simulation: Basic probability rules; sampling variability; introduction to simulation. Estimation and Hypothesis Testing: Point estimates; confidence intervals; z-tests and t-tests; p-values; Type I/II errors; power. Comparing Groups and Resampling: Confidence intervals for differences in means; chi-square tests; ANOVA; bootstrap concept. Exploratory Data Analysis: Scatterplots; covariance and correlation; data cleaning; missing values; transformations. Categorical Data Analysis: Contingency tables; proportions; association in categorical variables. Statistical Learning & Modeling: Supervised vs. unsupervised learning; linear regression; decision trees. Model Evaluation and Project: Accuracy; confusion matrix; cross-validation (basic); mini-project presentations.

Course References:

Text Books
1. Agresti, Alan, and Maria Kateri. Foundations of statistics for data scientists: with R and Python. Chapman and Hall/CRC, 2021.
Reference Books
1. Efron, Bradley, and Trevor Hastie. Computer age statistical inference, student edition: algorithms, evidence, and data science. Vol. 6. Cambridge University Press, 2021.
2. Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. "The Elements of Statistical Learning." (2009).
3. Mendenhall, William, Beaver, Robert, and Beaver, Barbara, Introduction to Probability and Statistics, 14th edition, Springer, 2013.

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Electives

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Electives

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Semester VI is kept "Free" to enable the students to do internships.

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Electives

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Electives / Projects

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Electives / Projects

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Electives

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Electives

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Electives

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Electives

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Electives / Projects

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Electives / Projects

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TBU

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TBU