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.

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

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.

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.

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.

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),.

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.

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.

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.

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.

Statements, Quantifies, Focus on negation. Sets and Relations.
Focus more on Equivalence relations and partition.
Countability. Order Relations, Axiom of Choice and its Equivalences.

NA

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.

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

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.

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.

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.

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

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.

Text Books
TBU
Reference Books
TBU

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.

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.

TBU

TBU

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

Choose from Electives - A only

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.

Text Books
TBU
Reference Books
TBU

TBU

TBU

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Kindly choose from MTech Core courses and the electives to be finalised.

Some basic topological notions including homotopy of maps, homotopy equivalence, convex sets and contractible spaces, with motivating examples from Euclidean spaces. Paths and path homotopy, path connected spaces, and the relationship between path connectedness and connectedness, including homotopy relative to a subspace. Definition of the fundamental group, loops and loop homotopy, the group operation on homotopy classes of loops, and change of base point. Induced homomorphisms on fundamental groups, functoriality, homotopy invariance of the fundamental group, and applications to deformation retracts. Computation of the fundamental group of the circle, degree of maps from the circle to itself, and related examples. Free groups and free products of groups, reduced words and normal forms, with motivation from topological constructions. Statement and proof of the Seifert–Van Kampen Theorem and its interpretation in terms of unions of spaces. Applications of the Seifert–Van Kampen Theorem to compute fundamental groups of wedge sums of circles, graphs, and basic surfaces. Introduction to covering spaces, covering maps, evenly covered neighborhoods, and standard examples of covering spaces. Path lifting and homotopy lifting properties of covering spaces, uniqueness of lifts, and consequences for fundamental groups. Relationship between covering spaces and the fundamental group, induced homomorphisms under covering maps. Classification of covering spaces and the correspondence between covering spaces and subgroups of the fundamental group, with illustrative examples. Deck transformations, group actions on covering spaces, normal coverings, and their relationship with normal subgroups of the fundamental group. Existence of universal covering spaces.

Text Books
1. James R. Munkres, Topology, 2nd Edition, Pearson (Classic/Modern Classics), 2024.
Reference Books
1. Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001.

Normed linear spaces, Banach spaces, Sequence spaces (C_00, C_0, C, I^p ), Space of continuous functions with p-norms, Quotient spaces, L^p-spaces. Properties of finite-dimensional normed linear spaces- equivalence norm, Heine-Borel theorem. Riesz lemma, (non)compactness of unit ball. Schauder basis, Separability with examples, Strict convexity. Inner product, Hilbert spaces, Cauchy-Schwarz and the Parallelogram Law. Orthonormal basis, Bessel inequality, Parseval identity. Best approximation, Projection theorem. Bounded Linear operator: examples, counterexamples. Hahn-Banach extension theorem. Open mapping theorem, Closed graph theorem. Uniform boundedness principle (Banach-Steinhaus Theorem). Bounded inverse theorem. Riesz representation theorem.

Text Books:
1. M. T. Nair, Functional analysis. (PHI-Learning, New Delhi, Fourth Print 2014).
2. Amol Sasane, A Friendly Approach to Functional Analysis, World Scientific, 2017.
Reference Books:
1. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Student edition
2. B.V. Limaye, Functional analysis (New Age International, 1996)
3. M. Fabian, P. Habala, P. Hajek, V. M. Santalucia, J. Pelant and V. Zizler, Functional analysis and infinite-dimensional geometry. (Canadian Math. Soc, Springer 2001)
4. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol I. (Academic Press, 1980).
5. B. Bollobas, Linear analysis (Cambridge Univ. Press 1999).
6. B. D. MacCluer, Elementary Functional Analysis, Springer 2009.

Review of floating point arithmetic. The bisection method, fixed point iteration method. Newton’s method and extensions. Error analysis in iterative methods. Interpolation: Problem statement, Lagrange polynomials, existence and uniqueness. Newton’s forward difference, -backward difference, -central difference formulae. Hermite interpolation: Problem statement, existence and uniqueness. Cubic splines. Numerical Methods for Linear Systems: Direct methods - Gaussian elimination. Operation counts, partial and complete pivoting. LU factorization, diagonal dominance and its implications, LDU factorization. Cholesky decomposition, Crout’s method. Iterative methods: Vector and matrix norms. Jacobi and Gauss-Seidel methods, Relaxation techniques, Error bounds and iterative refinement. Numerical Differentiation and Integration: Numerical differentiation – three point formula, five point formula, mid-point formula. Richardson’s extrapolation, round-off errors and instability. Numerical Integration: Trapezoidal rule. Simpson’s rule, Newton-Cotes formulae, composite numerical integtation, round-off errors and instability.

Text Books
1. R. L. Burden & J. D. Faires, Numerical Analysis, ninth Edition, Cengage Learning (India), 2010.
2. J. Stoer & R. Bulirsch, Introduction to Numerical Analysis by Springer (India), 2009.
Reference Books
1. D. Kincaid & W. Cheney, Numerical Analysis and mathematics of Scientific Computing, Brooks/Cole, 1999. 2. F. B. Hildebrand, Introduction to Numerical Analysis, Dover, South Asia Edition, 2008.
3. S. S. Sastry, Introductory Methods of Numerical Analysis, Prentice Hall, India, 2009.
4. S. D. Conte & C. De Boor, Elementary Numerical Analysis, Tata Mcgraw-Hill, 2010.

Simultaneous first order system of equations, first-order quasi-linear equations: Lagrange method, Reducible higher-order PDE. Method of characteristics for first-order quasi-linear equations. Pfaffian equations. Fully Nonlinear first order PDEs. Classification and Canonical forms of equations in two independent variables. One-dimensional wave equation: D'Alembert's solution for whole line, Reflection method for halfline. Heat equation, and infinite line, Semi-infinite line, inhomogeneous equation on the whole line. Separation of variables, non-homogeneous problems. Laplace equation in polar coordinates: circular regions, annular regions, Poison integral formula. Maximum-Minimum principle for heat and Laplace equations. Green’s function and boundary value problem.

Text Books
1. I. N. Sneddon, Elements of partial differential equations, McGraw-Hill, New York,1986.
2. Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhauser, 2006.
Reference Books
1. Russell L. Herman, https://people.uncw.edu/hermanr/pde2/PDE2notes/PDE1_Main.pdf2.
2. Jeffery Cooper, Introduction to partial differential equations with matlab, Birkhauser, 1998.
3. Clive R Chester, Techniques in partial differential equations, McGraw-Hill, 1971.
4. W. E. Williams, Partial differential equations, Clarendon Press, Oxford, 1980
5. Ioannis P Stavroulakis and Stepan A Tersian, Partial differential equations- an introduction with mathematica and maple, world - Scientific, Singapore, 1999.

Quick review of ring theory, ideals in polynomial rings, simple properties like R[x]/I[x] ~ R/I[x], PID, prime and maximal ideals, commutative ring containing identity has a maximal ideal. Spectrum, max-Spectrum, Zariski topology, identify topology on Spectrum of F[x], Z, in general PIDs. Local rings, localization with respect to a multiplicative set, homomorphism from the ring to its localization, description of the spectrum of the localization. Universal property of localization, Jacobson radical of a ring, Jacobson radical of a polynomial ring over a field is zero, Radical of a ring, characterization of radical, Jacobson radical and radical of an arbitrary ideal. Modules, homomorphism, Hom_R(M,N), quotient of modules, universal property of quotients, isomorphism theorems, generating sets of a module, minimal generating sets, non-uniqueness of cardinality of minimal generating sets. Direct sum and direct product of modules, inclusion and projection maps to/from the direct sum and product, identifying homomorphisms from the direct sum, Free module, universal property, Nakayama Lemma and applications. Exact sequences, short exact sequences, complexes, split exact sequences, section and retraction, connecting homomorphism and Snake Lemma. Finite presentation, finite generation and finite presentation under short exact sequences, Noetherian modules and rings. Characterization of Noetherianness in rings and modules, Noetherian modules in short exact sequences, Hilbert basis theorem. Primary ideals, primary decomposition of ideals, minimal primary decomposition, non-uniqueness of the decomposition, primary decomposition in Noetherian rings, first uniqueness theorem. Associated primes, isolated and embedded primes, associated primes as colon ideals, zero divisors of R/I as union of associated primes, second uniqueness theorem of the primary decomposition. Descending chain condition, Artinian modules and rings, Artinian modules under short exact sequences, Artinian iff Noetherian an dimension zero, Artinian module need not be Notherian. Integral extensions, lying over, going up and going down theorems. Noether Normalization, Hilbert’s nullstellensatz.

Text Books
1. Algebraic Geometry and Commutative Algebra, Siegfried Bosch, UTX, Springer, 2011.
Reference Books
1. Introduction to Commutative Algebra, M. F. Atiyah and I. G. Macdonald, Addison-Wesley Publishing Company, 1969.
2. A Term of Commutative Algebra, A. Altman and S. Kleiman, Worldwide Center of Mathematics, LLC 2017 (online availabe).

Random Variables and distributions: Random Variables and Vectors, Subfields, Distributions, Independence, Sequences of Random Variables, Convergence in Probability, The Glivenko-Cantelli Theorem. Expected Values: Expected Values and Distributions, Moments. Sums of independent random variables: Inequalities, Independence, Moment Generating Functions. Sums of independent random variables: Laws of Large Numbers, Kolmogorov’s Zero-One Law, Maximal Inequalities. The Poisson Process: Characterization of the Exponential Distribution,The Poisson Process, The Poisson Approximation, Stochastic Processes. Weak convergence: Convergence in Distribution,Convergence in Probability, Fundamental Theorems, Helly’s Theorem, Integration to the Limit. Characteristic functions: Definition, Moments and Derivatives, Independence, Inversion and the Uniqueness Theorem, The Continuity Theorem, The central limit theorem. Infinitely divisible distributions: Vague Convergence, The Possible Limits, Characterizing the Limit. Limit Theorems in Rk: The Basic Theorems, Characteristic Functions, Normal Distributions in Rk, The Central Limit Theorem. The Radon-Nikodym Theorem: Additive Set Functions, The Hahn Decomposition, Absolute Continuity and Singularity, The Main Theorem. Conditional Probability: The Discrete Case, The General Case, Properties of Conditional Probability, Conditional Probability Distributions. Conditional Expectation: Definition, Properties of Conditional Expectation, Conditional Distributions and Expectations. Martingales: Definition, Submartingales, Functions of Martingales, Stopping Times, Inequalities, Convergence Theorems. Kolmogorov’s Existence Theorem: Stochastic Processes, Finite-Dimensional Distributions, Product Spaces, Kolmogorov’s Existence Theorem, The Inadequacy of RT, The Hewitt-Savage Theorem. Brownian Motion: Definition, Continuity of Paths, Measurable Processes, Irregularity of Brownian Motion Paths, The Strong Markov Property, The Reflection Principle, Skorohod Embedding, Invariance.

Text Books
1. Patrick Billingsley, Probability and Measure, 4th edition, Wiley, 2012.
Reference Books
1. John B Walsh, Knowing the Odds: An Introduction to Probability, American Mathematical Society, 2012.
2. Jean Jacod and Philip Protter, Probability Essentials, Springer, 2004.
3. Leo Breiman, Probability, SIAM, 1992.
4. Geoffrey Grimmett, and David Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
5. Geoffrey Grimmett, and David Stirzaker, One Thousand Exercises in Probability, Oxford University Press, 2001.
6. Leonid Koralov and Yakov G. Sinai, Theory of Probability and Random Processes, Springer, 2007.

Graphs, Isomorphism: Theory: Definition of Graphs (Simple, Multigraphs, Digraphs). Isomorphism, Automorphisms, and the Labeling Problem. The Handshaking Lemma. Applications: Chemical Isomer Problem. Stolen circuit design. Identifying persons from anonymized network. Identifying terrorist leadership group from communication logs.
Connectivity and Cut Sets: Theory: Paths, Walks, Trails, Cycles. Cut-vertices, Bridges, and Blocks.
Vertex connectivity vs Edge connectivity. Whitney’s Inequality. Menger’s theorem. Applications:
Supply chain disruptions. Determining if the extinction of a "keystone" species splits a food web into isolated components. Designing a network that can survive k router failures. One way streets in a city.
Trees and Spanning Trees: Theory: Characterizations of Trees, Cayley’s formula via Prüfer Codes. Kruskal and Prim Algorithms. Applications: Single-linkage clustering, Wiring a set of pins on a motherboard using the minimum amount of copper track , Games in Single elimination tournaments. Placing a Mole in an organisation.
Eulerian Circuits and trails: Theory: Necessary and Sufficient conditions, Decomposition. Hierholzer’s Algorithm. Applications: garbage truck/snowplow route, Reassembling a DNA sequence from overlapping reads , A spray-painting robot covering a wireframe object without turning off the nozzle or retracing. Museum guard visit. Safe cracker - de Bruijn sequence.
Hamiltonial graphs: Theory: Hamiltonian cycles, Complexity gap with Euler tour, Closure, Dirac and Ore theorems. Applications: CNC drilling, Amazon delivery route, seating protocols. Knight’s tour.
Flow Networks: Theory: Flow networks, capacity, and conservation.
Max-Flow Min-Cut Theorem (Ford-Fulkerson). Applications: Streaming bitrate in constrained networks. Traffic control in rush hours. Enemy supply lines disruption.
Bipartite Matching: Theory: Bipartite graph characterization, Hall’s and König’s theorems. Gale shapely. Applications: Job fair. A general has k squads of soldiers and k distinct missions. Each squad can only do certain missions. How to convince commander that it is not possible to cover all missions. Cloud Computing: Maximising parallel reads.
General Matching and Factorization: Theory: 1-Factor theorem, Petersen’s theorem, Logic of Blossom algorithm. Applications: Roommate pairing. Kidney exchange ( cycle packing), Max dimer bonds in crystal lattice.
Vertex coloring: Theory: Cliques, Independence, Greedy coloring, chromatic number bounds. Brook’s theorem. Applications: Exam Scheduling, Compiler optimizations, Chemical storage room requirement, Frequency assignments in radio tower networks.
Edge coloring: Theory: Line graphs, Vizing’s theorem. Class 1 class 2. Applications: Fiber optics light wavelengths. Game weeks in round-robin tournaments. Scheduling jobs with shared resources.
Planarity: Theory: Planar graphs, Dual graphs, Euler identity, Kuratowski’s theorem. Applications: PCB design, Minimizing overpasses, Database visualization.
Coloring Planar graphs: Theory: 5 colour theorem. Ideas for 4 colour theorem. Discharging Method. Applications: Cartographer's Dilemma. Art gallery guard problem (cameras to cover polygonal room).
Random Graphs and Ramsey Numbers: Theory: Threshold functions, Emergence of Giant component, Ramsey numbers and bounds. Applications: How contagious ($p$) must a virus be to infect the "Giant Component" of a population? Small world phenomenon - six degrees of separation. Zero error capacity. Finding order in large chaotic systems.

Text Books
1. Jonathan L. Gross, Jay Yellen, and Mark Anderson ,Graph Theory and Its Applications, 3rd Edition, CRC Press, 2018.
Reference Books
1. Douglas B. West, Introduction to Graph Theory, 2nd Edition, Prentice Hall, 2001.
2. Fred Roberts and Barry Tesman, Applied Combinatorics, 2nd Edition, CRC Press, 2009.