Mathematics
From Niserwiki
M 301 - Real Analysis
M 302 - Algebra - I
M 303 - Complex Analysis
AM 301 - Foundations of mathematics, logic and elementary number theory
ML 301 - Math Lab
M 401 - Probability and Statistics
M 402 - Calculus of several variables
M 403 - Algebra – II
AM 401 - Discrete Mathematics and graph theory
ML 401 - Math Lab
M 501 - Topology
M 502 - Differential Equations
M 503 - Representation theory of finite groups
AM 501 - Elective (Stream)
AM 502 - Elective (Stream)
M 601 - Functional Analysis
M 602 - Differential Geometry
M 603 - Optimization and calculus of variations
AM 601 - Elective (Stream)
AM 602 - Elective (Stream)
M 701 - Advanced PDE
M 702 - Advanced Probability and Stochastic Process
EM 701 - Elective(stream): Information and coding theory/Representation of linear lie groups
EM 702 - Elective(stream):(Analytic/Algebraic) Number Theory
PM 701,702 - Project/Seminar
M 801 - Algebraic Topology
M 802 - Nonlinear analysis
EM 801 - Elective (stream): Algebraic graph theory
EM 802 - Elective (stream):Theory of computation/Mathematical Logic
PM 801,802 - Project/Seminar
EM 901 - Elective (stream)
EM 902 - Elective (stream)
DM 901-904 - Dissertation
EM 1001 - Elective (stream)
EM 1002 - Elective (stream)
DM 1001-1004 - Dissertation
Here OE stands for an optional subject which must be taken outside the core subject of student's choice and OI stands for an optional subject which is from within the Core subject of student's choice.
Syllabus for Semesters I - VI in detail :
Semester 1
M 101:
Linear Equations, matrices, Vector spaces, Groups,(15)
Elementary differential equations (solution techniques for first and second order ordinary differential equations (ODE))(17),
partial differentiation, line and surface integrals, elementary vector calculus (gradient, divergence and curl).(10)
ML 101: Languages
C++, MATLAB, MAPLE/MAXIMA/MATHEMATICA.
Semester 2
M 201:
Real and complex numbers(8)
Sequence, Series, Radius of convergence, (10)
Differentiation, Integration, Fundamental theorem of calculus,(14)
Taylor series, Binomial, Exponential, Logarithmic functions.(10)
ML 201:
Basic numerical methods; numerical linear algebra; direct and iterative methods; numerical solution of ODE; numerical differentiation and quadrature; nonlinear equations (Newton's method, etc.); least squares method; spline approximation.
Semester 3
M 301: Real Analysis
(i) Metric topology (compactness, connectedness, continuity).(10)
(ii) Riemann-Stieltjes integral, functions of bounded variation.(7)
(iii) Sequence and series of functions, uniform convergence.(7)
(iv) Arzela-Ascoli Theorem.(2)
(v) Fourier series.(4)
(vi) Lebesgue integral (12)
M 302: Algebra I
(i) Groups, subgroups, homomorphisms (5)
(ii) modular arithmetic (2)
(iii) quotient groups, isomorphism theorems (3)
(iv) groups acting on sets, permutation groups, matrix groups (10)
(v) Sylow's theorems (12)
(vi) Finite dimensional vector spaces, basis, eigenvalues and eigenvectors, canonical forms, diagonalization (10)
M 303: Complex Analysis
(i) The Cauchy-Riemann equations (2)
(ii) Power series and holomorphy (8)
(iii) Line integrals, the exponential map and the logarithm (3)
(iv) Cauchy integral formula and its consequences (4)
(iv) Cauchy's theorem (2)
(v) Zeros, poles and singularities of holomorphic functions (6)
(vi) The open mapping theorem and the argument principle (5)
(vii) Maximum modulus principle, Schwarz lemma (4)
(viii) Residues and the residue calculus (8)
ML 301:
Numerical linear algebra (advanced topic), solution of partial differential equations by finite difference methods.
AM 301: Foundations of mathematics, logic and elementary number theory
(i) Dedekind cuts (4)
(ii) Ordinal and cardinal numbers, countable and uncountable sets (4)
(iii) Propositional and quantified logic (4)
(iv) Divisor function (2)
(v) Euler Phi functions
(vi) Fermat's little theorem, Wilson's theorem, Euler's theorem (4)
(vii) Quadratic reciprocity law (2)
(viii) Primitive roots (2)
Semester 4
M 401: Probability and statistics
(i) Combinatorial probability and urn models (2)
(ii) Conditional probability, independence (2)
(iii) Discrete and continuous sample spaces (4)
(iv) Random variables, Distributions and density functions, mean and measures (5)
(v) Moment generating functions - probability laws (binomial, geometric, negative binomial, hypergeometric, Poisson, uniform, exponential, gamma) (6)
(vi) Standard discrete distributions uniform, binomial, Poisson, geometric, hypergeometric (3)
(vii) Independence of random variables, joint and conditional discrete distributions (3)
(viii) Densities: normal, exponential, gamma, Chi-square, beta, Cauchy (3)
(ix) Expectation and moments of continuous random variables (2)
(x) Transformation of univariate random variables (2)
(xii) Tchebychev's inequality and weak law of large numbers (2)
(xiii) Inferential statistics, estimation of parameters by method of moments and maximum likelihood. (4)
(xiv) Confidence intervals, simple and complex hypotheses and list of hypotheses, least square estimation, test of hypothesis, analysis of variance. (4)
M 402: Calculus of several variables
(i) Differentiable maps from Rn - Rm (6)
(ii) Df(w) as a linear map discussions (4)
(iii) Higher derivatives, chain rule, Taylor expansion, local maxima and minima, Lagrange multiplier (8)
(iv) multiple integrals, existence of Riemann integrals for sufficiently well-behaved functions on a rectangle (6)
(v) Change of variable formula with examples (3)
(vi) Inverse and Implicit function theorems (no proofs) (2)
(vii) Line and surface integrals (3)
(viii) Divergence, gradient, curl (5)
(ix) Green's, Stokes & Gauss theorems (5)
M 403: Algebra - II
(i) Ideals, Factorization in a ring, Euclidean domain, principal ideal domain, unique factorization domain (10)
(ii) Field of fractions, Gauss lemma (6)
(iii) Fields, field extension, Galois theory (8)
(iv) Finitely generated modules over a PID and their representation (6)
(v) Structure theorem for finitely generated Abelian groups (6)
(vi) Rational form and Jordan form of a matrix (6)
ML 401: Conjugate radient methods, various methods of optimization
AM 401: Discrete Mathematics and Graph theory
(i) Counting principles, recursion, generating functions, mathematical induction (10)
(ii) Graphs, sub-graphs, isomorphism, planar graphs, trees, graph coloring (12)
Semester 5
M 501: Topology
(i) Topological spaces, quotient spaces (12)
(ii) Separation axioms (Urysohn's lemma, Tietze's extension theorem) (6)
(iii) Filters and nets (4)
(iv) connectedness and compactness (6)
(v) Covering spaces, fundamental groups (14)
M 502: Differential Equations
(i) Second order linear equations with constant coefficients (2)
(ii) System of first order differential equations, equations with regular singular point (8)
(iii) power series methods special ordinary differential equations from physics, some special functions like Bessel's function, Legendre polynomial, gamma function (8)
(iv) Picard's theorem on existence and uniqueness of solution to first order ordinary differential equation (3)
(v) Oscillations - Sturm Liouville theory (4)
(viii) First order partial differential equations (5)
(ix) The Laplace, Heat and the Wave equations (10)
M 503: Representation theory of finite groups
(i) Multilinear forms, tensor products, wedge product, Grassmann ring, symmetric product (18)
(ii) complete reducibility, Schur's lemma, characters (6)
(iii) Projection formulae, inducted representation (4)
(iv) Frobenius reciprocity (4)
(v) Representation of permutation groups (10)
AM 501 : Stream Elective
AM 502 : Stream Elective
Semester 6
M 601: Functional Analysis
(i) Normed linear spaces and continuous linear transformations (12)
(ii) Hahn-Banach theorem (analytic and geometric versions) (4)
(iii) Baire's theorem and its consequence - three basic principles of functional analysis (open mapping theorem, closed graph theorem and uniform boundedness principle) (12)
(iv) Hilbert spaces - Riesz representation theorem, adjoint operator, etc. (4)
(vi) Compact operators (4)
(vii) Spectral theorem for self adjoint compact operators (6)
M 602: Differential Geometry
(i) Curvature and torsion for space curves (12)
(ii) Surfaces in R3 as 2-dimensional manifolds (4)
(iii) Tangent spaces and derivatives of maps between manifolds (4)
(iv) First & Second fundamental forms and the Gauss map (4)
(v) Differential forms (4)
(vi) Integration on surfaces (8)
(vii) Gauss-Bonnet theorem (6)
M 603: Optimization and calculus of variations
(i) Linear programming, simplex method, duality (8)
(ii) Applications - transportation problems (4)
(iii) Calculus in normed linear spaces (5)
(iv) Quadratic programming (4)
(v) Conjugate gradients, etc.(4)
(vi) Farkas - Minkowski lemma, Kuhn - Tucker condition (3)
(vii) Calculus of variations: Euler-Lagrange equation and some sufficient conditions (8)
(viii) Isoperimetric inequality (6)
AM 601 : Stream Elective
AM 602 : Stream Elective
- () Refers to number of lectures
