PAPER 1 - ALGEBRA

Syllabus

• Sets: Cardinals, ordinals. Countability. Zorn’s Lemma.

• Linear algebra: Finite-dimensional vector spaces, bases. Tensor product. Isomorphism of Mn(F) and End(Fn). Orthogonality, examples of classical groups. Diagonalization, Cayley-Hamilton theorem, spectral theorem, Jordan canonical form.

• Group theory: Significance of classification of finite simple groups. Central and derived series. Structure of finitely generated abelian groups. Group presentations. Representations of finite groups.

• Ring and module theory: Euclidean domain, principal ideal domain, unique factorization domain. Polynomial rings, reducibility. Noetherian rings, Hilbert basis theorem. Some noncommutative rings, e.g. matrix rings. Free and projective modules. Exactness.

• Field theory: Fundamental theorem of algebra, algebraic closure. Classification of finite fields. Examples of Galois groups.

• Category theory: Examples of categories, functors, natural transformations, adjoint functors.

Suitable textbooks

• G D Crown, M H Fenrick & R J Valenza, Abstract Algebra, Marcel Dekker (NY, 1986)

• A I Kostrikin, Introduction to Algebra, Springer Universitext (NY, 1982)

Suitable reference books

• T W Hungerford, Algebra, Springer Graduate Texts in Math 73 (NY, 1974)

• S Lang, Algebra, Springer (NY, 2002), rev. 3rd ed.

PAPER 2 - ANALYSIS

Syllabus

• Part I: Advanced Calculus

1. Properties of the reals such as Bolzano-Weierstrass, Heine-Borel and equivalent of norms in Rⁿ.

2. Differential calculus of R^m valued functions on subsets of Rⁿ. Continuity and uniform continuity, differentiability, partial derivatives, Jacobians, implicit and inverse function theorems.

3. Differential equations: existence and uniqueness theorem of initial value problems.

4. Infinite sequences and series of numbers and functions. Absolute and uniform convergence, equi-continuity, Arzela-Ascoli theorem, Weierstrass approximation theorem.

5. Riemann and Riemann-Stieltjes integrals, fundamental theorem of Calculus.

6. Line integrals, surface integrals, differential forms. The theorems of Stokes and Green and the divergence theorem. Change of variables in multiple integrals.

7. Metric spaces, completeness, limit and continuity.

•  Part II: Real Analysis

1. Functions of bounded variation and absolutely continuous function.

2. Definition and elementary properties of Lebesgue measure.

3. Borel measures, measurable functions and simple functions.

4. Lebesgue integral and its elementary properties.

5. Convergence theorems.

6. Various types of convergence such as almost everywhere, in measure, in mean.

7. Multiple integrals and changing the order of integration (Fubini’s theorem).

8. Lebesgue’s differentiation theorem, Vitali’s covering lemma.

9. Basic properties of L^p spaces, such as density of C^¥ functions, approximation identities, Riesz representation theorem.

10. Hilbert space and its basic properties.

• Part III: Complex Variables

Cauchy-Riemann equations. Analytic functions. Contour integration. Cauchy integral formula. Taylor series. Residues and poles. Laurent series. Isolated singular points, removable/essential singularities, poles, residues, residue theorem, improper real integrals and their evaluation using the residue theorem. The argument principle. The open mapping theorem and the maximum modulus principle. Conformal mapping and linear fractional transformations. Harmonic functions.

Recommended textbooks

• (For Part I of this syllabus) Walter Rudin: Principles of Mathematical Analysis (first 10 chapters), 3rd edition, McGraw Hill

• (For Part II of this syllabus) H.L. Royden: Real Analysis, 3rd edition, Macmillan

• (For Part II of this syllabus) R. Wheeden, A. Zygmund: Measure and Integral: An Introduction to Real Analysis, Marcel Dekker

• (For Part III of this syllabus) James W. Brown, Ruel V. Churchill: Complex Variables and Applications, 7th edition, McGraw Hill

PAPER 3 - COMPUTATIONAL MATHEMATICS

Syllabus

• Fundamentals of Computational Mathematics

  • Approximation theory:  polynomial interpolation; piecewise polynomial interpolation; orthogonal polynomials; least squares approximation

  • Numerical integration: trapezoidal rule, Simpson’s rule and  Newton-Cotes formulas; composite trapezoidal rule and Simpson’s rule; Richardson extrapolation and Romberg integration

  • Matrix computation: matrix norms and vectors norms; direct and iterative methods for linear system (basic and Krylov subspace iterative methods); eigenvalue problem; QR factorization; singular value decomposition; linear least squares problem; Iterative mthods for nonlinear systems: fixed point methods, Newton’s method;

   

• Numerical Solution of ODEs and PDEs

  • For initial value problems: Runge-Kutta methods; one-step methods; multi-step methods; consistency, stability and convergence

  • For two-point boundary value problems:  shooting and finite difference methods

  • Finite difference discretization for Poisson’s equation, heat equation and wave equation; consistency and stability analysis

   

• Optimization Theory and Computation

  • Optimization theory: convex sets, hulls, cones and polar cones; convex functions; subgradients; KKT-optimality conditions; Lagrangian dual problems; weak and strong duality theorems

  • Numerical optimization: Newton's method and conjugate gradient methods for unconstrained optimisation; exterior penality function methods; Barrier function methods; Methods of feasible directions

 Recommended textbooks

• Richard L. Burden and J. Douglas Faires, Numerical Analysis, 8th edition, Thomson/Brooks/Cole, 2005.

• L.N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, 1997.

• G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edition, Oxford University Press, 1985.

• J. Nocedal and S.J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.

Sample Papers

Algebra  |  Analysis 1  |  Analysis 2  |  Computational Mathematics

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