Topic: Modules over PID ( Speaker: S. Kannan)

Basic theory, applications to abelian groups and canonical forms.

Galois theory (Speaker: K. N. Raghavan)Splitting fields, Separable and normal extensions ,Algebraically closed fields Fundamental theorem of Galois theory with applications to fundamental theorem of algebra and constructibility of regular polygons. Galois groups of cubics and quartics Cyclotomic extensions with some number theory applications. Galois's solvability criterion, existence of Galois extensions of Q with given abelian group.

Applications to arithmetic and geometry (Speaker: P. Rath)
Finite fields with applications to quadratic reciprocity.
Norms and traces, Hilbert's theorem 90, Artin-Schreier theorem with
number theory applications. Luroth's theorem with applications to
algebraic curves. Galois group of $(T)/K$ .

1. M. Artin, Algebra, Prentice-Hall of India, New Delhi, 1994.

2. N. Jacobson, Basic Algebra, Vol. 1, Freeman & Co., USA, Hindustan
Publishing Corporation (India), Delhi, Reprint, 1991.

3. S. Lang, Algebra, 3rd edn., Addison Wesley Pub. Co., Inc., USA,
1993.

Review of the Riemann integral.
Construction of the Lebesgue measure on ${\Bbb R}^n$.
Abstract integration theory and the main convergence theorems.
Dimension Theory: Similarity and Hausdorff Dimension.'

1. W. Rudin: Real and Complex Analysis, Tata MacGraw Hill, New Delhi.

Conditional, unconditional and absolute convergence of a series in
a normed linear space; orthonormal basis for a Hilbert space
Trigonometric series, Fourier series,
Piecewise continuous/smooth functions, absolutely continuous
functions, functions of bounded variation
Generalised Riemann-Lebesgue lemma,
Dirichlet and Fourier kernels, Convergence of Fourier series,
Gibbs phenomenon, divergent Fourier series, term-by-term operations on
Fourier series, various kinds of summability, Fejer theory, multivariable
Fourier series

1. George Bachman, Lawrence Narici and Edward Beckenstein, Fourier and
Wavelet Analysis, Springer-Verlag, New York, 2000.

2. Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Marcel
Dekker Inc., New York, 1977.

3. Balmohan V. Limaye, Functional Analysis, New Age International
Ltd., New Delhi,

Existence of local solutions for first order systems, maximal time of
existence, finite time blow-up, global solutions. Gronwall inequality,
Continuous dependence on initial data and on the vector field on bounded
intervals. Examples of linear systems, Fundamental solutions.

Differential Equations, Dynamical Systemes and an Introduction to Chaos,
2nd Edn., by M. W. Hirsch, S. Smale, R. L. Devaney, Elsevier, 2004.

Manifolds, tangent space, immersion and submersion, regular and critical values, Sard's theorem. Applications.

Differential Geometry( Kingshook Biswas)Smooth maps, bump functions, smooth partitions of unity. Inverse and implicit function theorem. Examples. Classification of 2-manifolds. Differential geometry of curves in R2, R3 and surfaces in R3.

Introduction to algebraic topology (P. Sankaran)
Covering spaces and fundamental groups, simplicial complexes,
simplicial homology (axiomatic approach), some applications.

1. M. P. do Carmo, Differential Geometry of Curves and Surfaces,
Prentice Hall, Engelwood, NJ 1976.

2. A. Gray, Modern Differential Geometry of Curves and Surfaces with
MATHEMATICA, CRC Press, 1998.

3. W. Massey, Algebraic Topology, GTM Series, Springer Verlag.

4. I. M. Singer \& J. A. Thorpe: Lecture notes on elementary topology
and geometry, Undergraduate Texts in Mathematics,
Springer International Edition.