|R. R. Simha : Smooth manifolds
and maps, smooth vector bundles and tangent space, orientability. Normal
bundle, tubular nbds. Transversality theorems, oriented intersection number
and some applications.
|A. R. Shastri : Vector fields
and 1-parameter group of diffeomorphisms, Isotopies, collar theorem.
sum, Boundary connected sum, attaching handles, Cancellation lemma,
disc bundles: Example of a PL manifold which is not
|M. S. Raghunathan and I. Biswas
: Cobordism and Handle Presentation. Homology Data, Morse inequality,
Poincare duality, An application to 3-manifolds: Heegard splitting.
/applications to classification of Surfaces.
|Introduction to Number Theory
|Eknath Ghate : Introduction
to modular forms :upper half plane and it's automorphisms. Cusps,
subgroups, fundamental domain for SL(2,Z). Modular forms, Fourier
examples of modular forms, like Eisenstein series etc.
|C. S. Rajan and C. Khare : (1)
Ring of integers of a number field, failure of unique factorisation, notion
of Dedekind domains, unique factorisation of ideals, splitting behavior,
examples of quadratic and cyclotomic fields. Interpretation of quadratic
reciprocity and possible proof. (2) Ramified primes, discriminant and
trace fn., and prime divides the discriminant iff it is ramified. n=erf.
(3) Localization, dvr, local fields.
|Ravi Raghunathan : Dirichlet
series, Riemann zeta function, Dirichlet characters and their L-functions.
Dirichlet's theorem on infinitely many primes in arithmetic progression.
Prime number theorem.
|Commutative Algebra and Combinatorics
|Yogish Holla : Tensor products,
right derived functors, Ext and Tor Depth and projective dimension via
Ext and Tor, Koszul Complexes and depth, Hilbert syzygy
|N. Fakhruddin : Ext and depth,
basic properties of CM modules, Macaulay's theorem, graded depth,
formula, depth and local cohomology (rapid treatment)
|J. K. Verma : Simplicial
and face rings, Hilbert series of face rings, Macaulay expansion of
shellable simplicial complexes, Cohen-Macaulayness of shellable simplicial
|M. K. Srinivasan : Binomial
and Stirling reciprocity, Moebious inversion over posets, Dehn Summerville
equations and cyclic polytopes reciprocity for P-partitions, reciprocity
for linear Diopahntine equations, Anand-Gupta-Dumir conjecture for magic
squares, Erhart polynomials for lattice points of
|R. C. Cowsik and Raja Sridharan
: Reisner's theorem, Munkres theorem, proof of UBC, Hochster's formula.
Gorenstein property of K[Sd]