
Indian Institute of Technology, Bombay 
Differential Topology 
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 1parameter group of diffeomorphisms, Isotopies, collar theorem. Connected sum, Boundary connected sum, attaching handles, Cancellation lemma, Plumbing disc bundles: Example of a PL manifold which is not differentiable. 
M. S. Raghunathan and I. Biswas : Cobordism and Handle Presentation. Homology Data, Morse inequality, Poincare duality, An application to 3manifolds: Heegard splitting. hcobordism /applications to classification of Surfaces. 
Introduction to Number Theory 
Eknath Ghate : Introduction to modular forms :upper half plane and it's automorphisms. Cusps, congruence subgroups, fundamental domain for SL(2,Z). Modular forms, Fourier expansions, 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 different, 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 Lfunctions. 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 theorem. 
N. Fakhruddin : Ext and depth, basic properties of CM modules, Macaulay's theorem, graded depth, AuslanderBuchsbaum formula, depth and local cohomology (rapid treatment) 
J. K. Verma : Simplicial complexes and face rings, Hilbert series of face rings, Macaulay expansion of numbers, shellable simplicial complexes, CohenMacaulayness of shellable simplicial complexes 
M. K. Srinivasan : Binomial and Stirling reciprocity, Moebious inversion over posets, Dehn Summerville equations and cyclic polytopes reciprocity for Ppartitions, reciprocity for linear Diopahntine equations, AnandGuptaDumir conjecture for magic squares, Erhart polynomials for lattice points of polytopes 
R. C. Cowsik and Raja Sridharan : Reisner's theorem, Munkres theorem, proof of UBC, Hochster's formula. Gorenstein property of K[S^{d}] 