Group theory
Module 1 (Speaker: S. Senthamarai Kannan)

Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.

Module 2 (Speaker: Krishna Hanumanthu)

free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of PSL(V).

Module 3 (Speaker: S. Vishwanath)

Representations and characters of finite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.

Module 4 (Speaker: P. Rath)

Modules over PIDs: Modules, direct sums, free modules, finitely generated modules over a PID, structure of finitely generated abelian groups, rational and Jordan canonical form.

Analysis
Module 1 (Speaker: Amritanshu Prasad)

Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.

Module 2 Speaker: B. V. Rao)

Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.

Module 3( Speaker: M. Sundari)

Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem. Module 4 (Speaker: E. K. Narayanan)

Fubini’s theorem, Lp - spaces.

Differential Topology.
Module 1 (Speaker: Priyavrat Deshpande)

Review of differential calculus of several variables: Inverse and Implicit function theorems. , Richness of smooth functions; Smooth partition of unity, Submanifolds of Euclidean spaces (without and with boundary), Tangent space, embeddings, immersions and submersions, Regular values, pre-image theorem , Transversality and Stability

Module 2 (Speaker: Sukhendu Mehrotra)

Topological and smooth manifolds, partition of unity , Fundamental gluing lemma and classication of 1-manifolds, Vector bundle; Tangent bundle, Morse - Sard theorem, Easy Whitney embedding theorems.

Module 3 (Speaker: Vimala Ramani)

Orientation on manifolds. , Transverse Homotopy theorem and oriented intersection number , Degree of maps both oriented and non oriented case, Winding number, Jordan Brouwer Separation theorem, Borsuk - Ulam Theorem, Vector eld and isotopies (statement of theorems only) with application to Hopf Degree theorem.

Module 4 (Speaker: Mahan Mj.)

Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.