Annual Foundation School - Part I (2012)

1.1. Group Theory.

1. Speaker: S. Senthamarai Kannan
1.  Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.
2. Speaker: Krishna Hanumanthu
1. free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of PSL(V).
3. Speaker: Purusottam Rath
1. Representations and characters of finite  groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.
4. Speaker: Sanoli Gun
1. 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.

1.2. Analysis.

1. Speaker:  Amritanshu Prasad
1. Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.
2. Speaker: B. V. Rao
1. Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.
3. Speaker: E. K. Narayanan
1. Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.
4. Speaker: R. Srinivasan
1. Fubini’s theorem, Lp - spaces.

1.3. Differential Topology.

1. Speaker: Shiva Shankar
   
1. 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
   
2. Speaker: TBA
1. 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.
   
3. Speaker: TBA
1. 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.
   
4. Speaker: TBA
1. Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.
   

Tutors: Research Scholars and Post-Docs from CMI and IMSc. The list will be confirmed by the instructors.

References:
[G-P] V. Gullemin and A. Pollack, Differential Topology, Englewood Cliff,
N.J. Prentice Hall (1974).
[H] W. Hirsch, Differential Topology, Springer-Verlag.
[M] J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press,
Verginia.
[S] Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011.