Annual Foundation School - Part I (2011)

Speakers

 1. Prof. Ravi Kulkarni: Some subtle points in Differential Topology (2 lectures)

2. Prof. S. S. Sane: Finite simple groups (2 lectures)

Syllabus

1.   Group Theory.
1. Module 1: (6 lectures) Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.
2. Module 2: (6 lectures) free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of PSL(V).  
3. Module 3: (6 lectures) Representations and characters of finite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.
4. Module 4: (6 lectures) 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.
2.  Real Analysis.
1. Module 1: 6 lectures: Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, non-measurable sets.
2. Module 2: 6 lectures: Measurable functions, approximation bysimple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.
3. Module 3: 6 lectures: Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.
4. Module 4: 6 lectures: Fubini’s theorem, L^p - spaces.
3. Differential Topology. Numbers in the bracket refer to sections from [S]
1. (1) Module 1: 6 lectures: Review of differential calculus of several variables: inverse and implicit function theorems. (1.4), Richness of smooth functions; smooth partition of unity.(1.6, 1.7), Submanifolds of Euclidean spaces (without and with boundary) (3.1, 3.2), Tangent space, embeddings, immersions and submersions (3.3,3.4), Regular
   values, pre-image theorem (3.4), Transversality and Stability (3.5, 3.6) [The above material should be supported by exercises at the end of each section and examples matrix groups (9.1).]
2. (2) Module 2: 6 lectures: Abstract topological and smooth mani-folds, partition of unity (5.1,5.2), Fundamental Gluing lemma and classification of 1-manifolds (5.3,5.4), Definition of a vector bundle;tangent bundle. (5.5), Morse-Sard theorem. Easy Whitney embedding theorems.
3. (3) Module 3: 6 lectures: Orientation on manifolds. (4.1), Transverse Homotopy theorem and oriented intersection number (7.1,7.2), Degree of maps both oriented and non oriented case(7.3,7.4), Winding number, Jordan Brouwer Separation theorem. (7.5),Borsuk-UlamTheorem (7.6), Vector fields and isotopies (statement of theorems only) with appliaction to Hopf-Degree theorem. (6.3 and 7.7).
4. (4) Module 4: 6 lectures: Morse functions (8.1), Morse Lemma (8.2),Connected sum, attaching handles (8.3), Handle decompostion the-orem.(8.4), Application to smooth classification of compact smooth surfaces (8.5) (2 lectures)

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.

Link for how to reach the venue