AFS - I

Speakers and Syllabus 

1.1. Group Theory:

• Speaker: B. Ramakrishnan
  Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.
• Speaker: M. Manickam
  free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of  PSL(V).
• Speaker: M. Manickam

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

• Speaker: Sanoli Gun
  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:

• Speaker: R. Thangadurai
  Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.
  Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.

   

• Speaker: P. K. Ratnakumar
  Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.
• Speaker: K. Sandeep
  Fubini’s theorem, Lp - spaces.

1.3. Differential Topology:

• Speakers: A. J. Jayanthan
  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
• Speakers: P. K. Ratnakumar
  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.
• Speakers: P. Sankaran
  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.
• Speakers: G. Santhanam
  Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.

Tutors:
Karam Deo/Akhilesh/Bhavin Moriya (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.

Time Table:

Applicants/Participants