Overview: Basic focus in this workshop is a discussion of the topology of curves, surfaces, and to give an introduction to Algebraic Topology through a discussion of homotopy, fundamental group, covering spaces and simplicial homology. Main emphasis will be to explain concepts through examples.Below is a more descriptive information about the syllabus.
Theme 1: Definition of manifolds as submanifolds of Rn , and tie this up with the standard abstract definition by discussing a proof of ‘medium form’ of the Whitney embedding theorem for compact manifolds.
Theme 2: Proof of triangulation of closed surfaces, classification theorem for surfaces. Discuss orientability, and show that closed surfaces embedded in R3 are necessarily orientable.
Theme 3: A discussion of quotient topology, group actions, quotient spaces, projective plane and Klein bottle. Embeddings of projective plane and Klein bottle in R4 .
Theme 4: A discussion of homotopy, path homotopy, homotopy equivalence, fundamental groups, covering spaces. Discuss fundamental groups of sphere, projective plane, torus, Klein bottle and higher genus surfaces.
Theme 5: Simplicial homology, singular homology, homotopy invariance, equivalence of simplicial and singular homology, homology and fundamental group.
|Date||9:00- 10:00||10:30-11:30||11:45- 12:45||15:00-16:00||16:30- 17:30|
Ln indicates Lecture no. n, and Tn indicates Tutorial no. n
Tea Breaks: 10:00 - 10:30 and 16:00 - 16:30
Lunch Break: 12:45 - 15:00
1) Arijit Ganguly, School of Mathematics, TIFR, Mumbai.
2) Amith Shasthri, School of Mathematics, TIFR, Mumbai.
3) Sanjit Das, Vellore Institute of Technology, Vellore.
4) Binoy Raveendran, TIFR CAM, Bangalore.
|Names of speaker||Affiliation||Topics|
|C S Aravinda (CSA*)||TIFR CAM, Bangalore||
Brief review of multivariable calculus.
|Bankteshwar Tiwari (BT)||BHU-DST Centre, Varanasi||Bump functions, partition of unity, smooth
Urysohn lemma; quotient topology, Group actions, quotient spaces, Projective plane, Klein bottle.
|Anisa Chorwadwala (AC)||IISER, Pune||Definition of smooth manifolds as submani- folds of Rn, abstract definition by smoothness of transition functions, ‘medium form’ of the Whitney embedding theorem for compact manifolds, embeddings of Projective plane and Klein bottle in R4 .|
|C S Aravinda (CSA)||TIFR CAM, Bangalore||Proof of triangulation of closed surfaces, and Classification theorem for surfaces.|
|Suhas Pandit (SP)||IIT, Chennai||Homotopy, path homotopy, homotopy equivalence,
Fundamental groups, covering spaces.
|H A Gururaja (HAG)||TIFR CAM, Bangalore||Transversality, proof of orientability of closed surfaces in R3 .|
|Keerti Vardhan Madahar (KV)||Panjab University, Chandigarh||Simplicial homology, singular homology, homotopy invariance, equivalence of simplicial and singular homology, homology and fundamental group.|
Reference Books: The following books are the suggested references:
(1) A first course in Topology By Munkres.
(2) Elementary Differential Geometry By Christian B ̈ar.
(3) Differential Topology By V. Guillemin and G. Pollack.
(4) Algebraic Topology By A. Hatcher.