AIS Algebraic Topology (2016)

Venue: North-Eastern Hill University, (NEHU) Campus, Shillong
Dates:  13th June to 2nd July, 2016

 

Convener(s) Speakers, Syllabus and Time table Applicants/Participants

 

School Convener(s)

Name Himadri Kumar Mukerjee Goutam Mukherjee Angom Tiken Singh
Mailing Address Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-
793022, Meghalaya, India.
Indian Statistical Institute,
203 B.T. Road, Kolkata- 700108,
West Bengal, India.
Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-
793022, Meghalaya, India.

 

Speakers and Syllabus 

Speakers with their affiliations

S. No Name of Associates Affiliation
1. Swagata Sarkar (SS) CEBS, Mumbai
2. Keerti Vardhan (KV) Punjab University, Chandigarh
S. No Name of Resource persons Affiliation
1. Somnath Basu (S/B) RKMV Univ. Belur, Howrah
2. Samik Basu (SNB) RKMV Univ. Belur, Howrah
3. Amiya Mukherjee (AM) ISI Kolkata
4. Mahuya Dutta (MD) ISI Kolkata
5. Goutam Mukherjee (GM) ISI Kolkata
6. Himadri Kumar Mukerjee (HKM) NEHU, Shillong
7. Angom Tiken Singh (ATS) NEHU, Shillong

Topics actually covered:

  Lectures Detailed Syllabus
Topic 1 (HKM/SS) 6 (90 minute) Cohomology, cup products, orientations, Poincare duality. Steenrod operations. Generalized cohomology theories. K-theory as an example of a generalized cohomology theory.
Topic 2 (ATS/KV) 6 (90 minute) Definition of homotopy groups, Fibrations and induced exact sequences. Some elementary differential topology : smooth manifolds, regular values, critical values, Sard's theorem. Computation of \pi_k(S^n) for k<n. A short introduction to the theory of orientations of manifolds, degree and an idea of the Hopf degree theorem.
Topic 3 (AM/GM) 6 (90 minute) Spectral sequences, the Serre spectral sequence associated to a fibration, some elementary computation. Proof of Serre's theorem on the finiteness of the homotopy groups of spheres. Eilenberg MacLane spaces. The Steenrod algebra. Computation of H^*(K(Z/p,n);Z/p).
Topic 4 (MD) 6 (90 minute) (Morse theory) Non-degenerate smooth functions on a manifold (existence). Determination of the homotopy type in terms of the critical values. Examples. Morse inequalities. The Lefschetz theorem on hyperplane sections. Other applications.
Topic 5 (SB) 6 (90 minute) Adams spectral sequences. Easy computations with K-theory. The Hopf invariant one theorem. Computation of cobordism groups - unoriented, oriented, complex. Formal group laws and Quillen's theorem.
Topic 6 (SNB) 6 (90 minute) Free loop space fibrations, string topology. Loop spaces of surfaces and Birkhoff's curve shortening. Geodesics as critical points. Energy functional on the free loop space. Bumpy metrics and Morse-Bott functions. Gromoll-Meyer's result and Gromov's conjecture.

 

Time-Table

    Lecture 1 Tea Lecture 2 Lunch/ Library consultation Tutorials Tea Tutorials Snacks
Day Date 9.30- 11.00 11.00 - 11.30 11.30 - 13.00 13.00-15.00 15.00 - 16.00 16.00- 16.30 16.30-17.30 17.30 – 18.00
Monday 13-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Tuesday 14-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Wednesday 15-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Thursday 16-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Friday 17-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Saturday 18-06-2016 Topic 1   Topic 2   On topics 1 & 2   On topics 1 & 2  
Sunday 19-06-2016 Break   Break   Break   Break  
Monday 20-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Tuesday 21-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Wednesday 22-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Thursday 23-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Friday 24-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Saturday 25-06-2016 Topic 3   Topic 4   On topics 3 & 4   On topics 3 & 4  
Sunday 26-06-2016 Break   Break   Break   Break  
Monday 27-06-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  
Tuesday 28-06-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  
Wednesday 29-06-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  
Thursday 30-06-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  
Friday 01-07-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  
Saturday 02-07-2016 Topic 5   Topic 6   On topics 5 & 6   On topics 5 & 6  

 

References:

  1. Marvin J. Greenberg and John R. Harper - Algebraic Topology – a first course, Benjamin, 1981
  2. Edwin H. Spanier - Algebraic Topology, Tata-Mc Graw Hill
  3. Allen Hatcher - Algebraic Topology, 2001
  4. Robert M. Switzer - Algebraic Topology- Homotopy and Homology, Sringer-Verlag, 1976.
  5. R.E. Mosher and M.C. Tangora – Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, 1968.
  6. Adams, J.F. Stable Homotopy and Generalised Homology.
  7. Complex cobordism and stable homotopy groups of spheres, D. Ravenel. Academic Press.
  8. Milnor - Morse Theory.
  9. Gromoll, Meyer - Periodic geodesics on compact Riemannian manifolds.
  10. Bott - On the iteration of closed geodesics and Strum intersection theory.
  11. 11. Further reading:
  12. Gromoll, Meyer - On differentiable functions with isolated critical points.
  13. Abraham - Transversality in manifold of mappings.
  14. Anant R. Shastri: Basic Algebraic Topology, CRC Press, Taylor and Francis group, 2013.
  15. MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007. Surgery theory was initiated by Milnor in the period 1956-1960 for the purpose of classifying smooth manifolds that are homotopy equivalent to a sphere of dimension \geq 5. In addition to published research articles, this volume contains four sets of notes for lectures he gave in this period; these lectures had remained in mimeographed form until their formal publication in this volume. They still provide excellent instruction in differential topology.
  16. MILNOR J., Morse theory, (based on lecture notes by M. Spivak and R. Wells) Annals of Mathematical Studies No. 51, Princeton University Press, 1963.
  17. MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
  18. MILNOR J. and STASHEFF J., Characteristic classes, Annals of Mathematics Studies No. 76, Princeton University Press, 1974. This monograph is based on more succinct mimeographed notes by Milnor and Stasheff for lectures of Milnor given in 1959 and still available at <http://www.maths.ed.ac.uk/~aar/papers/milnorcc.pdf>.
  19. Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
  20. Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
  21. STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.

There are many more new books and resources in the internet.

Actual Participants 

 

Sr SID Full Name Gender Affiliation Position in College/ University University /Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A
1 7379 Mr. Lokenath Kundu Male Ramakrishna Mission Vivekananda University PhD Visva-Bharati 2013
2 7542 Mr. Aritra Bhowmick Male Indian Statistical Institute, Kolkata PhD Indian Institute of Technology, Kanpur 2015
3 7720 Mr. Ripan Saha Male IISER Mohali PhD IIT Kanpur 2013
4 7808 Mr Pravakar Paul Male Indian Statistical Institute MSc Student Indian Statistical Institute Awaiting Result
5 7829 Mr. Satyendra Kumar Mishra Male IIT KANPUR PhD IIT KANPUR 2013
6 7831 Mr. Abhishek Sarkar Male IIT, KANPUR PhD Student VISVA-BHARATI 2013
7 7876 Mr Rishi Kumar Male IIT Kanpur Project Associate IIT Kanpur 2015
8 7941 Mr. Sourav Kanti Patra Male Ramakrishna Mission Vidyamandira PhD Student Chennai Mathematical Institute 2015
9 7945 Mr. Bhaskar Palit Male Ramakrishna Mission Vidyamandira PhD Ramakrishna Mission Vidyamandira 2015
10 8002 Ms Pinka Dey Female IISER MOHALI PhD JADAVPUR UNIVERSITY, WEST BENGAL,KOLKATA 2014
11 8023 Mr. Suraj Prakash Yadav Male IIT Kanpur MSc Student IIT Kanpur Awaiting Result
12 8059 Mr. Abhra Abir Kundu Male Indian Statistical Institute MMath Student Indian Statistical Institute M.Math Awaiting Result
13 8084 Mr. Arijit Mukherjee Male Hyderabad Central University PhD University of Calcutta 2014
14 8126 Mr. Anubhav Mukherjee Male Indian Statistical Institute M-Math Student Indian Statistical Institute Awaiting Result
15 7648 Mr. Chinglensana Phanjoubam Male North Eastern Hill University Asst. Prof. Delhi University 2006
16 7649 Mr. Sainkupar Marwein Mawiong Male North Eastern Hill University Shillong Asst. Prof. North Eastern Hill University 2004
17 7672 Mr. Dipen Ranjan Panging Male North-Eastern Hill University M.Phil Tezpur University 2012
18 8066 Ms Swathi Krishna Female IISER Mohali PhD student University of Hyderabad 2013
19 8119 Mr. Ranendu Adhikary Male Jadavpur University PhD Jadavpur University 2013
20 7674 Mr. Sudhir Kumar Male IISER Bs-Ms student IISER Pune Awaiting Result

 


 How to reach

  1. Come to Guwahati Airport/Railway Station by Air/Train.
  2. Accommodation is booked in the New NEHU Guest House, NEHU Campus, Shillong-793022.
  3. Enter the campus by Gate no. 2 and at the gate ask the security guards for the direction to the New NEHU Guest House.
  4. Take a shared Taxi from Airport to NEHU campus, or take a shared Taxi/ Sumo from the Railway station to NEHU Campus. 
    (Most taxis go to town and NEHU  Campus is out of the way, so you may have to take a local taxi from town to the NEHU Campus, which will charge about Rs. 200/-. Sometimes the taxi from Guwahati agrees to drop you at New Guest House at the NEHU Campus for this they may charge some extra money say Rs. 150 to 200 in addition to usual fare)
  5. It may be raining some times, so carry umbrellas/raincoats. On a rainy day temperature drops so carry at least a half sweater.
  6. Important local telephone/mobile No.s :
  • 0364-2725800 (New NEHU Guest House),
  • 09863020684 (Angom Tiken Singh)
  • 09863022257 (Sainkupar Mawiong)
  • 09402951147 (Ph Chinglensana)