AIS Differential Topology (2018)

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Venue: North-Eastern Hill University,  Shillong, Meghalaya
Date:  2nd, Jul 2018 to 21st, Jul 2018

 

School Convener(s)

Name Himadri Kumar Mukerjee Angom Tiken Singh
Mailing Address Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-793022, Meghalaya, India.
Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-793022, Meghalaya, India.

 


 Speakers with their affiliations:

Sr. Speaker Affiliation
1 Prof. Hemant Singh University of Delhi
2. Prof. Anant R. Shastri IIT Bombay
3 Prof. Shilpa Gondhali BITS, Pilani, Goa
4 Prof. Amiya Mukherjee ISI, Kolkata
5 Prof. Angom Tiken Singh NEHU Shillong
6 Prof. Himadri Kumar Mukerjee NEHU Shillong
AM - Amiya Mukherjee (ISICAL) ARS - Ananta R. Shastri (IITB) ATS - Angom Tiken Singh (NEHU)
SSG - Shilpa Suresh Gondhali, BITS, Goa HKM - Himadri Kumar Mukerjee (NEHU) HKS - Hemant Kumar Singh (DU)

 


 Syllabus:

Speaker Lectures Detailed Syllabus
Topic 1 (HKS) 6
(90 minute)
Topological manifolds, differentiable structures, smooth manifolds, smooth functions and smooth maps, tangent vectors and tangent space at a point, derivatives of a smooth map, inverse, implicit and rank theorems, submanifolds, regular and critical points and values, Sard’s theorem (special cases), immersion and submersion, inverse image of regular values. (First week)
Topic 2 (ATS) 6
(90 minute)
Whitney’s embedding theorem, transversality, inverse image of transverse submanifolds, transversality theorem, vector bundles, tangent and normal bundles, Grassmann manifolds, universal bundles, universality of the universal bundle.
Topic 3 (HKM) 6
(90 minute)
Thom’s cobordism theory, definition of cobordism groups, Thom spaces, Thom’s isomorphism theorem.
Topic 4 (SSG) 6
(90 minute)
Review of singular homology of spaces and pairs, Mayer-Vietoris sequence, adjunction spaces, cell complexes, singular cohomology, calculation of (co)homology groups of cell complexes and standard manifolds, Cup and cap products.
Topic 5 (AM) 6
(90 minute)
Local (co)homology of manifolds, orientation covers, orientation of manifolds, fundamental class of a compact orientable manifold, the Poincare, Alexander and Lefschetz duality theorems (statements, explanations and applications only).
Topic 6 (ARS) 6
(90 minute)
Geometric and algebraic intersection theory, the Lefschetz fixed point theorem, the Hopf index theorem, framed manifolds, framed cobordism, Hopf degree theorem.

 


 Time Table

    Lecture 1 Tea Lecture 2 Lunch Tutorials Tea Tutorials Teaand Snacks
Day Date 9.30 -11.00 11 -13 11.30 -13.00 13 - 15 15.00 - 16.00 16 -16.30 16.30 – 17.30 17.30
Monday 02-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1), HKS,ATS,HKM   Tutorial (Topic 4), SSG,ATS,HKM  
Tuesday 03-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1), HKS,ATS,HKM   Tutorial (Topic 4), SSG,ATS,HKM  
Wednesday 04-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1), HKS,ATS,HKM   Tutorial (Topic 4), SSG,ATS,HKM  
Thursday 05-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1), HKS,ATS,HKM   Tutorial (Topic 4), SSG,ATS,HKM  
Friday 06-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1), HKS,ATS,HKM   Tutorial (Topic 4), SSG,ATS,HKM  
Saturday 07-07-2018 Topic 1 (HKS)   Topic 4 (SSG)   Tutorial (Topic 1)   Tutorial (Topic 4), SSG,ATS,HKM  
Sunday 08-07-2018 Break  
Monday 09-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2), ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, SSG  
Tuesday 10-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2), ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, SSG  
Wednesday 11-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2), ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, SSG  
Thursday 12-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2), ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, SSG  
Friday 13-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2), ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, ARS  
Saturday 14-07-2018 Topic 2 (ATS)   Topic 5 (AM)   Tutorial (Topic 2),ATS,HKM,SSG   Tutorial (Topic 5), AM, HKM, ARS  
Sunday 15-07-2018 Break
Monday 16-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  
Tuesday 17-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  
Wednesday 18-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  
Thursday 19-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  
Friday 20-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  
Saturday 21-07-2018 Topic 3 (HKM)   Topic 6 (ARS)   Tutorial (Topic 3), HKM, ATS, ARS   Tutorial (Topic 6), ARS, ATS, SSG  

References:

  1. BREDON G., Topology and Geometry, Springer 1993. This book introduces many of the tools needed for manifold surgery.
  2. GAULD David B., Differential Topology: An Introduction, Dover Publications, 2006.
  3. GREENBERG and HARPER, Algebraic Topology – a first course, 1981.
  4. GUILLEMIN Victor and POLLACK Alan, Differential Topology, AMS, 2010.
  5. HATCHER : Algebraic Topology, 2001.
  6. HIRSCH M.W., Differential Topology, Springer-Verlag, 1997.
  7. HUSEMULLER Dale, Fibre bundles, GTM-20, Springer, 1994.
  8. KOSINSKI, A.A., Differential manifolds, Academic Press, 1993 and Dover publications, 2007.
  9. MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007.
  10. MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
  11. Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
  12. Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
  13. Shastri, A.R., Basic Algebraic Topology, CRC Press, 2013.
  14. STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
  15. WALLACE A.H., Differential Topology: First Steps, Dover Books on Mathematics, 2006.

List of Participants

Sr. Name Affiliation
1. Ms. Anju Kumari (F) University of Delhi
2. Ms. Roma Pathak (F) Dayalbagh Educational Institute
3. Ms. Arifa Naz (F) Aligarh Muslim University
4. Ms. Umme Tuba (F) Aligarh Muslim University
5. Mr. Umar Mohammad Khan (M) Aligarh Muslim University
6. Mr Jasraj Anil Date (M) Visvesvaraya NIT
7. Mr. Debanil Dasgupta (M) Indian Statistical Institute
8. Mr. Ajay Raj (M) IISER Tirupati
9. Mr Abhishek Shrivastava (M) Aligarh Muslim University
10. Mr. Gaurav Kumar (M) University of Delhi
11. Mr Kohinoor Ghosh (M) ICTS-TIFR
12. Mr. Rampao K. Sangma NEHU, Shillong
13. Mr. Saikat Panja (M) IISER Pune
14. Mr Navnath Balasaheb Daundkar (M) Chennai Mathematical Institute
15. Mr. Joji Benny (M) IIT, Madras
16. Mr. Arindam Das (M) BITS PILANI
17. Ms Ponaki Das NEHU, Shillong
18. Mr. Bikramjit Kundu (M) RKMV Edu. and Res. Institute
19. Mr. Sachchidanand Prasad (M) IISER Kolkata
20. Mr. Nilkantha Das (M) NISER Bhubaneswar
21. Mr Anantadulal Paul (M) NISER Bhubaneswar
22. Dr. Avinoy Paul (M) Cachar College
23. Mr. Arijit Mukherjee (M) University Of Hyderabad
24. Mr Siddharth Panigrahi (M) Indian Statistical Institute
25. Mr. Puneet Kumar Gupt (M) University of Lucknow
26. Mr. Yashasvi Aulak (M) NISER Bhubaneswar

 


How to reach

Travel , Lodging and Boarding Instructions:

  • Accommodation is booked at the Old NEHU Guest House, NEHU Campus, Shillong, with effect from July 01 to July 22, 2018.
  • After reaching Guwahati Airport / Railway station take a shared taxi or Sumo to Shillong.
  • Normally the shared taxis (Rs. 500/- / Rs. 300/- per head) or Sumos (Rs. 150/- per head) drop you at the Shillong City, but sometimes they agree to drop you at NEHU on some extra charge.
  • If they drop you at the Shillong city, you have to hire a local taxi to come to NEHU, which may charge you Rs. 200/- or so.
  • If they agree to drop you at NEHU with extra charge, may be around Rs. 200/-, well and good.
  • On reaching NEHU Gate no. 2 you instruct the Taxi / Sumo to drop you at Old NEHU Guest House, the security personnel at the gate can give directions to go to the Old Guest House.
  • Carry an umbrella and/or a raincoat because you will be at the peak of the rainy season.
  • Bring a pullover because temperature varies quite a bit.
  • Although the guest house provide mosquito repellant (dependent on electricity), it is always good to carry your own mosquito repellant (independent of electricity), in case power goes off in the night. But don’t get overmuch alarmed because there are not too many mosquitoes in Shillong.
  • Venue of the lectures will be Physical Sciences Lecture complex, NEHU, Shillong,which is at a ten minutes walking distance from the Old NEHU Guest house.
  • Breakfast and dinner will be served at the guest house. Lunch on Sundays will be served at the guest house with prior notice.
  • On working days during July 02 to July 21, 2018, lunch and session tea/coffee shall be served at the lecture complex itself.

Important telephone numbers:

  • Old NEHU guest House : 0364-272- 2057/2058
  • Himadri Kumar Mukerjee : 09435118439
  • Angom Tiken Singh . : 09863020648
  • Ardeline Mary Buhphang: 09436161409
  • Sainkupar Mn Mawiong: 09863022257
  • Ph. Chinglensana : 09402951147
  • Michael George (Liaison Officer at the Guwahati Airport): 09435547335 / 09864018246