AIS Homotopy Theory (2019)

Venue: Indian Statistical Institute,  Kolkata, West Bengal
Date:  13th, May 2019 to 25th, May 2019


School Convener(s)

Name Samik Basu Somnath Basu
Mailing Address

Indian Statistical Institute,
203 BT Road, Kolkata -700108.

Department of Mathematics and Statistics,
Indian Institute of Science Education and Research,
Mohanpur 741246, West Bengal.


 Speakers/Tutors with their affiliations:

Sr. Name Affiliation
1 Samik Basu ISI Kolkata
2 Somnath Basu IISER Kolkata
3 Aritra Bhowmick ISI Kolkata
4 Debanil Dasgupta ISI Kolkata
5 Sachchidanand Prasad IISER Kolkata
6 Rekha Santhanam IIT Bombay
7 Debasis Sen IIT Kanpur
8 Ajay Singh Thakur IIT Kanpur



Aim of the school and syllabus

Homotopy theory has turned out to be one of the most prominent topics in Algebraic Topology starting from the 1950s. The celebrated computations of Serre and Toda of ho- motopy groups of spheres led to the systematic calculation of cobordism groups by Thom, Milnor, Wall, Stong and others. From the 1970s, it was clear that the stable homotopy cal- culations are very important and bear a lot of applications to problems in differential topology.

The homotopy theory for G-spaces (called equivariant homotopy theory), that is spaces with an action of a specific group G, has also been of great interest. One of the starting points of the subject was the spherical space form problem which was finally solved by Madsen, Wall and Thomas. The analogous statement for a product of spheres is one of the major unsolved problems in this subject called the rank conjecture. The techniques in equivariant homotopy theory also give a nice proof of the Atiyah-Segal completion the- orem, among many other such results. In the AIS, we studied homotopy theory with a view towards equivariant homotopy theory.

Generalized cohomology theories, cohomology operations and K-theory was discussed in detail in the first week, together with an introduction to Bredon cohomology. Some background in differential topology and basic homotopy theory was also reviewed. The second week comprised of three mini-courses on 1) Atiyah-Segal completion theorem, 2) Stable homotopy theory, 3) Computations in equivariant homotopy theory.

Name of the Speaker

Detailed Syllabus

Ajay Singh Thakur Vector bundles, K-Theory, characteristics classes
Debasis Sen Cohomology operations, equivariant cohomology
Aritra Bhowmick Tutorial Topic: (Basic homotopy theory)Fibrations, cofibrations, homotopy groups, cellular approximation theorem, White-head’s theorem, Hurewicz theorem, Freudenthal suspension theorem
Debanil DasguptaSachchidanand Prasad Tutorial Topic D (Basic differential topology)Differential topology, orientation, intersection theory, degree, Hopf degree theorem
Somnath Basu Atiyah-Segal completion theorem, equivariant K-theory
Rekha Santhanam Stable homotopy theory, equivariant stable homotopy theory
Samik Basu Computations in equivariant homotopy theory including π0 S0 ∼  = A (G), Segalconjecture, Sullivan conjecture


 Time Table

Date/Time 10-11:30 11:45-1:15 1:15-2:45 2:45-3:45 4:00-5:00
13/05 A1 (A. Singh) B1 (D. Sen) L
C1 D1
14/05 A2 (A. Singh) B2 (D. Sen) C2 D2
15/05 A3 (A. Singh) B3 (D. Sen) C3 D3
16/05 A4 (A. Singh) B4 (D. Sen) C4 D4
17/05 A5 (A. Singh) B5 (D. Sen) C5 D5
18/05 A6 (A. Singh) B6 (D. Sen) C6 D6


Date/Time 10-11:30 11:45-1:15 1:15-2:30 2:30-4:00 4:15-5:00
20/05 E1 (So.Basu) F1 (R.S) L
G2 (Sa.Basu) Tutorial
21/05 E2 (So.Basu) F2 (R.S) G2 (Sa.Basu) Tutorial
22/05 E3 (So.Basu) F3 (R.S) G2 (Sa.Basu) Tutorial
23/05 E4 (So.Basu) F4 (R.S) G2 (Sa.Basu) Tutorial
24/05 E5 (So.Basu) F5 (R.S) G2 (Sa.Basu) Tutorial
25/05 E6 (So.Basu) F6 (R.S) G2 (Sa.Basu) Tutorial


List of 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 Ph.D. Deg
1 24977 Mr. Suvadip Sana Male Indian Statistical Institute, Bangalore. B. Math Student.      
2 25077 Mr Pritthijit Biswas Male Chennai Mathematical Institute PhD. Ramakrishna Mission Vivekananda University 2017  
3 25172 Mr. Amit Kumar Paul Male IIT KANPUR PhD Visva Bharati 2014  
4 25219 Ms Poonam Ramnath Pokale Female IIT BOMBAY PhD Savitribai Phule Pune University 2015 13 Jul 2017
5 25220 Mr Soumyadip Thandar Male IIT Bombay PhD Pune University 2015  
6 25225 Ms. Anju Kumari Female Department of Mathematics, University of Delhi PhD student Hindu College, University of Delhi 2014  
7 25259 Dr. Shilpa Gondhali Female Birla Institute of Technology and Science, Pilani. K K Birla Goa Camus. Asst. Prof. University of Mumbai 2005 12 Jul 2012
8 25407 Mr. Adittya Chaudhuri Male IISER-TVM PhD Student IISER-TVM 2018  
9 25480 Mr. Bikramjit Kundu Male Ramakrishna Mission Vivekananda Educational and Research Institute PhD Jadavpur University 2017  
10 25534 Mr. Reebhu Bhattacharyya Male IIT, Bombay Integrated Masters Student (Final Year) IIT, Bombay Awaiting Result  
11 25559 Mr Anantadulal Paul Male School Of Mathematical Science NISER PhD University Of Pune Maharastra 2015  
12 25972 Mr. Bhargab Das Male Indian Statistical Institute MSc Student Indian Statistical Institute Awaiting Result  
13 25988 Mr Soumik Ghosh Male Indian Statistical Institute Kolkata M Math Student Indian Statistical Institute    
14 26015 Ms. Priyanka Tanaji Magar Female Department of Mathematics, IIT Bombay PhD student Pune university 2017  
15 26038 Mr. Sourav Holme Choudhury Male Institute of Mathematical Sciencea PhD Indian Statistical Institute, Kolkata 2018  
16 26070 Mr. Samiron Sadhukhan Male Tata Institute of Fundamental Research, Mumbai MSc student TIFR, Mumbai Appeared / Awaiting Result  
17 26150 Mr. Ritankar Nath Male Tata Institute of Fundamental Research Integrated PhD Tata Institute of Fundamental Research, Mumbai Awaiting Result  
18 26171 Mr. Partha Sarathi Ghosh Male Presidency University PhD Presidency University 2018  
19 26187 Mr. Prakhar Gupta Male Indian institute of science Undergrad 4th year      
20 26207 Mr Arya Abhijit Vadnere Male Chennai Mathematical Institute BSc 2nd Year      
21 26263 Mr Jitendra Rathore Male TIFR PhD IISER Mohali 2017  
22 26302 Mr. Vignesh S Male Indian Statistical Institute M Math Student Indian Statistical Institute    
23 26304 Mr Siddharth Gurumurthy Male Chennai Mathematical Institute BSc Student      
24 26353 Mr. Kumar Sannidhya Shukla Male NISER, Bhubaneswar Integrated MSc Student NISER, Bhubaneswar Awaiting Result  
25 26442 Mr. Pratit Goswami Male Indian Statistical Institute M.Math Student Indian Statistical Institute, Kolkata    
26 26474 Mr Sanjoy Chatterjee Male IISER kolkata PhD IISER Kolkata 2018  
27 26495 Ms Pinka Dey Female IISER Mohali PhD Jadavpur Unniversity 2014  
28 26506 Mr. Koushik Brahma Male IIT MADRAS PhD IIT Madras 2018  
29 26509 Dr. Swagata Sarkar Female UM-DAE Centre for Excellence in Basic Sciences Asst Professor Delhi University 2003 1 Mar 2010
30 26518 Ms Apurva Seth Female IISER Bhopal PhD IIT Kanpur 2017  
11 25547 Mr. Arkamouli Debnath Male Indian Statistical Institute Kolkata Student Indian Statistical Institute Kolkata    


How to reach

The ISI campus lies five kilometers North of the Shyambazar five-point crossing. If you are coming from the airport, drive towards Dunlop Bridge (via Nager Bazar and Chiria More). If you are coming from the downtown area or one of the railway stations, you will have to drive past Shyambazar and Chiria More. As you approach Dunlop Bridge, the boundary wall of the ISI campus will be visible on your left. The residential campus is at the Bus Stop called Bon Hooghly and the office campus is at the Bus Stop called ISI or Statistical. The Bus Stop further to the Statistical Bus Stop is Dunlop Bridge, which is an important landmark. Dunlop Bridge is also a useful way of describing the general destination when you are asking for directions.

If you miss the ISI gates and reach the Dunlop Bridge intersection, take a sharp left turn into Gopal Lal Tagore Road. The ISI campus will be on your left. The first entrance on your left leads you straight into the Library Building.
If you are coming from the airport or one of the railway stations, it is better to hire a taxi from the prepaid taxi booth. The prepaid taxi fare is marginally higher than normal (approximately Rs. 130 from the airport, Rs. 120 from Howrah, and Rs. 100 from Sealdah) but it is a surer and safer mode of transport. While hiring a prepaid taxi, mention Dunlop Bridge as your destination.

Meter information (as of December, 2004):

    Mechanical meters (an old box attached outside the cab):
    Intial Reading: Rs. 5.00
    Total Fare: Rs. 3 x meter reading
    Electronics meters (usually attached inside the cab):
    Intial Reading: Rs. 10.00
    Total Fare: Rs. 1.50 x meter reading
    Extra charges for taking a taxi to the airport: Rs. 10

The taxi driver is also supposed to carry a conversion chart with the validation date clearly marked on it.

There are a large number of buses and minibuses that pass by the Institute campus. Almost any bus that goes to Dunlop Bridge passes by the ISI gates. The buses that pass by the ISI gates are 11, 32A, 34B/1, 78, 78/1, 78/2, 78A, 78B, 78C, 201, 214, 230, 234, DN9, DN9/1, L9A, L11, L20, L20A, M1, M2, M6, S9, S11, S32, GL32, GLS32, and SS4. The minibuses plying from Howrah to Nimta/Belgharia/Rathtala, from BBD Bag to Dunlop Bridge, from Babughat to Sodepur/New Barrackpore and from Metiaburuj to Srirampur go past the ISI gates before reaching Dunlop Bridge. While riding the bus from the city, the major bus stops on your way (from South to North) would be Shyambazar, Chiria More, Sinthi More and Tobin Road. The Bon Hooghly Bus Stop is just North of the Bus Stop at Ananya Cinema. This is where you have to get down in order to enter the residential campus. The next Bus Stop is ISI/Statistical, which is located just outside the entrance of the main office campus at 204, B.T. Road.