AIS h -Principle (2017)

Venue: Indian Statistical Institute,  Kolkata, West Bengal
Date:  22nd, May 2017 to 10th, Jun 2017



School Convener(s)

Name Mahuya Datta A. Adimurthi Goutam Mukherjee
Mailing Address Indian Statistical Institute,
203 BT Road, Kolkata-700108.
TIFR Centre For Applicable Mathematics,
Chikkabommsandra, Bangalore 560065.
Indian Statistical Institute,
203 BT Road, Kolkata-700108.



Speakers and Syllabus

Partial differential equations and more general relations which appear in Topology and Geometry are very often under-determined and have many solutions. This is in contrast with the PDE’s which arise in Physics where the solutions are few. This phenomenon was observed quite early in the work of Nash-Kuiper on C 1 -isometric embeddings (1954) and Smale-Hirsch theory of immersions (1965). Around 1969, Gromov introduced the theory of h-principle which brought all these work under the same framework. The theory of
h-principle addresses the question of existence and homotopy classification of solutions of geometric partial differential relations. In the AIS, two important techniques of this theory - Holonomic approximation and convex integration - was studied in details and several important applications have been discussed, including Nash-Kuiper theorem and Smale-Hirsch theorem. Convex integration theory has some apparent similarity with Filippov’s relaxation theory for Differential inclusions (of first order) - though the techniques are completely different. Some analytical techniques were studied in parallel with convex integration theory. Basic Differential Topology and theory of vector bundles and fibre bundles were reviewed during the first week. Preliminaries of PDE to develop the relaxation theory were discussed in the third week.

Week I: 22 May - 27 May, 2017
Lecture 1 Differentiable manifolds. Examples. Smooth maps. (GM)
Lecture 2 Tangent space. Derivative of smooth maps. (SB)
Lecture 1 Immersions, embeddings. Submersions. Transversal maps. (SB)
Lecture 2 C k -topology on function spaces. Stability. (GM)
Lecture 1 Vector bundles. Tangent, cotangent and exterior bundles. (GM)
Normal bundle of a submanifold. Sections of vector bundles.
Lecture 2 Vector fields and flows. Lie bracket of vector fields. Frobenius theorem. (SB)
Lecture 1 Differential forms. Exterior differentiation. Interior product. (SB)
Lie derivation. Cartan formula.
Lecture 2 Jet bundles with emphasis on 1-jet bundles. Topology of jet bundles. (AM)
Lecture 1 Fibre bundles, covering homotopy property. Existence of global section. (GM)
Lecture 2 Jet bundles with emphasis on 1-jet bundles. Topology of jet bundles. (AM)
Lecture 1 Topology on function spaces. Fine topology and Baire property.
Denseness of C k functions in C 0 function space. (MD)
Lecture 2 Genericity. Thom tranversality theorem. Applications. (AM)
Week II: 29 May - 3 June, 2017
Lecture 1  Introduction to h-principle (MD)
Lecture 2  Holonomic approximation theorem. Statement only.Application to differential forms. (DP)
Lecture 1  Existence of symplectic and contact structures on open manifolds. (DP)
Lecture 2  Gromov’s theorem on open manifolds. Applications. (AM)
 Lecture 1 Proof of Holonomic approximation theorem. (DP)
 Lecture 2  Proof of Gromov’s theorem. (AM)
 Lecture 1  Closed manifolds. Microextension trick. Smale Hirsch theorem. (MD)
 Lecture 2  Introduction to contact and symplectic manifolds I. (DP)
 Lecture 1 Introduction to contact and symplectic manifolds II. (YD)
 Lecture 2 Relaxation of hypothesis of Gromov’s theorem and further extensions. (MD)
Lecture 1 Iso-contact immersion theorem. (MD)
Week III: 5 June - 10 June, 2017
Lecture 1  1-dimensional convex integration I. (MD)
Lecture 2  Differential inclusions I (AA)
Lecture 3 Differential inclusions II (AA)
Lecture 1  1-dimensional convex integration II. (MD)
Lecture 2  Convex analysis I (KS)
Lecture 3 Convex analysis II (KS)
Lecture 1  Convex integration theory for first order relations. (MD)
Lecture 2  Convex analysis III (KS)
Lecture 3  Differential inclusions III (AA)
 Lecture 1  Open ample relations and h-principle. (MD)
 Lecture 2  Convex analysis IV (KS)
 Lecture 3  Differential inclusions IV (AA)
 Lecture 1  Nash-Kuiper theorem on isometric C 1 -embeddings I. (MD)
 Lecture 2  Convex analysis V (KS)
 Lecture 3  Differential inclusions V (AA)
Lecture 1 Convex analysis VI (KS)
Lecture 2 Differential inclusions VI (AA)
  • Goutam Mukherjee (GM)
  • Samik Basu (SB)
  • Amiya Mukherjee (AM)
  • Mahuya Datta (MD)
  • Dishant Pancholi (DP)
  • Yash Deshmukh (YM)
  • K. Sandeep (KS)

List of speakers
1. Adi Adimurthi, TIFR Centre for Applicable Mathematics
2. Samik Basu, RKMVU, Belur
3. Mahuya Datta, ISI Kolkata
4. Amiya Mukherjee, ISI, Kolkata
5. Goutam Mukherjee, ISI, Kolkata
6. Dishant Pancholi, IMSc, Chennai
7. K. Sandeep, TIFR Centre for Applicable Mathematics

Yash Uday Deshmukh, a participant from CMI, Chennai gave one lecture in the AIS.

 List of tutors. The speakers themselves conducted their tutorials. Mr. Aritra Bhowmick, ISI, Kolkata conducted some of the tutorials in the first and second week.



Actual Participants 


Sr.n SID Full Name Gender Affiliation Position in College/ University University/Institute M.Sc./ M.A. Year of Passing M.Sc./M.A Ph.D. Degree Date
1 10809 Mr Pravakar Paul Male Indian Statistical Institute M.Math Student Indian Statistical Institute Awaiting Result  
2 10961 Mr Firoj Sk Male IIT KANPUR PhD IIT KANPUR 2015  
3 11029 Mr. Aritra Ghosh Male IISER Bhopal PhD 1st year Ramakrishna Mission Vivekananda University 2016  
4 11046 Mr Venkata Sai Narayana Male IIT Bombay MSc Student IIT Bombay Awaiting Result  
5 11118 Mr Anantadulal Paul Male NISER PhD University of Pune 2015  
6 11427 Mr. Kiran Kumar A S Male IISER Thiruvananthapuram Int Msc student IISER Thiruvananthapuram    
7 11467 Mr. Prashanta Garain Male IIT Kanpur PhD IIT Kanpur 2014  
8 11535 Mr Yash Uday Deshmukh Male CMI MSc Student CMI    
9 11541 Mr. Ranadip Gangopadhyay Male Banaras Hindu University PhD Netaji Subhas Open University 2013  
10 11552 Mr. Abhra Abir Kundu Male Indian Statistical Institute M.Math Student Indian Statistical Institute Awaiting Result  
11 11572 Mr. Chandrahas Piduri Male IISER Kolkata MS IISER Kolkata Awaiting Result  
sr.n SID Full name Gender Affiliation                           
Position in College/ University University/Institute M.Sc./M.A.                      
Year of Passing M.Sc./M.A Ph.D. Degree Date
12 11733 Mr. Mrinmay Biswas Male IISER-Kolkata PhD TIFR-CAM, Bangalore 2009  
13 11735 Mr. Aritra Bhowmick Male ISI, Kolkata PhD IIT Kanpur 2015  
14 11749 Mr. Tushar Bag Male IIT Patna PhD Student IIT Kanpur 2014  
15 11930 Mr. Santanil Jana Male IISER, Kolkata MS Student IISER KOLKATA Awaiting Result  
16 11931 Mr. Sambit Senapati Male CMI BSc Student      
17 11957 Mr. Harshit Yadav Male IIT Kanpur B.S.      
18 12063 Mr. Dhriti Sundar Patra         Male Jadavpur University Senior Research Fellow IIT Bombay 2011  
19 12059 Ashis Pati   IIT, Kanpur        

Local Participants Full Name        Affiliation      
1 Debnil Dasgupta ISI, Bangalore
2 Surojit Ghosh RKMVU, Belur
3 Gobinda Sau RKMVU, Belur
4 Apurba Das ISI Kolkata
5 Aparajita Karmakar University of Burdwan


How to reach


ISI Kolkata Campus

The Kolkata campus of the Indian Statistical Institute is located in a sprawling 30-acre estate on the Barrackpore Trunk Road (B.T. Road) in the Baranagore suburb of Greater Kolkata. It consists of two approximately equal parts - the office complex and the residential complex, - separated by a public road. This road (Girish Chand Ghosh Street) connects B.T. Road with Gopal Lal Tagore Road, a road which runs along the western boundary of the main campus. The office complex bears door numbers 203, 204, and the residential complex, 205. There is a connection between the two parts of the campus through a subway; to move between the residential and office complexes.
How to Reach ISI

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.


Kolkata has a huge fleet of Black and Yellow taxis, most of which are Ambassadors. They run on Electronic meter. The Electronic meter starts at Rs. 25/-. Please ensure that the meter is started at the beginning of the journey.

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. Mention the destination as Dunlop Bridge at the pre-paid Taxi booth counter. In this case you have to pay the total fare at the taxi booth. The prepaid taxi fare is marginally higher than normal, but it is a surer and safer mode of transport.


There are a large number of buses and minibuses that pass by the Institute campus. 3, 32A, 34B/1, 78, 78/1, 201, 214, 222, 230, 234, 234/1, DN9, DN9/1, 9A, S11, S32 and GL32. AC 6, AC 23A, AC 25, AC 50, AC 54. The minibuses plying from Howrah to Nimta/Belgharia/Rathtala, from BBD Bag to DunlopBridge, from Babughat to Sodepur/New Barrackpore, and Garia to Belur Math 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 where you have to get down in order to enter the residential campus at 205 B T Road. The next Bus Stop is ISI / Statistical, which is located just outside the entrance of the main office campus at 203 B.T. Road.

Please visit for more information.