ISL Topology (2013)

Venue:  CEMS, Almora.
Dates: 30th Sept to 12th Oct ,2013


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


School Convener(s)

Name Anant R. Shastri         H.S.Dhami 

Sanjay Kumar (Pant)
Mailing Address Professor,
Department of Mathematics,
IIT Bombay, Powai,
Professor and Head,
Department of Mathematics,
SSJ Campus, Kumaun University
Almora, Uttarakhand-263601
Associate Professor,
Department of Mathematics,
Deen Dayal Upadhyaya College,New Delhi-110015


Speakers and Syllabus 

 Central Theme:– Topology as the study of Convergence and Continuity, with emphasis on Examples and Exercises.


  • Confirmed speakers/Instructors

(i) B. V. Rao , (Adjunct Professor), CMI, Chennai
(ii) S. M. Srivatava , ISI, Kolkata
(iii) H. Sarbhadhikari , ISI, Kolkata
(iv) Sanjay K. Pant , DDU College, DU and CEMS, Almora
(v) Anant R.Shastri , IIT Bombay

  • Special Lectures/Guest Speakers

(i) M S Raghunathan, NCM,IIT Bombay
(ii) Ravi S Kulkarni, BIM and Pune University

  • Course Associates

(i) Pawan K. Tamta, Kumaun University, Almora
(ii) Hemlata Pande, Kumaun University, Almora
(iii) Kushal Lalwani, Delhi University
(iv) Gopal Datt, Delhi University

  • Detailed speaker-wise Syllabus
    Part-I Speaker: B. V. Rao (BVR) (6 lectures)

1. Major examples: R, Rn, [0, 1], [0, 1]n.
2. Spaces of continuous functions with various notions of convergence. e.g.poitwise convergence, uniform convergence, uniform convergence on compacta (closed and bounded sets), which leads to compact-open topology. [Initially, examples should be taken from usual analysis]
3. The notion of (usual) distance on the above cited spaces, unit ball etc.
4. Can uniform convergence be covered as a usual notion of convergence using distance? As an answer, introduce metric spaces, continuous functions between them and go on to show that uniform convergence and uniform convergence on compacta of Rn can be described as a convergence using a metric.
5. Denseness of Q and Qn and applications, separability of metric space.
6. Can pointwise convergence be covered as a notion of convergence using metric? As an answer, emphasize the need to introduce a notion of continuity in terms of open sets and closed sets. Define topological spaces and show that further axioms need to be introduced to remove some garbages such as discreet and indiscreet spaces etc.
7. Show that every bounded sequence in Rn has a convergent subsequence.Prove Weierstrass’ theorem and related results.

Part-II S. M. Srivastava(SMS) (6 lectures)

8. Introduce sequential compactness.
9. Capture compactness in terms of closed sets and open sets. Heine-Borel with proof.
10. Basic properties of compact sets. Lebesgue covering lemma. Product of compact spaces.
11. Complete Metric Spaces.

Part-III H. Sarbhadhikari(HS) and Sanjay K. Pant (6 lectures)

12. Covering axioms, compactness, Lindel ̈ off.
13. I and II countability, separability.
14. Separation axioms: T0 to T5. Urysohn’s lemma and Tietze extension theorem.
15. Embedding and metrization.

Part-IV Anant R. Shastri(ARS)(6 lectures)

16. Intermediate value theorem with proof (for interval and then foe convex subset of Rn). Extract the definition of connectedness and define connected spaces. Relation with path-connectedness. Equivalence of LUB property and connectedness.
17. Quotient spaces. Separation properties under quotient topology.
18. Playing with the M ̈obius band.
19. Paracompactness and partition of unity.
20. Induced and coinduces topology, compactly generated topology and function spaces.

 Tentative Time - Table

Date  9.30-11.00 11.30-13.00 14.30-15.30


30-09-2013 BVR HS T1 T2
01-10-2013 BVR HS T3 T4
02-10-2013 BVR Hs T5 T6
03-10-2013 BVR SMS T7 T8
04-10-2013 BVR SMS T9 T10
05-10-2013 BVR SMS T11 T12
07-10-2013  ARS SMS T13 T14
08-10-2013  ARS SMS T15 T16
09-10-2013  ARS SMS T17 T18/S1
10-10-2013  ARS SKP T19 T20/S2
11-10-2013  ARS SKP T21 T22/S3
12-10-2013  ARS SKP T23/S4  Valedictory

[Abbreviations of names used in the time-table above are explained in the detailed- syllabus
Tn— nth tutorial
Sn—nth special lecture
Tea Breaks—11.00-11.30 and 15.30-16.00
Lunch Break— 13.00 - 14.30
6 October 2013 Sunday— Trip to Binsar/Jageshwar
11 October Friday— Faculty Dinner.



Selected Applicants

Sr. SID Full Name Gender Affiliation State Position University/ Institue M.Sc./ M.A Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 3889 Ms. Sandeep Kaur Female Punjabi University Punjab PhD Punjabi University Patiala 2009  
2 3958 Mr. Ankit Gupta   Male Delhi University Delhi M.phil M.Sc 2011  
3 3987 Mr Nanasaheb Mhatarba Phatangare Male Fergusson College Pune Maharashtra Asst. Professor Pune University 2006  
4 4005 Dr Manob Kumar Ghosh Male Dumkal College, Basantapur West Bengal Asst. Professor University Of Kalyani 2003 27 Nov 2009
5 4023 Mr Dilip Jain Male Gurukul Institute Of Engineering And Technology Rajasthan Asst. Professor Iit Delhi 2012  
6 4038 Ms Jita Parida Female Priyadarshini Indira Mahavidyalaya Orissa lecturer Sambalpur University 2004  
7 4055 Mr. John Paul Kharbhih Male North Eastern Hill University Meghalaya Asst. Professor North Eastern Hill University 2002  
8 4056 Mr. Sainkupar Marwein Mawiong Male North Eastern Hill University, Shillong Meghalaya Asst. Professor North Eastern Hill University, Shillong 2004  
9 4087 Mr. David Rosangliana Male Govt. Serchhip College Mizoram Asst. Professor North Eastern Hill University, Shillong 2007  
10 4120 Ms. Garima Tomar Female Central University Of Rajasthan Rajasthan PhD Research Schola Mdu, Rohtak, M. Sc. Mathematics 2004  
11 4133 Mr. Sukhdev Singh Male Lovely Professional University, Punjab Punjab Lecturer Panjab University, Chandigarh / Govt. College Hoshiarpur 2008  
12 4138 Mr. Shivaji Ashok Tarate Male New Arts,Commerce And Science College Ahmednagar Maharashtra Asst. Professor Pune 2002  
13 4141 Dr. Sunil C Mathew Male St. Thomas College Palai Kerala Asst. Professor Mahatma Gandhi University, Kottayam. 1994 17 Nov 2003
14 4148 Dr. N. M. Madhavan Namboothiri Male Government College, Kasaragod Kerala Asst. Professor M. G. University, Kerala 2000 30 Nov 2012
15 4177 Mr. Vishal Gupta Male Swami Vivekanand Subharti Universi Uttar Pradesh Asst. Professor Iit Guwahati 2010  
16 4189 Ms. Kondapalli Krishna Sowmya Female Osmania University College For Women Andhra Pradesh Asst. Professor National Institute Of Technology, Calicut 2010  
17 4239 Mr Shangerganesh Lingeshwaran Male Bannari Amman Institute Of Technology Tamil Nadu Asst. Professor Bharathiar University 2008 20 Feb 2013
18 4249 Dr. Ghanshyam Singh Thakur Male Maulana Azad National Institute Of Technology ,Bhopal Madhya Pradesh Asst. Professor Pt Ravi Sankar University Raipur C.G. 2003 16 Jun 2009
19 4210 Mr. Vinay Gautam Male Indian School Of Mines Jharkhand Ph.D M.Sc 2008  
20 4223 Ms. Rana Noor Female Banaras Hindu University Uttar Pradesh Research Scholar Banaras Hindu University, Varanasi 2010  
21 4224 Mr. Vijay Kumar Yadav Male Indian School Of Mines, Dhanbad Jharkhand Ph.D Dr. Ram Manohar Lohiya Avadh University, Faizabad 2003  
22 4234 Mr Dillip Kumar Pradhan Male National Institute Of Technology Orissa PhD Sambalpur University 2011 1 Jan 2013
23 4241 Mr. Varun Jindal Male IIT Delhi Delhi PhD Student IIT Delhi 2010  
24 4242 Ms. Anubha Jindal Female IIT Delhi Delhi PhD Student IIT Delhi 2010  
25 3876 Priyambada Tripathi Female University of delhi Delhi PhD Student Banaras Hindu university 2008