IST Topology (2017)
Venue:  IIT, patana, Bihar 
Date:  1st, May 2017 to 13th, May 2017 
School Convener(s) 

Name  Dr. Ashish Kumar Upadhyay  Dr. Om Prakash 
Mailing Address  Associate Professor, Department of Mathematics, IIT Patna 
Associate Professor, Department of Mathematics, IIT Patna 
Sr.n  Name and Affiliation  Detailed syllabus  
1  Prof. Ramji Lal, Adjunct Professor, HRI 
The lecture series was based on Geometry and Calculus. The lectures were started with some elementary definitions of calculus and briefly reviewed Calculus of Several Variables, Linear Algebra Curves and Isoperimetric Inequality. Then move forward to the definition of Surface and some properties of surfaces. Some good examples were introduces and discussed briefly in the first two lectures practice problems were also given. In the Tutorial session, some problem had been solved and some problems were discussed. In the next few lectures Gauss Map, Principle Curvatures and Gauss Bonnet Theorem were introduced and related results were discussed and some examples introduces. In the tutorial sessions, more problems were solved based on these topics for a good understanding of these topics. All the students took part in the tutorial session to solve the given problems.Hyperbolic Geometry has been introduced in the last class and some related results and examples were discussed and regarding problems were given and solved in tutorial session and tutorial were took place as planned. 

2  Prof. S. Deo Honorary Visiting Professor, HRI Allahabad 
These lecture series were mainly based on Homology Theory. The lectures were started with Simplicial Homology. Simplex Simplicial Complex, Polyhedra were introduced in the first lecture. Various examples were presented in the class and practice problems were also given. In the tutorial session some problems were solved regarding on the properties of Simplicial complexes and polyhedral complexes. In the next few lectures triangulation, orientibility were introduced. After that simplicial homology was introduced and some problems based on that were solved for better understanding of the topic. In the tutorial, more technical problems were solved, fellow students took part in the session to solve the assigned problems. Browers Fixed Point Theorem, Leftschetz Fixed Point Theorem, BorsukUlam Theorem and similar important topics were presented nicely using the homology theory. In the last part of the week, singular homology was introduced and related problems were given and Tutorials were organized as planned.Participants enjoyed these lecture series. They took part solving the assignment problems. They used to come up with their own conceptual ideas to solve the assignment. 

3  Prof. A R Shastri, Retd. Professor, IIT Bombay, Mumbai 
Prof. A, R. Shastri got valuable feedback from participants about the discussion of five days.Then he started his lectures on Saturday and soon found out that the participants have a poor mathematical background but all of them were making sincere attempts. After spending some time on `What is Topology?' and `What is Algebraic Topology?', he introduced the notion of homotopy , relative homotopy, homotopy equivalences, contractible spaces, a brief review of quotient topology, cone construction and some other examples etc. Then the notion of Fundamental group, and computation of the fundamental group via the exponential map were discussed. Later, he introduced covering spaces, unique path lifting property and homotopy lifting property, relation with fundamental group, actions of the group on topological space and their relation with covering spaces, and plenty of related examples. Often the discussion hours went far beyond the stipulated time because of the enthusiasm of the participants. We also had sessions of making Platonic solids, in which he discussed the ever important notion of Euler characteristic. He engaged six lectures of 1.5 hours each and as many hours of tutorial. We had a session in which the participants made Mobius bands and examined critically. Also a session was organized in which participants made Platonic solids and learned the important concept of Euler characteristic. 

4  Prof. A Tiken Singh, NEHU, Shillong 
This Lectures series were based on Smooth Manifolds. The lectures were started with the basic definition of topological space and some topological properties of these spaces and then move forward to what is concept of Manifolds? Some examples of Manifolds, definition of Smooth Manifolds and some examples of Smooth Manifolds were introduced in the first two lectures. Various examples were presented in the class and practice problems were also given. In the Tutorial session, some problems had been solved to find that the given topological space weather Smooth Manifolds or not. In the next few lectures Smooth functions, Atlas, Smooth Atlas, Maximal Atlas were introduced.After that Tangent Spaces, Tangent Bundle, Regular and critical values, Immersions, Submersions, Embeddings, Submanifolds were introduced and some problems based on that were solved for better understanding of the topic. In the tutorial sessions, more problems were solved based on these topics for a good understanding of these topics, all the students took part in the tutorial session to solve the given problems. At last Topological Groups and Lie Groups were introduced very nicely and related problems were given and solved in tutorial session and tutorial organized as planned. Every participant enjoyed these lectures. They participated solving the assignment problems. 
Time Table
Registration :May.01,2017, 08.30AM–09:00AM
Inaugural Session :May.01,2017, 09.00AM–09:30AM
High Tea :May.01,2017, 09.30AM–10:50AM
Venue forlectures :Tutorial Block, IIT Patna
Day 
Date 
Lecture 1 9:30 – 11:00 

Lecture 2 11:30  1:00 

Tutorial 14:30 – 15:30 

Tutorial 
1 
01/05/2017 
SD (11:00 – 12:00) 
T 
RL (12:00 – 01:00) 
L 
SD, AP,TB 
T 
RL, TB, AP 
2 
02/05/2017 
SD 
RL 
SD, AP, TB 
RL, TB, AP 

3 
03/05/2017 
SD 
RL 
SD, AP, TB 
RL,TB, AP 

4 
04/05/2017 
SD 
RL 
SD, AP, TB 
RL, TB, AP 

5 
05/05/2017 
SD 
RL 
SD, AP, TB 
RL, TB, AP 

6 
06/05/2017 
ARS 
TS 
ARS, AP, TB 
TS, TB, AP 

7 
Sunday 







Day 
Date 
Lecture 1 

Lecture 2 

Tutorial 

Tutorial 
7 
08/05/2017 
ARS 
T 
TS 
L 
ARS, SP, DB 
T 
TS,DB, SP 
8 
09/05/2017 
ARS 
TS 
ARS, SP, DB 
TS,DB, SP 

9 
10/05/2017 
ARS 
TS 
ARS, SP, DB 
TS,DB, SP 

10 
11/05/2017 
ARS 
TS 
ARS, SP 
TS, SP 

11 
12/05/2017 
ARS 
TS 
ARS, SP 
TS, SP 

12 
13/05/2017 
ARS 
TS 
ARS, SP 
TS, SP 
SD : Prof. S. Deo, Honorary Visiting Professor, HRI Allahabad
RL : Prof. Ramji Lal, Adjunct Professor, HRI Allahabad.
ARS : Prof A. R.Shastri, Retd Professor, IIT Bombay, Mumbai
TS : Prof. A.Tiken Singh, NEHU, Shillong
AP : Mr. Abhyendra Prasad, Research Scholar IIT Patna
TB : Mr. Tushar Bag, Research Scholar IIT Patna
SP : Mr. Sachin Pathak, Research Scholar IIT Patna
DB : Mr. DebasishBhowmick, Research Scholar IIT Patna
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. Degree Date 
1  11548  Ms Gunjan Singh  Female  University Of Delhi  PhD  M.J.P.R.U. Bareilly  2009  
2  11576  Dr. Avinoy Paul  Male  Cachar College  Asst. Prof.  Gauhati University  2004  7/1/2014 
3  11633  Ms. Barkha Kashyap  Female  Department of mathematics , university of delhi  Research scholar  Indraprastha college for women  2013  
4  11711  Mr. Tushar Bag  Male  IIT Patna  PhD Student  IIT Kanpur  2014  
5  11728  Dr. Asif Khan  Male  Aligarh Muslim University  Asst. Prof.  M.Sc  2005  12/16/2013 
6  11920  Dr Avijit Sarkar  Male  University of Kalyani  Asst. Prof.  University of Kalyani  2002  
7  12076  Mr. Chaitanya G K  Male  NITK Surathkal  PhD Student  Mangalore University  2014  
8  12125  Dr. Rajesh Pratap Singh  Male  Central University of South Bihar  Asst. Prof.  Kumaun University of Nainital  1999  5/26/2010 
9  12367  Dr. Suman Kumar  Male  Banasthali University  Asst. Prof.  University of Hyderabad  2010  11/22/2016 
10  12368  Dr. Malik Rashid Jamal  Male  Integral University  Asst. Prof.  Aligarh Muslim University, Aligarh  2005  12/31/2013 
11  12383  Mrs. Ramdayal Singh Kushwaha  Male  Banasthali University  Asst. Prof.  IIT Kharagpur  2010  
12  12556  Dr Shine C Mathew  Male  St Berchmans College Changanacherry  Asst. Prof.  Mahatma Gandhi University Kottayam  2004  2/5/2012 
13  12640  Mr. Rushabh Bhagwati Parikh  Male  M.S.University  MSc Student  M.S.University  
14  12797  Mr. Ram Krishna Verma  Male  IIT Patna  Ph.D  University Of Lucknow, Lucknow  2014  
15  12822  Mr. Ravindra Kumar  Male  IIT PATNA  PhD  IIT Roorkee  2013 
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