IST Topology (2017)

School Convener(s)

Speakers and Syllabus

The lecture series was based on Geometry and Calculus. The lectures were started with some elementary definitions of calculus and briefly re-viewed 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.
Every participant enjoyed these lectures. They participated in solving the assignment problems.

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, Borsuk-Ulam 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.

Prof. A, R. Shastri got valuable feed-back 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.

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.

Day

Date

Lecture 1

9:30 – 11:00

Lecture 2

11:30 - 1:00

Tutorial
(Topology)

14:30 – 15:30

Tutorial
(Diff. Geometry)
16:00 – 17:00

1

01/05/2017

SD (11:00 – 12:00)

T
e
a

RL (12:00 – 01:00)

L
U
N
C
H

SD, AP,TB

T
e
a

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
(Topology)

Tutorial
(Diff. Geometry)

7

08/05/2017

ARS

 T
e
a

TS

L
U
N
C
H

ARS, SP, DB

T
e
a

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