MTH413 - Differential Topology
References :
1. An introduction to Differentiable Manifolds and Riemannian Geometry : Boothby
2. Introduction to Smooth Manifolds : Lee
Further Reading :
A. If you are interested in Lie groups then please look into one of the following:
A1. Lie groups by Adams
A2. Lie groups by Bump
A3. Differential Topology and introduction to Lie groups by Warner
B. If you are interested in differential equations please look into one of the following:
B1. Dynamical Systems by Hirsch and Smale
B2. Differential Equations by Arnold
C. If you are interested in Differential Geometry and More topology:
C1. completely read the book by Boothby which we have been following in this course
C2. Smooth Manifold by Lee
C3. Topological manifold by Lee
C4. Differential Topology by Guillemen and Pollack
C5. Calculus to Cohomology be Madsen
D. Further follow the following topics: Reimannian geometry, Hyperbolic geometry, Dynamical Systems etc.
Instructor : Dr. Anupam Singh
Schedule : Monday, Wednesday and Friday 10 AM -11 AM
Evaluation Method : 4 exams
and assignment + tutorials
Goal of this course :
a. Get use to thinking about n-dimensional spaces not just 2 or 3.
b. learn things about new geometric objects
c. learn some more group theory (Lie groups)
Proposed Course Content
Review of function of several variable, Topological Manifolds, Differentiable Manifolds, Tangent Space, Lie Groups, Vector Fields, Lie Algebras.
(if time permits) Tensors, integration, Stoke's Theorem.
03 August 2010 : Topological manifolds and examples
04 August 2010 : Example of 2 dimensional surfaces Assignment-I
06 August 2010 : Charts, Atlas and Differentiable Manifolds and examples
09 August 2010 : Tangent space in Euclidean space (intuitive approach)
11 August 2010 : Germs of functions and Tangent space Assignment-II
12-31 August 2010 : No classes as I am away to attend ICM
01 September 2010 : Tutorial
03 September 2010 : Tutorial
06 September 2010 : Tangent Space, Differential of a map
07 September 2010 : Immersion and Imbedding
08 September 2010 : Submanifold and Regular submanifold
10 September 2010 : Holiday
13 September 2010 : Examples of Submanifold
14 September 2010 : Lie groups and examples Assignment - III
15 September 2010 : (Lie) subgroups and closed subgroups
16 September 2010 : TEST - I
Remark : We are going too fast!
17 September 2010 : Tutorial (exercises from test)
20 September 2010 : Tutorial (exercises from Assignment - II)
22 September 2010 : Tutorial (exercises from Assignment - III)
24 September 2010 : Tutorial (exercises from Assignment - III)
27 September 2010 : Action of Lie group on a Manifold
29 September 2010: Orbit Spaces
01 October 2010 : Tutorials
04-08 October 2010 : Mid Sem Exam
10-16 October 2010 : Mid Sem break
18 October 2010 : Discrete subgroup of a Lie group Assignment -IV
20 October 2010 : Properly Discontinuous action
22 October 2010 : Tutorial
25 October 2010 : Tangent Spaces
27 October 2010 : Vector Field
29 October 2010 : Left Invariant vector field on a Lie group
01 November 2010: One parameter group action
02 November 2010 : Test - III
03 November 2010 : Local one parameter group action, Infinitesimal generator, Flow
05 November 2010 : Holiday for Diwali
08 November 2010 : Tutorial Assignment - V
10 November 2010 : Vector Field <--> local one parameter group actions
12 November 2010 : For a Lie group : one parameter group action <--> elements of Tangent space <--> left invariant vector fields <--> (local=global) one parameter group action.
15 November 2010 : Lie Algebra of a Manifold and a Lie group, Lie(GL_n)=gl_n as an example
17 November 2010 : Holiday
22 November 2010 : End Sem exam