2018/2019
Research Seminar "Elements of Algebraic and Differential Topology"
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Language:
English
ECTS credits:
5
Contact hours:
60
Course Syllabus
Abstract
We will start by introducing the basic notions and objects such as manifolds and tangent spaces. After that we will discuss the inverse and implicit function theorems from the viewpoint of tangent spaces, Sard's lemma and the Morse lemma, as well as Whitney's immersion and embedding theorems. In the topology part of the seminar we will see CW-complexes, approximation theorems, simplicial homology, the fundamental group and higher homotopy groups. We will also discuss the de Rham cohomology which forms a connection between the differential and topological approaches. This course is optional. Pre-requisites : Undergraduate calculus and linear algebra
Learning Objectives
- To introduce Elements of Algebraic and Differential Topology
- Students have to learn to prepare and give a talk
Expected Learning Outcomes
- Can participate in research projects on Algebraic and Differential Topology
- Know fundamental constructions, concepts and methods in Algebraic and Differential Topology
Course Contents
- Smooth manifolds and tangent bundles
- Inverse and implicit function theorems
- Sard's lemma
- Morse lemma
- Whitney's immersion and embedding theorems
- Transversal intersections and transversality theorems
- CW-complexes
- Cellular and simplicial approximation theorems
- Complexes and exact sequences
- Simplicial homology
- The fundamental group and higher homotopy groups
- De Rham cohomology
Bibliography
Recommended Core Bibliography
- Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C
Recommended Additional Bibliography
- James, I. M. Handbook of Algebraic Topology: North Holland: p.1324 , 1995. - ISBN 978-0-444-81779-2