Algorithmic Topology
Time:
Lecture: Wednesdays 9.45-11.15 PRÜ 1103, public holiday at Nov 19th
Example class: to be moved.
Part 1: Point set topology (until Nov 26th).
You should be familiar with:
- Definition of a topological space and related notation such as: open, closed, compact, connected, path-connected, continuous)
- Basic examples of topological spaces: discrete Topology, subspaces of Rn
- Subspace and Quotient topology, Product topology
- Hausdorff separation property
Further topics:
knot theory
embedding problems for simplicial complexes
classification of compact 2-manifolds
Topics of lectures:
29.10. -- Introduction to Topology
05.11. -- Definition Topological Space, Basis of Topology, Subspace Topology (explicitly and via universal property), Quotient Topology (explicitly and via universal property), continuous, connected, plus lots of examples
12.11. -- Product Topology and statement of Tychnov's theorem plus examples, compactness principle and proof that an infinite graph is k-colourable if and only if all finite subgraphs are k-colourable
19.11. (public holiday)
26.11. -- Proof of Tychonov's theorem
03.11. -- Introduction to topological graph theory
10.12. -- Kuratowski's theorem
17.12. -- Embedding 2-complexes in 3-space
7.1. -- Introduction to knot theory (definition of knots, equivalence of knots, how knots are displayed by knot diagrammes, we proved used FOX tri-colourability that the trefoil knot is nontrivial. This is a prime example of a nonexistence proof by invariant)
14.1. -- Introduction to Fundamental groups (definition homotopy equivalent, fundamental group, basic examples)
21.1. -- Fundamental groups of knot complement
28.1. -- Introduction to Homology theory
4.2. -- classification of 2-manifolds
11.2. Wrap up and "Prüfungsvorgespräche"
The examination will be a 30 minutes oral exam. Dates will be discussed towards the end of the semester.
Literature for the course:
For Topology, I recommend the book of Armstrong.
For Grapg Theory, I recommend the book of Diestel
For advanced topics later in the course, I will provide notes here.