Excellent video for the curious minds! Who cares about Topology such as Torus (aka donut) or Mobius Strip ? They can be used to prove difficult math such as the unsolved problem “Inscribed square/rectangle inside any closed loop”.
To understand the Topology on Loops, please view the lecture here : Homotopy (同伦) and the Fundamental Group (群) of surface.
Continued from Computational Topology (1 ~4):
MATH 496/696 2016/02/10 Lecture
Homology from another angle:
= coKernel ○ kernel
- Define Chain Space C•(X)
- Define Boundary Map d•
- Define Simplicial Chain Complex (C•(X), d•)
Ref: Simplicial Complex by Wildberger
NJ Wildberger AlgTop24: The fundamental group
Homotopy 同伦: When playing skip rope, the 2 ends of the rope are held by 2 persons while a 3rd person jumping over the “swings of rope” – these swings at any instant are homotopic.
1) Row reduction, row-echelon form and reduced row-echelon form
2) Rank of Matrix = Rank (A)
Homology 同调 is better than Euler Characteristic to differentiate manifolds with holes (eg. Torus, Klein Bottle, …)
[Revision – Lecture 1 ~4: Foundation of Applied Algebraic Topology
Lecture 6: Creating Simplicial Complex]
Lecture 5: 4/9/2013 (三) Clustering Via Persistent Homology
Lecture 7: 6/9/2013 (五) Calculating Homology using matrix
Lecture 8: Column Space and Null Space of a matrix
Lecture 9: 9/9/2013 (一) Create your own Homology: (Important lecture in Applied Algebraic Topology)