Excellent Advanced Math Lecture Series (Part 1 to 3) by 齊震宇老師
（2012.09.10) Part I:
History: 1900 H. Poincaré invented Topology from Euler Characteristic (V -E + R = 2)
Motivation of Algebraic Topology : Find Invariants of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.
- Abelian Fundamental Group (so as to manipulate by + -);
- Vector Space (to + – , × ÷ by multiplier Field scalars);
- Ring (to + x) in co-homology
then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants (homology, co-homology, etc).
A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调  – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
Note:  Analogy of an”Invariant” in Population: eg. “Age” is an invariant can be added in the “Population Space” as the average age of the citizens.
Side Reading (Very Clear) : Invariant and the Fundamental Group Primer
Note : Homology 同调 = same “tune”.
南朝 刘宋 谢灵运山水诗:
同调 = Homology
(希腊 homo = 同, -logy = 知识 / 调)
– “Reading an ancient text allows us to think “in tune” (or resonant) with the ancient author.”
[温习] Category Theory Foundation – 3 important concepts:
- Natural Transformation
[Skip if you are familiar with Category Theory Basics: Video 16:30 mins to 66:00 mins.]
[主题] Singular Homology Groups 奇异同调群 (See excellent writeup in Wikipedia) (Video 66:20 mins to end)
- Singular Simplices 奇异 单纯
- Singular Chain Groups 奇异 链 群
- Boundary Operation 边界
- Singular Chain Complex 奇异 单纯复形
Part 1/3 Video (Whole) :
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.
Fundamental Group of Surface
Fundamental Group of Torus
Fundamental Group of Projective Plane (Torsion )
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)
◇ Affine Line:
Six Representations of a Circle:
1) Euclidean Geometry (O-level Math)
Unit Circle :
2) Curve: (A-level Math)
Transcendental Parameterization :
Rational Parameterisation :
3) Affine Plane (French Baccalaureate – equivalent A-level – Math)
1-Dim Sub-spaces = Projective Lines thru’ Origin
(Honors Year Undergraduate / Graduate Math)
[Using Quotient Group Notation]:
Are Circle and Line the same 1-dimensional object, i.e. are they Homeomorphic (同胚) in Topology ?
Answer: To be continued in the next blog “Homeomorphism” …