Fundamental Theorem of Algebra Proof With Algebraic Topology

Algebraic Topology

https://jeremykun.com/2012/01/22/the-fundamental-theorem-of-algebra-with-the-fundamental-group/

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代 数拓扑 Algebraic Topology (Part 1/3)

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 [1]of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.

  • Vector Space (to + – , × ÷ by multiplier Field scalars);
  • Ring (to + x) in co-homology
  • etc.

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 [2] – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
Note: [1] 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 [2]: Homology 同调 = same “tune”.

南朝 刘宋 谢灵运山水诗:
“谁谓古今殊,异代可同调
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:

  • Categories
  • Functors
  • 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)

  1. Singular Simplices 奇异 单纯
  2. Singular Chain Groups 奇异 链 群
  3. Boundary Operation 边界
  4. Singular Chain Complex 奇异 单纯复形

Part 1/3 Video (Whole) :

Darcy Lecture 5 ~ 9: Applied Algebraic Topology

[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)

Ref:
http://slideplayer.com/slide/9473277/

Circle in Different Representations

1-Dimensional Objects:
Affine Line: {\mathbb {A}^1}
Circle: {\mathbb {S}^1}

Six Representations of a Circle: {\mathbb {S}^1}
1) Euclidean Geometry (O-level Math)
Unit Circle : x^2 + y^2 = 1

2) Curve: (A-level Math)
Transcendental Parameterization :
\boxed { e(\theta) = (\cos \theta, \sin  \theta) \qquad  0 \leq \theta \leq 2\pi }

Rational Parameterisation :
\boxed { e(h) = \left(\frac {1-h^2} {1+h^2}  \: , \: \frac {2h} {1+h^2}\right) \quad \text { h any number or } \infty }

image

3) Affine Plane (French Baccalaureate – equivalent A-level – Math) {\mathbb {A}^2}
1-Dim Sub-spaces = Projective Lines thru’ Origin

image
4) Polygonal Representation (Undergraduate Math)

5) Identifying Intervals: (closed loop) (Undergraduate Math)
image

6) \text {Translation } (\tau, {\tau}^{-1}) \text { on a Line } {\mathbb {A}^1}
(Honors Year Undergraduate / Graduate Math)

[Using Quotient Group Notation]:
\boxed {  {\mathbb {S}^1} \simeq {\mathbb {A}^1 } \Big/ { \langle \tau , {\tau}^{-1} \rangle} }
{\mathbb {S}^1} = \text { Space of all orbits}

image

Question:
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