# 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”.

“谁谓古今殊，异代可同调

(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) :

# Simplicial Homology 单纯同调

Continued from Computational Topology (1 ~4):

MATH 496/696 2016/02/10 Lecture

Homology from another angle:

$\displaystyle H_{k}$ = coKernel $(m)$ ○ kernel $(d_{k}) \:$${(C_{k})}$

Simplicial Homology:

1. Define Chain Space C•(X)
2. Define Boundary Map d•
3. Define Simplicial Chain Complex (C•(X), d•)

# Homotopy 同伦 & Fundamental Groups

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 $\pi (M; \alpha)$

Fundamental Group of Torus $\pi (T) = Z.Z$

Fundamental Group of Projective Plane (Torsion ) $\pi (P) =Z_{2}$

# Darcy Lecture 5 ~ 9: Applied Algebraic Topology

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)

# Topological Data Analysis

Three Key Ideas:

1. Cloud of Points
2. Filter function: f (x,y,z) –> x
3. Cluster: Overlapping bins
4. Draw network:
• Vertices: clusters
• Edges: connecting lines between clusters.

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

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

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

6) $\text {Translation } (\tau, {\tau}^{-1}) \text { on a Line }$ ${\mathbb {A}^1}$
$\boxed { {\mathbb {S}^1} \simeq {\mathbb {A}^1 } \Big/ { \langle \tau , {\tau}^{-1} \rangle} }$
${\mathbb {S}^1} = \text { Space of all orbits}$