# Homology

Part 1: “Homology” without Pre-requisites, except “Function” (he un-rigourously interchanges with “Mapping”, although Function is stricter with 1-and-only-1 Image) .

Part 2: Simplex (单纯形)

Topology History : Euler Characteristic eg. (V – E + R = 2) ,
Poincaré invention

This video uses Algebra of point, line, triangle… to explain a Simplex (plural: Simplices) in R^{n} Space, that is organizing the n-Dimensional “Big Data” data points into Simplices, then (future Part 3, 4…) compute the “holes” (or pattern called Persistent Homology).

Part 3: Boundary

Part 3 justifies why triangles (formed by any 3 data points) called “Simplex” 单纯形(plural: Simplices) are best to fill any Big Data Space.

# Is There a Multi-dimensional Mathematical World Hidden in the Brain’s Computation?

Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) &  Computers.

Homology = compute the number of “holes” in multi-dim space.

Neurons formed in the brain can be modeled in Math (Topology) by Simplex 单纯 (plural : Simplices), billions of them interconnected into a complex – “Simplicial Complex” (单纯复体)。

https://singularityhub.com/2017/06/21/is-there-a-multidimensional-mathematical-world-hidden-in-the-brains-computation/?from=timeline#.WUvPNsvmjqD

# 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) :

# Homology: Why Boundary of Boundary = 0 ?

Homology and co-homology are the Top 10 Toughest Math in the world (1st & 8th topics in the list, of which 3rd, 4th, 9th and 10th topics received the Fields Medals). Like most Math concepts, which were discovered few decades or centuries ago, now become useful in scientific / industrial / computer applications never thought before by their discoverers.

Examples: Prime numbers, Chinese Remainder Theorem 韩信点兵 / Fermat’s Little Theorem / Gauss Modular Arithmetic / Elliptic Curve in Cryptography; Homology in Big Data Analytics for Epidemic Medicine, Pharmaceutical Drugs, Consumer Behavior Study, Stock Market, Economy, etc.

This Homology Fundamental Equation puzzles most people. WHY ?
$\boxed {{\delta}^2 = 0 { ?}}$

It is analogous to the Vector Algebra:
Let the boundary of {A, B} =
$\delta (A,B) = \overrightarrow{AB }$

$\overrightarrow{AB } + \overrightarrow {BA} =\overrightarrow{AB } - \overrightarrow {AB} = \vec 0$

Note: (co)-homology: (上) 同调

Euclid Geometry & Homology:

References: (Videos)

1. Isabell Darcy Lecture: cohomology

2. 同调代数 Homological Algebra

References: (Books)

# Intuition of Homology, Cohomology and Homotopy

Homology 同调 is better than Euler Characteristic to differentiate manifolds with holes (eg. Torus, Klein Bottle, …).

Cohomology: ​上同调

Homology => Free Abelian Group on Set (ie Add /Subtract structure)

Co-homology => Ring Function on Set (ie Product structure)

# NJ Wildberger Lecture Series : Simplicial Complex (单纯 复形)

Simplex : 单纯 (plural Simplices)
0-dim (Point) $\triangle_0$
1-dim (Line) $\triangle_1$
2-dim (Triangle) $\triangle_2$
3-dim (Tetrahedron) $\triangle_3$

Simplicial Complex: 单纯復形 built by various Simplices under some rules.

Definitions of Simplex : $S (v_0, v_1, ..., v_n)$
Face
Orientation
Boundary ($\delta$)
$\displaystyle \boxed { \delta(S) = \sum_{i=0}^{n} (-1)^i (v_0 ...\hat v_i ...v_n)}$

Theorem: $\boxed { \delta ^2 (S) = 0}$ SO SIMPLE !!!

Follow the entire Algebraic Topology from University of New South Wales (3rd / 4th Year Math) :

Algebraic Topology: a beginner’s course – N J Wildberger:

# Homology (同调 ) in Geometry & Topology

https://frankliou.wordpress.com/2011/10/07/幾何與拓樸簡介/

“同态” (同样形态homomOrphism), 就是Same-Shape-ism. eg. (相似) Similar Triangle.

=> Endomorphism.

=> Automorphism “自同构” (镜里的影子和自己同样结构)

WW2 后, 美国人Sanders MacLane 更上一层楼, 把Set/Group/Ring…等structures 再归类成Category (范畴), 研究其共通的性质 (Morphism 动态), 能够 举一反十。应用在IT 里, 其 Category 就是Functional programming, Types…

# Algebraic Topology: Homology vs Homotopy

Homology (同调) is easier to compute than Homotopy (同伦) , hence Computational Topology uses the Algebraic Topology of Homology.