Tag Archives: Algebraic Topology

Hardest Math

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Algebraic Topology

World’s Hardest Math: Algebraic Topology (study Homology 同调)

Answer to What is the hardest type of math? by Zakary Jay Nicholls https://www.quora.com/What-is-the-hardest-type-of-math/answer/Zakary-Jay-Nicholls-1?ch=3&oid=1477743635795466&share=8379c7a9&srid=oZzP&target_type=answer

He also said “Algebraic Topology” is the hardest math, which requires “Writing Proof”, different from high school computational Calculus.

大脑与数学 第二部分——拓扑学中的对象：单纯复形

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大脑与数学 第二部分——拓扑学中的对象：单纯复形

https://m.toutiaoimg.cn/group/6899390396544057864/?app=news_article&timestamp=1606415697&group_id=6899390396544057864&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

AI & Algebraic Topology

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The first Chinese “Turing Prize” (equiv Nobel Prize in IT ) Winner Prof YAO 姚 (Taiwanese-born, now invited by 杨振宁 to teach in 清华) on

“New Trend in AI Theory“.

[At 14:00 mins] Prof YAO showed an AI Theorem using Algebraic Topology (Betty Number).

https://m.toutiaoimg.cn/a6855411637365506571/?app=news_article_lite&is_hit_share_recommend=0

AI with Advanced Math helps in discovering new drugs

AI, Business, Data Science, IT, Medicine, Modern Math, Science 1 Comment

https://theconversation.com/i-build-mathematical-programs-that-could-discover-the-drugs-of-the-future-110689?from=timeline

Advanced Mathematical Methods with AI is a powerful tool:

  • Algebraic Topology (Persistent Homology)
  • Differential Geometry
  • Graph Theory

https://sinews.siam.org/Details-Page/mathematical-molecular-bioscience-and-biophysics-1

代 数拓扑 Algebraic Topology (Part 1/3)

Modern Math 1 Comment

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

Modern Math 1 Comment

[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

Modern Math 1 Comment

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” …

Algebraic Topology 代数拓扑与孙悟空72变

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How to turn a doughnut to a coffee mug ? They are the ‘same’ (homeomorphic 同胚) when the former is deformed continously into the later (the hole in doughnut becomes the cup’s handle).
image

这是孙悟空和二郎神杨戬斗72变法, 老孙输在变庙时, 尾巴变旗杆, 旗在庙后, 被杨戬识破。
image

Algebraic Topology: using algebra (eg. Group Theory) to study geometrical shapes.

Difference between Algebraic Topology and Algebraic Geometry (see the previous blog):

image

AlgTop1: One-dimensional objects: http://youtu.be/oYFZaqArf54

AlgTop2: Homeomorphism and the group structure on…: http://youtu.be/R_gDV17X7pc

AlgTop3: Two-dimensional surfaces: the sphere: http://youtu.be/tv0XlHfX9r0

AlgTop4: More on the sphere: http://youtu.be/GQ0torqFx8Y

AlgTop5: Two-dimensional objects- the torus and g…: http://youtu.be/4U9XzZjxMFI

AlgTop6: Non-orientable surfaces—the Mobius band: http://youtu.be/vlcdqPWg34k

AlgTop7: The Klein bottle and projective plane: http://youtu.be/ibg5KiG46nk

AlgTop8: Polyhedra and Euler’s formula: http://youtu.be/IY1VyUb44yE

AlgTop9: Applications of Euler’s formula and grap…: http://youtu.be/WxXg49jLkWc

AlgTop10: More on graphs and Euler’s formula: http://youtu.be/AfSBSucXFDM

Continue… (YouTube: search AlgTop11, etc)