新数启蒙 (1-5) – 给16岁以上的中学生

This “Modern Math” Introductory Course is based on the French Baccalaureate “Modern Math” for 16+ years old, simplified & customized for local students in Chinese.

https://v.ixigua.com/e6mRFnP/

(1)集合: 等价关系 Equivalence Relation

https://v.ixigua.com/e6mDFhw/

(2)集合与集合的关系:映射 Mapping

https://v.ixigua.com/e6mfC4C/

(3) 新数 历史: Bourbaki 学派

https://v.ixigua.com/e6mjedR/

(4)由外向内看代数结构 : 域Field,环 Ring,群 Group,向量空间 Vector Space

https://v.ixigua.com/e6mj5sd/

(5) 域 Field : 足球场

https://v.ixigua.com/e6mkau7/

Modern Math Education : Time for Reform

Modern Math Foundation : Set + Logic (Equivalence Relation). 

A level (H2 level) / International Baccalauréat Math should study the big red square box (Number System, Set Theory, Logic).


All STEM university Year 1 & 2 study these 3 Math foundations : Algebra, Analysis & Vector Space.

The Math major university Year 3&4 study more : real /complex analysis, Topology, Galois Theory, graph theory, etc…

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mRNA剪接Splicing 原理 – 施一公

3 Step Splicing : first & last steps are linear, 2nd step 3D syructure (non-linear) .

施一公 returned from USA, established the private 西湖大学 (funded by Alibaba, Baidu, Huawei etc), a first pure research university modeled on Caltech. His team first decoded the COVID19 genetic code sequence just 1 month after the Wuhan outbreak & announced the result FREE to the world.

The Hungarian lady scientist Katalin Karikó jumped on his newly published DNA code to invent the mRNA PfizerBioNTech vaccines.

https://m.toutiaoimg.cn/a6936774993665589790/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

数学与物理的关系

(小学生与诺奖得主杨振宁的对话)

Math was a tool for Physics (Newtonian Physics)

Math & Physics were independently developed ( Physics Gauge Field = Math Fibre Bundle )

Physics is a tool for discovering Math (Quantum Physics : String Theory)

杨振宁 Yang-Mills Conjecture is one of the unsolved Millenium Math Problems.

https://m.toutiaoimg.cn/a6936190545714741798/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Feymann Calculus Trick

理查德·费曼非常聪明的求导 differentiate 技巧

Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.

https://m.toutiaocdn.com/i6933218183247626755/?app=news_article&timestamp=1614316375&use_new_style=1&req_id=2021022613125401019410001308000E75&group_id=6933218183247626755&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

林群:先通过案例学道理,后才证明

中科院院士著名数学家林群为大家分享数学怎么学

林群:先通过案例学道理,后才证明

中科院数学院士 林群:"如何学高深 抽象 数学":
先通过案例 (case / motivation) 学道理,后才学证明 (Theorem Proofs) 。

eg. Learn “Group Theory”
Case / Motivation: Symmetry

https://m.toutiaoimg.cn/a6783678419574981120/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Ellipse Proof by Circle : Affine Transformation

Cumbersome Ellipse proof below :

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ

数学中的冷知识:圆幂定理,现在初中课本中都不讲了,但是非常有趣?

圆幂 定理 II: The Affine Transformation FROM Circle to Ellipse. Also applicable to all other 2 Theorems.

https://m.toutiaoimg.cn/a6580683485986423309

Proof : Ellipse Area from Circle :

函数 Function 概念

【数学科普】:

Function = f : R – > R
(with 1& only 1 image)

Mapping 映射 (l’Application) , can be non-value to non-value

仿射变换 Affine Transformation (解析几何 Analytic Geometry 的 Modern Math).

Affine 是 德文 (coined by Euler)

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丘成桐 的雄心计划: 培养800个数学天才苗子

当年汉朝 霍去病 带 800 骑士 深入漠北驱赶匈奴。

丘成桐 (属牛)的雄心计划: 培养800个数学天才苗子, 每年从中国&世界选16岁起的100个 华人,在清华少年班 8年培训:本科4年+硕2年+博士 2年.

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Fermat’s Last Theorem for Regular Primes

This FLT (for Regular Primes) is only first part of proof up till Ideal’s Ring Theory by Kumma.

The final complete proof by Andrew Wiles (1994) used more tools: Elliptical curve + Galois Theory.

The complete proof FLT by Andrew Wiles taking him 7 years in solitude, still a short time compared to 350 years before him but failed by the grandmasters Euler/Gauss etc. Today Andrew Wiles is hailed as the greatest 21CE Mathematician, even Fields Medal gave a Special Award to him (even he was > limit age of 40 years old ).

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.

《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明:

1. 哲学:希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴, 达芬奇, Golden Ratio

3. 工业革命:Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

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吴文俊:中学数学教育

吴文俊:数学教育(上)

要点:

1. 文化大革命 / 数学教育革命 灾难

2. 东方数学 : 小学 – 中学/高中数学 (机械化)eg. 四则运算

3. 西方数学 :大学”新”数学 (非机械化)

https://m.toutiaocdn.com/i6917519470810366472/?app=news_article&timestamp=1610825278&use_new_style=1&req_id=202101170327580100080431050833ACBD&group_id=6917519470810366472&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

吴文俊 批评法国的1970s Bourbaki 数学进入中学/Bac 课本。美国也同时推行 New Math。

问题出在 数学老师的训练不够,太仓促。比如 Singapore Math 1980s 刚推行 也是很"乱",后来新加坡教育部 (MOE) 要求家长们也去学习,才能在家里辅导孩子。

吴文俊:数学教育(下)

他提倡中学数学教育 2 要点:
1) 教少一点 Euclidean Geometry, 用 Analytic Geometry取代, 才可以 computerise (机械化).

2) 中国古代是有 Geometry, 但 用天元(x), 地元(y) 代数(Algebraic) 化。

https://m.toutiaocdn.com/i6917528352391807499/?app=news_article&timestamp=1610902867&use_new_style=1&req_id=20210118010107010204051088185887A1&group_id=6917528352391807499&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share