Intuitive Group

Intuitive Group & Action.

Group Action is an alternative, more powerful than Combinatorics (排列组合) when calculating complex combinations.

If the Math profs had used this intuitive teaching of Group Theory, then Abstract Algebra would be not so scary to all students – even 40 years after “Classe Préparatoire” University Math !

IMO 2020 (6th Question)

2020 IMO (6th problem) only one Chinese 李 student scored full 7 marks.

https://v.ixigua.com/emUQpn1/

This question is similar to Analysis finding epsilon-delta value of N:

(Epsilon-Delta Analysis, aka Advanced Calculus ) definition for “Limit of Series” :
It exists N such that, for all n > N…

[Solution] :

[See also] IMO 2020 (4th Question) : https://m.toutiaoimg.cn/i6881188536880464384/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

3 Math Foundation Skillsets for Engineering / Finance Students

【普通人学数学系列-下集:学到什么程度】

理工/金融科Engineering/Finance 的三个扎实数学基础 :
1) Calculus
2) Linear Algebra
3) Probability / Statistics

Prerequisites for the 3 above in
1) O/A level : Trigonometry, Algebra, Geometry
2) Vector Space (with some basic Algebraic Structures : Group, Ring, Field )
3) O/A level : Permutation/Combinatorics

https://m.ixigua.com/video/6967105499342832166/?app=video_article&timestamp=1622200486&utm_source=native_share&utm_medium=android&utm_campaign=client_share

French Baccalaureate Math Paper 2 (Analysis)

【「给我进来做题啦!」法国高考数学题难度如何?居然只有四道题?】

Baccalaureate Scientifique Math 2019 (Analyse).

Yet to see the Baccalaureate (S) Paper 1 “Algèbre” which is tougher on Abstract Algebra (Group, Ring, Field, Vector Space / Linear Algebra,…)

https://m.ixigua.com/video/6867459333400035851/?app=video_article&timestamp=1621316174&utm_source=native_share&utm_medium=android&utm_campaign=client_share

吴文俊 Wu WenJun

吴文俊,陈省身在 WW2昆明西南联大的弟子, 留法Strasbourg University, 兼任 博士生的tutor, 教导 法国未来的 大师Grothendieck (两人都是新数Bourbaki 学派 最后一批会员) 。

吴文俊在1975 文革后才研究 中国古代数学,从中得灵感,发明 电脑 机器化 AI 证明axiom-based 几何定理。得 1st batch Run- Run Shaw (东方Nobel Prize) $1 m Prize.

数学家清心寡慾不爱斗争,一心专注 “数学之美” ,其他生活繁琐的事(文革 批斗) 都看得开。所以多长寿 (Newton, 陈省身, Hadamard, 杨振宁, 也都90+) ,吴文俊也高寿活到98岁。

https://v.ixigua.com/efrQhx8/

Princeton 《Calculus Lifesaver》

This is a free video lecture series of the Princeton Math Textbook 《Calculus Lifesaver》(link Amazon.com) for university non-math majors.

He explains the “funny” but smart Calculus basics : First & 2nd Fundamental Theorems of Calculus (by Newton’s co-inventor Leibniz), some how never taught in the 1970s GCE A level (now also ? ) bcos not Newton’s Anglo-saxxon invention. Neither the French textbooks teach so (bcos Leibniz was German ?). They only appear in pure American/German Calculus books.

“Funny” bcos by “1st Theorem” you could ‘d/dx’ any intergral to get the anti-derivative . Richard Feymann self-learnt this trick in High school to solve complex integrations.

Also why the “Coset” (+ C) is explained by the 2nd Theorem in this video.

2021人口普查的契机:大学普及改革先废高考

2021人口普查数据 :14.1亿,新生婴儿减少,男多于女,老人(>60岁)佔20%…

教育需要改革来应对新的人口趋势:
(1)缩短基本 小学-中学教育 :废除高考,省下2年准备高考的时间,12年减2年。

(2)高中到 大学本科 :直通车(看高中学校成绩)。

(3)名牌大学(清/北/211/918 等):不设本科undergrads,直接 研究生 Masters/PhD , 像法国Grandes Ecoles 制度 (2nd Cycle Tertiary education, 本科设在普通 universities / 工程科预备班 Classes Préparatoires).

(4)所有 普通大学universities 只 教 本科undergrad 学生为主。

这样普及全国人口教育素质=
10年 (强迫 中/小学) +2 or 4年 (专校 or 大学), 即20岁 / 22 岁 专科/大学本科毕业, 解决“人才荒”的问题。

笔者认为

– 大学改革不可操之过急,要按步就班, 先试点 一些top高中附属大学的 “直通车”(例如:人民大学附属中学), 这样就省下2年高考准备时间。慢慢推行到全国各省内大学和结盟高中。十年完成直通车,百年树人才。

– 名牌大学集中精力/资金在培养 硕士/博士科研人才,把本科教育的资源分散给各省普通大学,直接提升后者的教育水平。这样也可留住中国人才,防止外流去美国读博深造而不回国。

https://m.toutiaocdn.com/i6960914813585392158/?app=news_article&timestamp=1620752197&use_new_style=1&req_id=20210512005637010212205224381BC609&group_id=6960914813585392158&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

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…

https://m.toutiaocdn.com/i6743479520755450380/?app=news_article&timestamp=1615286356&use_new_style=1&req_id=202103091839150102120751990F03C964&group_id=6743479520755450380&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

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.

https://m.toutiaoimg.cn/i6928283118424031751/?app=news_article&is_hit_share_recommend=0&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

《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” .

https://m.toutiaoimg.cn/group/6919012894134764046/?app=news_article&timestamp=1611057637&group_id=6919012894134764046&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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

吴杰:拓扑世界的架构师

https://mp.weixin.qq.com/s?__biz=MzA5OTA2MTIzNw==&mid=2656737111&idx=2&sn=e494f6bb6aec2da14ace83220ff103ad&chksm=8b262bcdbc51a2dbcdbc310b96209ba3c47b07478e8db37acecfec050e110896850e5825be07&mpshare=1&srcid=0105hADjmNvluUUjsqfpjNx8&sharer_sharetime=1609899719523&sharer_shareid=82d451518f239e8840f13fc06d759dff&from=timeline&scene=2&subscene=1&clicktime=1609998317&enterid=1609998317&ascene=2&devicetype=android-28&version=27001639&nettype=WIFI&abtest_cookie=AAACAA%3D%3D&lang=en&exportkey=AhuDDklJOctFUKK4k%2Bsfn5o%3D&pass_ticket=gfaC%2F0RZPoeZYEePorezdPv248UbY30mbPoB8ygMtLtw1AlQf1GZnSWDfVaxHpDd&wx_header=1

IMO Geometry Techniques 几何理论基础:分角定理、张角定理,推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.

数学竞赛几何理论基础:分角定理、张角定理,推理证明

https://m.toutiaoimg.cn/a6910816390312100360/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Dr. Eugenia Cheng: “How to Bake Pi”

Key Points of the Talk:

1) Math is interesting but is only so after undergraduate school. Before that, Math is taught as computation subject from Elementary to High school.

2) Braid : Bach music, Juggling 3 balls

3) Platonic Icosahedral (20面体) Structure discovered by ancient Greek Plato 2000 years ago, but can’t find a real world Icosahedral object until in Viruses found by Louis Pasteur in 20th century – also now in Sars, Covid19.

4) Group Theory : Battenberg Cake, Bed Mattress Rotate/Flop/ Flipping

5) Mobius Strip & Donnut Cutting.

6) Fermat’s Last Theorem : Andrew Wiles in 1994 proved in 7 years still with a “hole”, but fixed a year later by himself & his student.

7) MacLane Pentagon : Higher-Dimension Categories (PhD Math)

为啥法国数学这么强,平民的算数这么烂 ?

结账的算数:€ 313.5

华人:[给] €323.5 – [找] €10 =€313.5

法国人: [给] €320 – [找] =€ 313.5

{?€ 6.5 是这样算的 :313.5+0.5 [凑整]=314.0, 然后 314+6=320,得[找] 0.566.5}

https://m.toutiaocdn.com/i6896389485832389134/?app=news_article&timestamp=1607277607&use_new_style=1&req_id=202012070200070100140540151610B97E&group_id=6896389485832389134&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Chinese Remainder Theorem 《韩信点兵 》& Ideal

听说过“韩信点兵”的歌谣吗?李永乐老师讲中国古代数学定理

Chinese Remainder Theorem

By 1930, even Bourbaki founder Dieudoné didn’t know “Ideal” when he read 《Algebra》from Noether’s lecture-note compiled by her student Van der Waeden

https://m.toutiaoimg.cn/a6559856147912720909/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

日本战后数学教育家 远山启

远山启:用数学照亮人性与自由

(1909-1979)

日本战后的"新"数学:把抽象化转成 平易化。出了一群 Fields Medalists, 数学教育家 (这位 远山启,Shimura, ), 科普作家 结城浩 (《Math Girls》Series : Galois Theory, Fermat’s Last Theorem, etc. )…

https://m.toutiaocdn.com/i6895034925657883144/?app=news_article&timestamp=1605514504&use_new_style=1&req_id=2020111616150301020405504612025E05&group_id=6895034925657883144&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

听杨振宁讲物理课

听杨振宁讲物理课(七):

[7.1] 好书推荐

1. E. T. Bell: 《Men of Mathematics” 》 – [Comments:] Irish Hamilton & Quartenions – Bell said Halmiton later’s life was a tragedy in trying to make quartenions universally applied in Physics but failed. Prof Yang : Quartenions will be “universally useful” in future.

Note : Quartenions (1,i,j,k) & Special Relativity – Does it ring a bell to you the UNIVERSAL 4 FORCES (Gravity, Electric, Magnetic, Particles Weak force) :

https://tomcircle.wordpress.com/2019/07/18/division-algebras-and-physics/

2. E. Segrè (吴健雄的博士导师) & the story of the missing Periodic Table Element ’43’ (Tc) discovered by accident from unwanted trash.

[7.2] 重力势能和弹性势能

https://m.toutiaoimg.cn/group/6608369844179960327/?app=news_article&timestamp=1605036314&group_id=6608369844179960327&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

听杨振宁讲物理课(二):速度和加速度

https://m.toutiaoimg.cn/a6606089315711713796/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

中国数学家之乡 – 浙江 温州

温州为何会成为数学家之乡?答案就在这里!

姜立夫 (子:姜伯驹):庚子赔款(第二届) 哈佛大学,中国第二个数学博士(第一位是 杨武之 – 杨振宁之父) 。南开大学数学系 创办人。学生:陈省身江泽涵 (翻译中文名词:Topology 拓扑,Homology 同调, Homotopy 同伦) etc.

苏步青 :留日,浙江/复旦大学数学系 创办人。

李秉彝:南洋大学数学系(第一届) 毕业生,苏格兰大学博士。新加坡南洋理工大学国立教育学院数学教授,曾任东南亚数学会理事长。闻名世界的 “Singapore Model Math” 发明者。

https://m.toutiaocdn.com/i6686319487479710220/?app=news_article&timestamp=1604069365&use_new_style=1&req_id=202010302249240102020851974C100485&group_id=6686319487479710220&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

丘成桐:我研究数学的经验

丘成桐:我研究数学的经验

路漫漫其修远兮,吾将上下而求索‘ – 屈原 《离骚》(340BC)

丘成桐:我研究数学的经验

‘路漫漫其修远兮,吾将上下而求索‘ – 屈原 《离骚》(340BC)

第一个华人拿 Fields Medal, 而且世界上少数人27岁得奖,ST Yau (岳 – 山=丘) 的成功不是偶然:

  1. 在香港中文大学遇到美国人年轻数学讲师,引导他进"数学之大门",并推荐他去美国拜他的教授 陈省身 ( 微分几何 的 “Gauss II” 宗师) 读PhD。
  2. 14岁前父亲给他打下古文学/哲学的基础,”真/善/美” 艺术的直觉 Intuition, 对高深数学理解有大帮助。
  3. “Read directly from Masters” (Abel 说的) . 读经典原著。
  4. 多听别人的讲座,多看 journal papers. “抛砖引玉”,“他山之石可以攻玉”。Get inspiration from others.
  5. 高中/ undergrad 基础功夫要打好,Algebra, Analysis, Geometry, Physics ,都要 “懂” (只有多作习题才知道)。

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

https://m.toutiaocdn.com/i6888616174393754123/?app=news_article&timestamp=1603895425&use_new_style=1&req_id=20201028223025010202055140190F8C1C&group_id=6888616174393754123&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share