丘成桐:我研究数学的经验

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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

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如何学数学 How to Learn Math

Math Online Tom Circle

林群院士:

https://m.toutiaoimg.com/group/6783678419574981120/?app=news_article_lite&timestamp=1579537218&req_id=2020012100201801001404702306D0DBEB&group_id=6783678419574981120

道理 (X 定理)
发明 (X 证明)

林群院士 评学校数学教法颠倒:不教 "道理"(Intuition ) 只教 定理,不教 “发明” 的背景(Discovery background)只教”证明” (Proof) 。

他的数学教法 强调 “all maths go back to the fundamentals.”

He has redefined the abstract “Cauchy ε-δ definition of a limit” in terms of only elementary maths (trigonometry, inequality, gradient).

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化学:金属活动性顺序表

Math Online Tom Circle

中三化学刘老师教我们背这个金属活动性顺序表, 排前面的元素比排后面的活跃,写化学反应方程式时,非常有用。基本上整个 Physcial Chemistry 就是浓缩在这表里,朗诵上口,终身难忘,50年后还记得!- 化学靠记忆,用中文学化学最方便 。(我在法国精英大学 Classe Preparatoire 念化学,打败全班法国人得第一名,中学用中文读化学的基础, 功不可没。)

铅 (氢)

K Na Ca Mg Al Zn Fe Sn Pb (H)

[嫁 倩 (轻)]

Cu Hg Ag Pt Au

[统 金]

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数学历史大纲by丘成桐

Math Online Tom Circle

28 Dec 2019 @中国科学技术大学研究生院

Key Points:

  1. 中国数学从古到今只注重计算和解题没有提出自己的问题。 (这是 科举 /Concours / 高考 / IMO Math Olympiad 的心态mindset)。大问题: eg. 法国人 Group Symmetry, Cauchy’s Limit, Poincare Topology, 英国Newton / Leibniz’s Calculus, Russell’s Logic, 德国 Cantor’s Set Theory, Infinity,…
  2. 清朝数学家花200+年 考古研究古代中国数学 (《九章算数》… ), 这时西方文艺复兴开始搞近代数学 (意大利 虚数 i, Fermat 数论,Descartes 解析几何,Pascal 或然率, Newton Calculus,… ) 。中国数学从此落后西方。
  3. 要读数学历史,尤其是17/18CE Fermat/Newton 之后的近代数学史, 才有宏观思想,继承他们的精神,从而开拓新的领域 (eg. AI 数学: HoTT = Homotopy Type Theory?)
    https://m.toutiaoimg.cn/a6864500678341952013/?app=news_article_lite&is_hit_share_recommend=0
  4. 他选出 41位 17/18世纪之后至今的伟大数学家, 属于开创新领域的宗师:第一个是 Fermat, 第二个是 Newton…. GALOIS, ABEL,… LIE, KLEIN, HILBERT, Minkowski, Riemann, Euler, Gauss, Cauchy, POINCARÉ, Noether, Weyl, E. Cartan, André Weil, 陈省身,丘成桐, Von Neumann, Turing, … [没有 Bertrand Russell, Ramanujian, Hardy, Terence Tao,… ? ]
  5. 提到重要的数学:Lie Group, Lie Algebra, Guage Theory, Homology, Co-homology, Homotopy, Riemann Geometry, Zeta Function, etc.
  6. 他将出书写这些数学家的历史。
  7. [Q&A 1] : 当明白数学来龙去脉,才知道历史,看未来发展。司马迁写《史记》,司马光 写《资治通鉴》的目的:以古鉴今!
  8. [Q&A 2] : 数学读得越多,一切数学就变得简单。(这跟学厨艺,钢琴等一样,起初一切都难,但练习煮/弹得多,就越来越容易上手,而且能够创新,举一反三)。40年前在法国大学读抽象代数(Abstract Algebra) 满头雾水,考试心生恐惧。15年前温习并读其他更多的数学书, 顿然开窍,现在爱不释手。

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《Galois Theory》 – Cousera by Ecole Normale Supérieure

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Complete Free COUSERA Course 《Galois Theory》 (French) from Ecole Normale Supérieure (ENS 巴黎师范) , the university which kicked out Galois as a student in1830, but apologized to him 150 years later.

Today ENS ranked top 3 globally (after Harvard, Princeton IAS) in Math research & education, producing 1/3 of the world’s Fields Medalists by a single university.

For English Text translation (choose Google translate)

https://www.coursera.org/learn/theorie-de-galois

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IMO Coach : Drill vs Think Style

Math Online Tom Circle

The USA IMO coach Prof Loh is the son of 2 Singaporean Math Prof (dad) & JC Math teacher (mom).
He uses the Chinese “drill” style + American “think” style to beat China IMO Team last few years.

https://m.toutiaocdn.com/group/6764245433339347460/?app=news_article_lite&timestamp=1574935791&req_id=201911281809500100100542211C369EF1&group_id=6764245433339347460

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Music = Math + History

Math Online Tom Circle

To really appreciate music, you must understand this Math & history :
5 notes to 7 notes to 12 notes.
& fractions: 3/2, 3/4, 2/1 (octave)

Music = Math + History

1. Ancient China 600 BC 管仲 (5 notes 宫商角徵zhǐ羽)

2. Pythagoras 450 BC (7 notes, Chords 和弦)

3. Modern Music: 十二平均律 (明朝 1600 AD 朱载堉- > Bach 1700 AD)

https://m.toutiaoimg.com/group/6683391962764018183/?app=news_article_lite&timestamp=1581773659&req_id=20200215213418010194100050312DF0B9&group_id=6683391962764018183

Pythagoras did not believe in Irrational numbers (sqroot 2) , so only 7 integer notes.

17CE Chinese 明朝 Prince invented 12 notes music, using a giant 81-row Abacus to compute the12th root of 2 (picture below) :

https://chinoiseries2014.wordpress.com/2019/02/07/%e5%8d%81%e4%ba%8c%e5%b9%b3%e5%9d%87%e7%8e%87/

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Ideal in Ring

Math Online Tom Circle

Historical Background:

Ideal (理想) was a by-product by mathematicians in the 350-year proof of the 17CE Fermat’s Last Theorem, wherein they found a violation of the existing “Fundemental Law of Arithmetic” (Unique Prime Factorization) . Since it is a Law, there must be an alternative ideal number to satisfy it, hence the birth of the “Ideal”.

Read here: the raison d’être of Ideal : What is an Ideal ?

Note:

Why Integer (Z) is called “Ring” (Dedekind coined it using the German word “Der Ring”) ? because
{1, 2, … , 11, 12 = 0} is clock number “Z/12Z” like a Ring-shaped Clock 🕜

Application:

The ancient “Chinese Remainder Theorem” (aka 韩信点兵 ) since 200 BCE is explained by 19CE Ideal Theory.

[Solve] : “The Problem of 6 Professors

Whatever inside multiplies outside, still comes back inside.

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Where to park your car, according to math

Math Online Tom Circle

This “Simple Car Parking” is the Decision Making Strategy demonstrated in our daily life, even in the micro-cell biological level in nature.

In the current USA-China Trade War, China Xi is taking the Prudent (中庸) approach of a partial 80% deal, while Trump is too Optimistic (or gung-ho) , insisting to accept either a “Full 100%” deal or None. Math tells you the smartest man’s approach is China Xi “Prudent” strategy.

Opportunity Score:

Meek : 1/5 score = ‘Kiasu’ (怕输 )

Prudent : 3/5 score = (中庸 )

Optimist : 5/5 or 0/5 score = (Gung-Ho)

Differential Equation:

https://m.phys.org/news/2019-09-car-math.html

In 三国演义 Three Kingdoms, the smartest military strategist Zhuge Liang (诸葛亮) was attacking the biggest northern enemy Wei (魏). He chose the Prudent (‘P’) route longer distance but less ambushes, rejected his General 魏延 ‘s Optimistic (‘O ‘) approach of taking the risky “子午谷” mountainous but shortest route.

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11 of the Best YouTube Channels for Understanding Mathematics

Math Online Tom Circle

https://interestingengineering.com/11-of-the-best-youtube-channels-for-understanding-mathematics

Below are some of the best and Free Math channels from Secondary School to University to Public Math Enthusiasts.

1. Khan Academy (School Math, Sponsored by Bill Gates)

2. Numberphile (Math Curiosity)

3. WooTube (School Math)

4. Linear Algebra (Undergrad Math)

5. Rigourous French Math (Classe Prepa, Undergrad Math)

Set Theory Basics (Baccalaureate Math Revision)

Espace Vectorielle (French Abstract Version of Linear Algebra)

Rigorous French Math (Classe Prepa)

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Data Science: 5 Statistics & Math tools

Math Online Tom Circle

【掌握这五大统计学知识,让你笑傲数据科学界】
https://m.toutiaocdn.com/group/6723020943871967751/?app=news_article_lite&timestamp=1565375984&req_id=20190810023944010152028205708822C&group_id=6723020943871967751&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

1. MEAN / MEDIUM

2. PERCENTILE

3. SKEWNESS

4. STANDARD DEVIATION

5. PCA = Pricinpal Component Analysis

【学习大数据,我不会数学怎么办?还要不要学啦?】
https://m.toutiaocdn.com/group/6706386149763449356/?app=news_article_lite&timestamp=1565378178&req_id=20190810031618010152042134003992B&group_id=6706386149763449356&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

1. Probability & Statistics

2. Linear Algebra (Matrix, Eigenvectors, Eigenvalues)

3. Optimisation : eg. Gradient Descent, etc.

4. Discrete Math in Computing

(想看更多合你口味的内容,马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link

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林群院士:怎样学好数学?How to learn Math well ?

Math Online Tom Circle

【中国科学院林群院士告诉你,怎样才能学好数学?】复制这条信息€80cKc€ZjlaWx€后打开👉今日头条极速版👈

林群院士很谦虚,自称很笨, 看到200页的数学教科书头也会晕:

他用学习Calculus例子,批评"教科书不教真理,只教证明 (Proof)" :

…把简单道理讲得很复杂:

…其实一个日常生活的案例(火车 Case), 就可以解释清楚。

Note: 林群院士是第一个把复杂吓人的Cauchy Epsilon-Delta Analysis (aka Advanced Calculus) 用中学数学简单化。

Video: 大自然的悟性发现数学:

… 文学 《The War & The Peace》: 托尔斯泰用Calculus 证明Napoleon 必败。

"不是你听不懂数学,是数学老师教得不好。"

https://baike.baidu.com/wikisecond/videoflow?secondId=332699&lemmaId=11364&from=lemma&feedInlemma=1

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Singlish Math : Toa-Cah-Soh大脚嫂

Math Online Tom Circle

In Chinese school Sec 3 we had to memorize by hard these 3 Trigonometric formula:

正弦 sin, 馀弦cos, 正切 tan,
对边比斜边= sin,
邻边比斜边= cos,
对边比邻边= tan,

凶数学老师叫一个个学生轮流从座位上站起来背诵,不会背就罚站。

她说学高数”三角” (Trigonometry) 第一步就是死记这个公式。我最反感无趣味的学习,违背”数学精神” (Mathematical Spirit) 。尤其讨厌 “三角恒等式” (Trigonometric Identities Proof) 证明,用死记的公式纠缠地 (messy) 变来变去,"剪不断, 理还乱",考试差点不及格 (60%)。

80年代 英校数学老师发明一个记忆妙招 :Singlish TOA CAH SOH 大脚嫂👍

Who says Singlish is useless? Our Sec 3 Math students are Top PISA (#1) in the World !

Another smart Singlish Math Chant:

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Math High-Pay Career in Data Science

Math Online Tom Circle

https://towardsdatascience.com/data-science-jobs-with-their-salaries-171acd3bf9be

Gone are the days where Math graduates were destined to low-pay teachers in the 60s to 90s.

Now Math career is the hottest high pay job in the 2 key engines of the “4th Industrial Revolution” , ie Mind Automation by Machine Learning and its ‘Mother’ Artificial Intelligence.

The 3 top jobs (by order of pay) :

1. Data Scientist (US $120K) :

Skills : Math (Stats, Linear Algebra, Calculus, Probability, & potentially Algebraic Topology / Homological Algebra) + A. I.

Mathematician PhD preferred.

2. Data Architect / Engineer (US $100K)

Skill: IT, especially in Big Data technologies

3. Data Analyst / (US $65K)

Skills : IT, Business

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A Programmer’s Regret: Neglecting Math at University – Adenoid Adventures

Math Online Tom Circle

Advanced Programming needs Advanced Math: eg.

Video Game Animation: Verlet Integration

AI: Stats, Probability, Calculus, Linear Algebra

Search Engine : PageRank: Linear Algebra

Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad

Program “Proof” : Propositions as Types, HoTT

https://awalterschulze.github.io/blog/post/neglecting-math-at-university/

Abstraction: Monoid, Category

Category

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IMO 2019

Math Online Tom Circle

【打破美国35年纪录的天才,与中国奥数、信息学双料冠军的巅峰对决】
https://m.toutiaocdn.com/group/6714952524924715531/?app=news_article_lite&timestamp=1563521442&req_id=20190719153042010152029169696E814&group_id=6714952524924715531&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

USA Team (2016 & 2018 World Champion) :

China IMO Team :

【2019国际数学奥赛结果公布 中美两国并列团体冠军】
https://m.toutiaocdn.com/group/6715594110440309261/?app=news_article_lite&timestamp=1563682066&req_id=20190721120745010025067136905FD11&group_id=6715594110440309261&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Two observations:
1. Why almost all Chinese in top China, USA, Canada, NZ, Singapore teams?
Either Chinese chase the wrong Math (IMO) education, or the western Math Power countries (UK, France, Germany…) are RIGHT to ignore IMO education?
2. Singapore team skewed heavily in 2 schools: RI (5) + HCI (1). Also 0 girl.

【解读中美数学奥赛并列第一,敬请期待对话美国队总教练 | 袁岚峰】
https://m.toutiaocdn.com/group/6716112894641046024/?app=news_article_lite&timestamp=1563733425&req_id=20190722022345010152042134236B8C3&group_id=6716112894641046024&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(想看更多合你口味的内容,马上下载 今日头条)
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10部关于数学的顶级纪录片,带你进入奇幻之门

Math Online Tom Circle

【10部关于数学的顶级纪录片,带你进入奇幻之门】
https://m.toutiaocdn.com/group/6719737122653209101/?app=news_article_lite&timestamp=1564575985&req_id=201907312026250101520222026997CB8&group_id=6719737122653209101&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

10 Math Videos:

  1. The Joy of Math
  2. The Joy of Stats
  3. The Joy of Logic
  4. The Code
  5. A Brilliant Madness
  6. The Great Math Mystery
  7. How Long is a piece of string
  8. Hunting the hidden dimension
  9. Dimension: A walk thru mathematics
  10. The secret life of chaos

(想看更多合你口味的内容,马上下载 今日头条)
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New Math applied in Modern Physics

Math Online Tom Circle

【文小刚:物理学的新革命——凝聚态物理中的近代数学 | 众妙之门】
https://m.toutiaocdn.com/group/6712424299841454595/?app=news_article_lite&timestamp=1562923076&req_id=20190712171755010152027098774DD5D&group_id=6712424299841454595&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Keywords:

  • Quantum Entanglement (QE) : 量子纠缠
  • Topological Order (Long-range QE) : 拓扑序
  • Topological Insulator (Short-range QE) : 拓扑绝缘体
  • Rigid State : 凝聚态
  • Cohomology : 上同调
  • Tensor : 张量
  • Category Theory : 范畴学
  • Group Theory : 群论
  • Algebraic Topology : 代数拓扑

….

1. Four “Physics” Revolutions aided by Math

Math is applied to explain the Nature (Physics) :

Calculus (Differential Equation) : Newtonian Physics

Riemann Geometry : Einstein General Relativity

Partial Differential Equation: Maxwell Electro-Magnetic Field

Linear Algebra (Matrix) : Quantum Mechanics

Group Theory: Physics / Chemistry Symmetry

Next…

Category Theory : Topological Order of the long-range Quantum Entanglement

Algebraic Topology: Topological Insulator (2016 Nobel Prize Physcis)

2. The 5th Physics Revolution aided by New Math tool : Category Theory

3. Category Theory = “Structural Relationship” Math

4. Algebraic Topology

(想看更多合你口味的内容,马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link

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5G Polar Codes Dr. Erdal Arikan

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Dr. Erdal Arikan (1958 -) of Bilkent University, Turkey, recieved his PhD in MIT at the age of 27.

Huawei Award: The Father of Polar Codes (极化码)

Industry watchers believe that the selection of polar coding as the channel coding technique for control channels for 5G communications system may have put Huawei at the forefront in the 5G race(27 Gbps in 2016).

1 Gbps = 1000 Mbps

5G Basic Explained:

Part 1: The Math behind 5G Polar Codes

..

Part 2:

Part 3:

Notations:

..

$latex displaystyle F = begin {bmatrix}
1 &…
1 & 1
end{bmatrix}$

$latex A =
begin{pmatrix}
a_{11} & a_{12}& ldots & a_{1n}
a_{21} & a_{22} & ldots & a_{2n}
vdots & vdots & ddots & vdots
a_{n1}& a_{n2} &ldots & a_{nn}
end{pmatrix}
$$latex B =
begin{pmatrix}
b_{11} & b_{12}& ldots & b_{1n}
b_{21} & b_{22} & ldots &…

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“Poisson Distribution Intuition (and derivation)”

Math Online Tom Circle

“Poisson Distribution Intuition (and derivation)” by Aerin Kim 🙏

https://link.medium.com/bho611jxSX

Poisson was a 19CE French mathematician, or Statistician, who invented the Poisson Distribution.

Note: “Applied” Math vs “Pure” (aka Modern / Abstract) Math

Poisson, being an Applied Math expert in the French Science Academy, did not understand the “abstract” Math paper submitted to him by the 19-year-old Evariste Galois in “Group Theory” for the “Insolvabilityof Quintic Polynomial Equations“. Poisson rejected the paper commenting it was too vague, destroying the final hope of Galois to make his invention the “Modern Math” known to the world. The dejected poor Galois died at 21 after a duel during the chaotic era of the French Revolution.

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丘成桐:数理与人文(原创完整版)

China Gaokao 2019 Interesting Math Question

Math Online Tom Circle

Gaokao = Baccalaureate = GCE A Level

…[Solution scrolled below]

Let Ф = 0.618

… Golden Ratio Ф = 0.618

defined as:

“Head to Throat” vs “Throat to Navel” = a/b = Ф ,

also

“Head to Navel” vs “Navel to bottom of Legs” =(a+b) / c= Ф

Knowing :

Head to Neck = 26 cm &

Legs = 105 cm

Find the height (H) ?

Note: The tricky part is to know “the neck is below the throat” & “the legs start some distance below the navel”, hence 2 inequalities :

a <…

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数学大师的数学启蒙书

Math Online Tom Circle

【大师的数学启蒙书】
https://m.toutiaocdn.com/group/6712914606001488391/?app=news_article_lite&timestamp=1562988400&req_id=201907131126390100180730727967601&group_id=6712914606001488391&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Abel / Gauss 自修数学经验:

Read Directly from the Masters – 自接读大师的书 (而非读其他人注解的书)

下面这2本数学经典 "葵花宝典"(有中译版) 是WW1&2 前 世界第一流 宗师 (Noether’s ‘Abstract Algebra’ , Hardy’s ‘Number Theory’ ) 的杰作,值得阅读,收藏。

1. Modern Algebra 史上第一本抽象代数的书: by Van der Waerden (Noether的学生: Artin, Waerden)

2. Number Theory 史上第一本数论启蒙书:

(想看更多合你口味的内容,马上下载 今日头条)
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Simple groups, Lie groups, and the search for symmetry

Math Online Tom Circle

Math History | NJ Wildberger

Summary:

  1. Normal subgroup
  2. Quotient Group
  3. Finite Simple Group: Z5, D5,…
  4. Tesselation (regular)
  5. Platonic solids

… Two families of Group :

  1. Symmetric Groups: Sn
  2. Linear Group: GL(n) = M, det M ≠0

An = Alternating Group

[even permutation, det = +1]

n>=5, An is a Simple (non-commutative) Group

Lie Groups :

SO(2)

SO(3)

Simple Lie Groups : Matrix Group M

  1. Special Linear Group: SL(n), det M = 1
  2. Orthogonal Group: O(n), $latex M. M^T = I$
  3. Symplectic Group : Sp(2n)

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Alibaba “Damo” Global Math Competition

Math Online Tom Circle

https://m.damo.alibaba.com/alibaba-global-mathematics-competition/

Jack Ma is a kongfu fan, his Alibaba offices all named under the Kongfu heroes, eg. Damo Academy.

Damo” 达摩 (Bodhidharma) was an indian Buddhist monk who came to China in 5th Century, established ShaoLin Monastery, also invented Chinese Kongfu.

Problem 1

Hint: Answers in the above link.

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Homotopy Type Theory: Vladimir Voevodsky

Math Online Tom Circle

A self-study Math genius Vladimir Voevodsky, Fields Medalist, Russian.

Voevodsky borrowed idea from Computer Science (“Interface” =hide implementation details) to Math, and vice-versa, from Math to Computer Science :
Homotopy Type Theory (HoTT)

Result: Computer to prove math theorems.

Computer Science Idea:

  • Hiding Implementation Detail
  • Black Box “Interface”
  • Type (类) : Boolean, Integer, String, … (stronger structure than "Set" 集合)

Math Idea (Abstract Geometry : Algebraic Geometry) :

  • Homotopy (同伦)
  • eg. Cup ~ Donut, both are homotopic or same “Type” (in Computer)
  • Isomorphism is Equality “=”
  • Homotopy Type Theory (HoTT):
    • Univalent Theorem: both objects are considered “=” if they are isomorphic (prove if 1-to-1 correspondence 对应)

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Learn Calculus With These Four Online Courses

Math Online Tom Circle

https://lifehacker.com/learn-calculus-with-these-four-online-courses-1836377912/amp

Four Online Free Calculus Courses:
1. http://www-math.mit.edu/~djk/calculus_beginners/

2. “Calculus Made Easy” (1910 )

http://calculusmadeeasy.org/

3. “Essence of Calculus” (12 short videos, with Chinese subtitle)

4. [Best👍] “Calculus” (MIT Prof Gilbert Strang)

https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/

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Chinese Math Olympiad (Primary) Trick vs Singapore Modeling Math

Math Online Tom Circle

Both Methods (I) & (II) below does not use Algebra, which is not taught to primary school kids until secondary school (above 12 or 13 years old).

(I) Ancient Chinese Arithmetic Method:

例2:The sister is 13 years old this year, while the brother is 9 years old.
When the total of their ages is 40, how old will they be?

【这位妈妈太绝了!竟把小学6年奥数化为13句口诀】

Trick: Age difference does not change over time, they + or – in tandem. When 2 ages change, the multiple between them also changes.

https://m.toutiaocdn.cn/group/6636154201682477582/?iid=65171531104&app=news_article_lite&timestamp=1552573996&group_id=6636154201682477582&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(II) Singapore Modeling Math:

[例2]

Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):

  • (II) is better for young kids to understand visually the logic behind,
  • (I) is more a memorised trick but poor math education pedagogy. In [例 2] it is very hard for kids to understand why the 2 steps (40+4) & (40 – 4), although they lead to…

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English Fun Math

Math Online Tom Circle

Different Maths:
1. English (UK/USA) Math is playfully tricky (Math Olympiad, Math Puzzles eg. this Numberpile video)

2. French Math is abstractly theoretical (Group theory explains Symmetry, Rubik’s cube game,… )

3. Chinese Math (Abacus, “Chicken-Rabbits Problem” 鸡兔问题, Singapore Modeling Math) is applied & procedural (明朝 13CE invented “Determinant” 行列式 before the 19th CE “Matrix” by UK Sylvester,Chinese Remainder Theorem 韩信点兵).

4. Indian Math (Vedic Math , Factorization, Ramanujian-Style Math) is mysteriously spiritual.

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AI with Advanced Math helps in discovering new drugs

Why French excel in math ?

Math Online Tom Circle

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront…

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The Laws of Mathematics Zermelo-Fraenkel Axioms Set Theory

Math Online Tom Circle

The Foundation of Modern Math which is based on the “Set Theory”, hit a crisis due to Bertrand Russell “Barber’s Paradox”, until the Zermelo-Fraenkel Axioms came to its rescue.

Here is a humorous example of an ISP Internet Plan violating the 9 Z-F Axioms:

How ISPs Violate the Laws of Mathematics:

Video:

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Group Theorems: Lagrange, Sylow, Cauchy

Math Online Tom Circle

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

However,
if h = p (prime number),
=>
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
ie.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)

Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H…

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陈省身 SS Chern – “The 2nd Gauss”

Math Online Tom Circle

陈省身香港科技大学 演讲经典 (Highly Recommended – spoken in English & Chinese)

http://www.bilibili.com/video/av5473133

他的女婿是香港科大校长。现任校长是 Prof 陈繁星 Tony F. Chan , the ‘Search Algorithm Thesis’ supervising Prof of the 2 Google founders in Stanford University, Applied Math Department)

Key Points:

  1. SS Chern won the “Wolf Prize” in the same year with the Hungarian “Vagabond” 流浪汉 Mathematician Erdos Paul.
  2. He was appointed the Scientific Advisor of President Reagan, during which Chern recommended to build the USA Center of Math in Berkeley University. After retirement, Chern built a similar center in China at his alma mater Nankai University 南开大学 where he graduated in 1930.
  3. SS Chern was an assistant lecturer of Prof Yang WuZhi 杨武之, who liked to invite Chern to his house for dinners, where he saw his 8-year old son Yang ZhenNing 杨振宁 (Nobel Prize Physics 1958).
  4. SS Chern has guided great PhD students eg. James Simons (Billionaire Financial Investor applying…

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