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：
林群院士：
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).
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。)
钾 钠 钙 镁 铝 锌 铁 锡 铅 (氢)
K Na Ca Mg Al Zn Fe Sn Pb (H)
[嫁 那 个 美 女 身 体 细 倩 (轻)]
铜 汞 银 铂 金
Cu Hg Ag Pt Au
[统 供 一 百 金]
28 Dec 2019 @中国科学技术大学研究生院
Key Points:
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)
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.
These 2 smart Singaporean students invented “Silver Ratio” , “Bronze Ratio” … equally good as the Golden Ratio.
Recap: Golden Ratio
https://plus.maths.org/content/silver-ratio
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)
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/
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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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.
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)
She is a Cambridge PhD in Quantum Computing.
How to learn Quantum Mechanics on your own :
1. MEAN / MEDIUM
2. PERCENTILE
3. SKEWNESS
4. STANDARD DEVIATION
5. PCA = Pricinpal Component Analysis
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
【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
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:
Given: 3 circles of radius 2
Find: Shaded Area?
Hint:
Sum of 3 angles in the triangle
= π = 1/2 (2π) ＝1/2 circle
=> its Area = 1/2(π. 2^2) = 2π = Sum of 3 arcs’ area
Prof Loh (CMU) is the 2019 USA IMO coach whose team (5 ABC, 1 White) co-won the IMO Championship with China.
…
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
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
【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×tamp=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×tamp=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×tamp=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)
http://app.toutiao.com/news_article/?utm_source=link
10 Math Videos:
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Find the last 2 in Singapore NLB libraries:
“The Joy of X” , “How to bake Pi”
【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×tamp=1562923076&req_id=20190712171755010152027098774DD5D&group_id=6712424299841454595&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
Keywords:
….
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
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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)” 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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Gaokao = Baccalaureate = GCE A Level
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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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Gaokao 高考 ＝ Baccalaureate = Cambridge GCE A-Level
Hint:
f(x+1) = 2f(x)
tells the height = f(x) double when x moves to x +1
=> periodicity is 1
Like Sine or Cosine curves have periodicity of 2π.
sin (2π + A) = sin (A)
Ref: More complicated curve drawing by French Method.
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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Math History | NJ Wildberger
Summary:
… Two families of Group :
…
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
…
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.
https://dev.to/renegadecoder94/understanding-the-number-theory-behind-rsa-encryption-1pdo
Background knowledge:
1. Group
2. Cardinality : Euler Totient Function
Note:
A linguistic remark: The Latin word “totiens” (or “toties”) means “that/so many times”.
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:
Math Idea (Abstract Geometry : Algebraic Geometry) :
Unicode UTF-8:
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 )
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/
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.
(II) Singapore Modeling Math:
［例2］
Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):
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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.
Advanced Mathematical Methods with AI is a powerful tool:
https://sinews.siam.org/Details-Page/mathematical-molecular-bioscience-and-biophysics-1
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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Urbain Le Verrier, Who Discovered Neptune
Example:
4% of 75 = 75% of 4
Mental Steps:
75 x 4 – > 300 – > / 100 – > ＝ 3 (ANS. )
Is it easier ?
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:
Challenge: Put ’15’ and ’14’ in the right sequence (Impossible! )
以下无解：群论
“Essential mathematical GIFs that will make mathematics finally make sense” by Ali Kayaspor https://link.medium.com/z1nWgHm9GU
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:
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