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

Octonions & United 4 Forces

剑桥数学物理学家以八元数打开“万有理论”研究思路

This Cambridge Mathematician explains why need so many number system :
N Natural,
Z Integers
R Real,
Complex,
Quartenions (1,i,j,k) ,
Octonions

https://m.toutiaoimg.cn/group/6584683147512775182/?app=news_article&timestamp=1607604783&group_id=6584683147512775182&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

She believes the Octonions is the secret behind united field of the 4 forces :

G gravitational,
E electrical,
M magnetic,
W weak force in particules.

Maxwell United EM,
Einstein United GEM,
She believes her research in Octonions can unite GEM+W.

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

Particle Physics solved by Algebraic Topology with Co-Homology structure

物理学家已经确定了粒子碰撞的代数结构,这将诞生新的物理理论

Keywords: Algebraic Topology 代数拓扑, Co-Homology structure 上同调结构

https://m.toutiaocdn.com/i6886832762934788620/?app=news_article&timestamp=1603662776&use_new_style=1&req_id=20201026055255010014047206182262C3&group_id=6886832762934788620&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Wolf Prize : Donaldson

The Modern Math is now proved by Physics, unlike the past history in reverse direction.

The first Math Fields Medals awarded to a Quantum Physicists Edward Witten in String Theory, now the 1st Wolf Prize to Donaldson who proved Math with Physics 杨振宁 Yang-Mills Equation (which is still one of the 7 Millenium Math Conjectures worth $1 m Clay Prize, but proven true in Physics Experiment) .

Wolf Prize is S$1m for 数学终身奖, unlike $100K Fields Medal for < 40 years old. Only Harvard Dean 丘成桐 (HK) received both prizes, while his mentor 陈省身 only Wolf (same year as Paul Erdos).

Simon K. Donaldson

《万物皆数》

[中文书评]

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

Audio book ]: It all adds up – the story of people & Mathematics

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

Nuclear Physics Leads to Linear Algebra Non-Traditional Method

Strange Nuclear Physcis leads to Math discovery of Linear Algebra new method.

Traditional Method:

Characteristic Polynomial – >Find Eigenvalues – > Solve Simultaneous Linear Equations – > Eigenvector

https://m.toutiaocdn.com/group/6759397458687558151/?app=news_article_lite&timestamp=1573802277&req_id=201911151517560100140481302E37987B&group_id=6759397458687558151

https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/

Traditional Way : Find eigenvalues first, then eigenvector.

Terence Tao’s Comments:

https://m.toutiaocdn.com/group/6759905079915971085/?app=news_article_lite&timestamp=1573994548&req_id=20191117204228010014041161230B3A0B&group_id=6759905079915971085

Relativistic Space Time, Fengshui, Isometry

风水 Fengshui (aka Geomancy) is Einstein Relativistic using Compass (东西南北上下 =Space) & Timing (吉凶日/时间 = auspicious date & time).

\Yu (Space) \Zhou (Time) = Space-Time
eg.
宇环 (universal surrounding)
宙时 (instant timing)

https://www.asiatimes.com/2019/08/opinion/why-quantum-physics-needs-asian-philosophy/

Axonometry = Isometry ( dengjiao toushi 等角透视,ie Angle Preserved Projection )

Chinese art is axonometric using isometry geometry, scrolling from right to left along time. Example : this digital animated 12-century painting

清明上河图 Qingming Festival River Scene(北宋)

“Why the world relies on a Chinese ‘perspective’” by Jan Krikke, China, AI and Quantum Physics

https://link.medium.com/z5LaoL0tdZ

Paul Dirac: Physcis must have Math Beauty

【物理理论离不开数学之美!一个划时代的方程式!】

Paul Dirac : Unified Relativity with Quantum Mechanic

https://m.365yg.com/group/6713849761440465416/?app=video_article&timestamp=1563627447&utm_source=native_share&utm_medium=android&utm_campaign=client_share
(想看更多合你口味的内容,马上下载 西瓜视频)
https://www.ixigua.com/app/?utm_source=native_share&utm_campaign=client_share&utm_medium=android&app=video_article

Group Theory in a Nutshell for Physicists

Anthony Zee 徐一鸿 (1945- 上海) Ameican-Chinese.

2016 University of Paris-Saclay (Grouping of University of Paris-South, etc, with the Grandes Ecoles : Ecole Polytechnique, Ecole Centrale-SupElect).

Prerequisites:

  • Group definition
  • Continous group, Lie Group
  • Group Representation (Matrices)

Mystery:

Division Algebras and Physics

Why 4 Physical forces, 4 Number Systems?

  • Gravitatiinal force, Electrical force Magnetic force, Particle weak force
  • Real number, Complex number, Quartenion number, Octonion number

Division algebras and physics – version for the public: (2 mins)

Ideal = Algebraic “Black Hole”

[Note: What is an Ideal? ]

Minimum Left Ideal = Left Ideal with no trivial ideals

Quartenions = Associative

Clifford Algebra = Associative

Octonions = Non-Associative

Result Summary :

New Math applied in Modern Physics

【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

In Math We Trust: Quantum Computing, AI and Blockchain – The Future of IT

In memory of Prof Zhang Shoucheng 张首晟教授 who passed away on 1 Dec 2018.

Key Points :

  1. Quantum Computing with “Angel Particle” (no anti-particle) : [Analogy] Complex Number (a + i.b) , ‘Anti’ = Conjugate = a – i.b, ‘No anti’ = Real number = a
  2. A. I. Natural Language Algorithm : “Word To Vector” eg. King / Queen (frequently appear together) , etc.
  3. Data Privacy and Big Data Analytics with A. I. : Homomorphic Encryption, ie reveal data but not privacy. (eg. Millionaire Problem)

Joseph Fourier is Still Transforming Science

Key Words: 250 years anniversary

  • Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
  • Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
  • Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
  • Taught at the newly established Military Engineering University “Ecole Polytechnique”.
  • Scientific Research: Short period but intense.
  • Before Fourier died (he wrapped himself with thick blanket in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

https://news.cnrs.fr/articles/joseph-fourier-is-still-transforming-science

Symmetry, Algebra and the “Monster”

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.

https://www.quantamagazine.org/symmetry-algebra-and-the-monster-20170817/

Summary:

  • Symmetry of a Square
  • Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
  • What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
  • Monster Group = God ?
  • String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry),是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

Prof ST Yau’s 丘成桐 Talk to Chinese Youth on Math Education 


Prof ST Yau 丘成桐 , Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal 1982 (at 27, proving Calabi Conjecture), Crafoord Prize (1994) , Wolf Prize (2010).

Key Takeaways :

1. On Math Education
◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc.
◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after lecture do the problems to enhance understanding.
◇ Read Math topics even though you do not understand in first round, re-read few more times,  then few days / months / years / decades later you will digest them. (做学问的程序).
◇ Do not consult students in WHAT to teach, because they don’t know what to learn.
◇ Love of Math beauty is the “pull-factor” for motivating  students’ interest in Math.
◇ Parental Pressure.

2. “3D” facial photo using Math

3. Pi-Music: 1 = “do”, 2 = “re”, 3 =”me”…
Pi =3.1415926…

4. Math Olympiad: Prof ST Yau had criticised publicly it as a bad Math training, not the “real” Math. 

An audience tested Prof ST Yau on a Math (Accounting) Puzzle which he couldn’t  solve on the spot. He said Mathematicians are poor in +-×÷ arithmetic. 

5. Chinese students in USA: China sends over 200,000 students to USA universities. They are good in secondary / high school Math with known solutions,  but poor in graduate PhD Math which requires “out-of-the-box” independent thinking skill for finding unknown solutions. Recent few years Chinese students (eg. Stanford Prof 李骏 : 1989 Harvard PhD)  in USA have improved standard in PhD research.

6. Research is not for fame. It takes many years to think through an interesting topic.

Reference:

1. Prof ST Yau’s Best Seller Book 《The Shape of Inner Space》avail @ NLB (Ref #530.1) 11 copies in most NLB branches@ AMK, Bishan etc.

2. Interview Prof ST Yau by HK TV (Cantonese)

3. 丘成桐 (2008) 评中国 和 美 国的教育 : 中国学生不爱看课外书, 因为考试太重, 课余时间花在玩电脑游戏。

4.  丘成桐 (2016): 中国大学本科要注重基础教育, 才能培养世界级一流人才

Curious Thoughts in Math & Science 

1. Statistical Mechanics: e^ {- Ht}

Quantum Mechanics: e^{iHt}

2. Ramanujian:

1 +2 + 3 + …+ n = –  1/12

Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

Tau Special Function:

\boxed {\displaystyle \sum_{n=1}^{\infty}\tau (n) x^{n} = x \{(1-x)(1-x^{2})(1-x^{3})... \}^{24}}

3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843).

AND\boxed {x.y} 

NOT\boxed {1-x} 

XOR\boxed {x+y-2x.y} 

Extra constraints ”  \boxed {x^{2}=x} 

 4. Solomon Golomb, Sol: “Linear Feedback Shift Register” (LFSR)  — shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

Maximal Length = The shift register of size n will repeat every 2^{n}-1 steps (exclude all ‘0’ sequence).

Which arrangement of “Taps” will produce the maximal length ? 

Solomon applied Pure Math : represent the above sequence of  registers algebraically  by: 

\boxed {x^{7}+x^{6}+ 1} 

in reducible modulo 2 (prime in polynomial, ie can’t be factored).

=> the sequence is Maximal length

LFSR Applications: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance,  etc.