3 Math Foundation Skillsets for Engineering / Finance Students


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




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.


Octonions & United 4 Forces


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


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.



[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) :


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




Particle Physics solved by Algebraic Topology with Co-Homology structure


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


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




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


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



Traditional Way : Find eigenvalues first, then eigenvector.

Terence Tao’s Comments:


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
宇环 (universal surrounding)
宙时 (instant timing)


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


Paul Dirac: Physcis must have Math Beauty


Paul Dirac : Unified Relativity with Quantum Mechanic

(想看更多合你口味的内容,马上下载 西瓜视频)

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).


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


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】


  • 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


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)

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”.


Symmetry, Algebra and the “Monster”

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



  • 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.


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.