Analysis -> (Topology) -> Algebra

Mathematics is divided into 2 major branches:
1. Analysis (Continuity, Calculus)
2. Algebra (Set, Discrete numbers, Structure)

In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.

Topology specialised in
–  ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.

Example:
The local data of consumer behavior uses ‘Point-Set Topology'; the global one is ‘BIG Data’ using Algebraic Topology.

Kolmogorov Complexity

What are the differences of these strings of characters?

1111111111111…
010101………..01…
1001100100001….

The 1st string is all ‘1’
The 2nd string is all ’01’
The 3rd string is complex: random ‘0’, ‘1”

Kolmogrov complexity deals with randomness.

https://en.m.wikipedia.org/wiki/Kolmogorov_complexity

http://www.amazon.com/product-reviews/0387339981/ref=dp_ob_custreviews_bk_cm_cr_acr_img?showViewpoints=1

Goldbach Conjecture & Theorem Chen

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It said: “(The above) Prof Yang was a long time colleague of Chen Jingrun (陈景润). He revealed that Chen was, by nature, not a Math genius, his success in climbing the Math Summit was through sheer determined hardwork — especially during the decade long cruel Cultural Revolution when the intellectuals were persecuted — sustained by a very strong interest in Math.”

“Math journey is not a short 100-meter sprint but a 20-year marathon run.”

“He critised the crazy rush of Math Olympiad training in whole China, not only adding extra school burden on kids, it also kills their interest in Math.”

Note: Recently China government bans IMO training in schools.

Chen Jingrun (陈景润), a student of Hua Luogeng (华罗庚), proved in 196x during The Cultural Revolution: (1+2) Goldbach Conjecture.

Goldbach Conjecture:

Any Even Integer = 1 prime + 1 prime

eg. 12 = 5+7 = “1+1″

Thinking in Philosophy:
Any prime (odd numbers except 2) together with another prime make an Even number:

Two single odd (prime) persons together make a couple (even).
二个单(odd)身人 结合成 偶 (even)。

This is the biggest secret in Nature (Natural Number). To prove it is extremely impossible, unless guidance from God.
这是自然数(Natural numbers) 的奥秘, 天意 ! 要证明它, 难若登天, 除非上帝给你指引。

Theorem Chen:

Chen’s proof is the world’s best approximate one (so far):
Any Even Integer = 1 prime + (1 prime x 1 prime) = “1+2″

Note: (1 prime x 1 prime) = semi-prime

陈景润证明:
1 单(男) + 2 单(妻x妾) = 对偶
。还是非天意 :)

陈景润电影:

IMO 2015 USA beat China after 20 Years

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The result is not surprising to China but to USA:
♢Recently China government bans IMO training in schools.
♢Obama was surprised that the USA IMO team consists of predominantly Chinese American students.

IMO Math is like ‘Acrobatics’ to real ‘Kung-fu’, it is not real Math education, but special ‘cute’ techniques to solve tough ‘known’ solution problems. Real Math is long R&D solving problems with UNKNOWN solution (eg. Fermat’s Last Theorem, Riemann Conjecture,…)

2 types of Math: Algorithmic or Deductive (演绎). Chinese long traditional ‘abacus’ mindset, procedural computational Math is Algorithmic, applied to special cases (eg. astronomy, calendar, agriculture, architecture, commerce,…). European Greek’s Euclid deductive, step-by-step axiom-based proofing, is theoretical, generalized in all cases (Geometry, Abstract Algebra,…)

Look at the Fields Medal (aka ‘Nobel Prize’ of Math) super-power – France – which has produced 1/3 of the Fields Medalists, but performing so-so in IMO. In contrast, China has ZERO Fields Medalists, albeit dominating IMO championship for more than 2 decades!

IMO 2015:
https://www.imo-official.org/results.aspx
USA 1st,
China 2nd,
South Korea 3th,
North Korea 4th,
Vietnam 5th,
Australia 6th
Iran 7th
Russia 8th
Canada 9th
Singapore 10th [2012 Individual World’s Champion ]
Ukraine 11th
Thailand 12th
Romania 13th
France 14th

United Kingdom 22th

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