数论的世界 Number Theory

【数学之美 | 数论的世界】
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20世纪初数学界的三国混战时代,最终被他一统天下】

20世纪初数学界的三国混战时代,最终被他一统天下
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The Math we learnt in High school is pre-19th Century up to Newtonian Calculus of 17CE.

The birth of Modern Math since 19CE till WW2 is the “Abstract Algebra” from French Revolution Galois “Group Theory” 群论.

After WW2 till now, Mathematics faces the crisis of “Truth”: whether its Foundation is correct.

3 schools of fight on the Fondation of Mathematics:

1. Russell (Logic with Types to fix the “Russell Paradox” in Set Theory Crisis)

2. Hilbert (Axiomatization of all Mathematics)

3. Brouwer (Intuitionism) against "排中律" (Law of the Excluded Middle)

Winner: Godel “The Incomplete Theorem” (不完备定律)

However, the by-product of these 3 school fights give rise to new Math discovery in Machine Proofing (2010s):

Homotopy Type Theory (HoTT) = Logic (Proof) + Type (Intuitionist) + Topology (Homotopy).

ie. Math Proof = Computer Program

函数概念并不难,理解“函”字是关键——函数概念如何理解】

【函数概念并不难,理解“函”字是关键——函数概念如何理解】
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清. 李善兰 翻译 Function 为函数。函,信也。只能有一个收信人,所以 只有一个 f(x) 值。

The unique 1 single output of a function becomes very important for subsequent development in Math & IT:
functions are composable, associative, identify function,etc (distributive,… ) => it can be treated like vector => structure of a Vector Space “Vect”

Extended to..

“Vect” is a bigger structure “Category” in which “function of functions” is a
Functor” (函子)F:F(f)

Example : F(f) = fmap (in Haskell)

fmap (+1) {2,7,6,3}

=> {3,8,7,4}

here F = fmap, f = +1

The Math branch in the study of functions is called “functional” 泛函。

IT : Functional Programming in Lisp, Haskell, Scala, ensure safety of guaranteed output by math function property. Any unexpected exception (side effects: IO, errors) is handled by a special function called “Monad” (endo-Functor).

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张益唐 天才的野心: 《朗道-西格尔零点》猜想

【张益唐 天才的野心】
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Dr. Zhang YiTang (1955 -) the “Subway Sandwich” mathematician who, before 50 was still a temporary hourly worker… until 2013 at 58 proved the “70-million Twin Prime Gap“.

He becomes overnight famous worldwide, now a tenure professor.

His next ambition is the “Landau-Siegel Zero Conjecture”, a weaker form of the Millenium Problem “Riemann Conjecture”.

下一个“野心

Landau–Siegel zeros and zeros of the derivative of the Riemann zeta function

https://www.sciencedirect.com/science/article/pii/S0001870812001600

只差技术问题…

专注入神

够了,该放就放

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许晨阳谈菲尔兹奖趣闻:如何打破“数学诺奖”魔咒?

【许晨阳谈菲尔兹奖趣闻:如何打破“数学诺奖”魔咒?】

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数学最高奖沃尔夫数学奖的历史 Wolf Prize

【数学最高奖沃尔夫数学奖的历史】
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