Tag Archives: Andrew Wiles

Andrew Wiles FLT: Galois Representation, Deformation Theory

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@corneliusgoh110

Fermat’s Last Theorem #Andrew Wiles

♬ original sound – Cornelius – Cornelius

[Deepseek 解释]

好的！我用一个超简单的例子来解释 “变形理论”（Deformation Theory） 在数学里是干啥的，尤其是它怎么和 “伽罗瓦表示”（Galois Representation） 扯上关系。

例子：橡皮筋上的蚂蚁（直观理解“变形”）

想象你有一根 橡皮筋，上面趴着一只蚂蚁。

  • 初始状态（未变形）：橡皮筋是放松的，蚂蚁的位置在点 A。
  • 拉长橡皮筋（变形）：现在你把橡皮筋拉长，蚂蚁的位置变成了 A’，但它还是那只蚂蚁，只是位置“变形”了。

这里的：

  • 橡皮筋 = 数学对象（比如一个方程的解，或一个伽罗瓦表示）。
  • 拉长/扭曲 = “变形”（Deformation），即连续改变它的某些性质（但保持核心结构不变）。

数学例子：解方程的“变形”

设我们有一个超级简单的方程：
[ x^2 – 1 = 0 ]
它的解是 x = 1 和 x = -1。

(1) 初始伽罗瓦表示

  • 这个方程的对称性（伽罗瓦群）是 {交换1和-1}，记作 (G = \mathbb{Z}/2\mathbb{Z} )。
  • 我们可以用矩阵表示这个对称性，比如：
  • 恒等变换（不动）：\rho(e) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
  • 交换1和-1： \rho(g) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

(2) 变形：让解“稍微变化”

现在，我们让方程 “变形”，比如加一个小扰动：
x^2 - (1 + \epsilon) = 0
解变成 x = √(1+ϵ) 和 x = -√(1+ϵ)。

  • 伽罗瓦群还是 ( \mathbb{Z}/2\mathbb{Z} )（因为还是交换两个解）。
  • 但矩阵表示可能变成 \rho_\epsilon(g) = \begin{pmatrix} 0 & 1+\epsilon \\ 1+\epsilon & 0 \end{pmatrix}

这里，“变形”就是让矩阵里的数从 1 变成 1+ϵ（一个微小的变化）。

为什么这有用？

数学家研究这种“变形”是为了：

  1. 分类所有可能的对称性：比如所有能让方程解“连续变化”的表示。
  2. 连接不同的数学对象：比如证明某些变形后的伽罗瓦表示一定来自模形式（这是费马大定理证明的核心）。

说人话总结

  • 伽罗瓦表示 = 用矩阵描述方程解的对称性。
  • 变形 = 让这些矩阵“微微调整”，看看能变成啥样。
  • 用途 = 就像用橡皮筋的拉伸来研究蚂蚁能爬到的所有位置，数学家通过变形研究伽罗瓦表示的所有可能性。

这个例子虽然简化，但希望让你直观感受到“变形”在干啥！ 😊

Weil’s Proof of Fermat Last Theorem

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Canadian Prof Darmon was the post-doc student of Prof Andrew Wiles.

This lecture on Fermat’s Last Theorem marvelous proof is “simplified” into key concepts accessible to Math (honors) undergrads or Classe Préparatoire (spécialisé M/M’):
Elliptic Curve /Modular Form, Galois Field,
Langlands Programme,
etc.

困扰了人类358年 费马大定理 Fermat’s Last Theorem

Modern Math 1 Comment

Keywords:

  1. Fermat’s Last Theorem (FLT): \boxed {x^{n} +y^{n} = z^{n} \; \; \forall n >2 }
  2. Pierre de Fermat (France 1637 AD): FLT Conjecture or Prank ?
  3. Euler (n= 3)
  4. Taniyama(谷山)-Shimura(志村)-(André) Weil Conjecture (now Theorem)
    Modular Form = Elliptic Curve
  5. Galois Group Symmetry
  6. Andrew Wiles (UK Cambridge 1994):
    (Modular Form = Elliptic Curve) <=> FLT (q.e.d.)

Notes:

1. Do not confuse Prof Andrew Wiles (proved FLT) with (French/American) Prof André Weil (Founder of Bourbaki School of Modern Math in POST-WW2 universities worldwide).

2. When Shimura heard that FLT was proved finally by Andrew Wiles using his Conjecture (Theorem), he was very calm, said, “I told you so”.

3. There is another “Fermat’s Little Theorem” used in computer cryptography.

4. The journey of proving FLT in 358 years by all great mathematicians worldwide had created many new math tools and Number theories (eg. “Ideals” in Ring Theory, etc).

费马大定理 Fermat’s Last Theorem

History, Modern Math 1 Comment

费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来近代数学(Modern Mathematics)的突飞猛进。

1977秋 ~1979秋 笔者在法国-图卢斯(Toulouse, Southern France, Airbus 产地)费马学院 (College Fermat, aka Lycée Pierre de Fermat: Classe Préparatoire, 178th Batch)读两年的大学近代数学 (Mathématiques Supérieures et Spéciales), 尝过一生读书的”地狱”生活, 严谨(Mathematical Rigor)的思考训练, 像地鼠般(法国人戏称taupe)不见天日, 废寝忘食的煎熬。 当年对数学的恐惧, 终生牢牢铭记在心; 30年后”由惧转爱”, 数学竟然成为半退休后的业余嗜好, 享受数学的美 — 也是造物者宇宙天地的美!

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FLT 350年数学长征英雄人物:
1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

“数风流人物, 还看今朝”集大成者 :
8. Andrew Wiles (怀尔斯) 证明 (1994 -1995)”盒外思路” (Think Out of The Box): The Great Moment of 1994 Proof (YouTube)

\boxed {(1) = (2) = (3) }
(1). Elliptic Curve (椭圆曲线)
(2). Modular Form (模形式)
(3). Fermat’s Last Theorem (费马大定理)

费马大法官品尚清高, 讨厌政界官僚逢场作戏的应酬, 工余爱躲在家里玩数学, 然后写信和好友(巴斯卡 Pascal, 笛卡儿 Descartes,…)讨论, 无心中发明了物理(Optics)定律, 或然率 (Probability – 和Pascal合作), 解析几何 (Analytical Geometry – 和Descartes合作)…尤其他是近代数论(Number Theory)的开山鼻祖 (他的另一个Fermat’s Little Theorem今天用在电脑密码RSA Encryption)。
他偶然读到3,000年前希腊数学家Diaophantine的书 (10世纪阿拉伯人保存, 16世纪拉丁文翻译自阿拉伯文)。他心血来潮, 在书眉写道: “我找到一个漂亮的证明这题Diaophantine Equation, 但此书旁地方太小, 不能写下”。 他死后, 儿子整理遗作而发现此书, 就成为350年来的数学疑案。

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Shimura-Taniyama-Weil Conjecture (Modularity Theorem)

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Shimura and Taniyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.

The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.

It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:

Example:
\boxed {y^{2} + y = x^{3} - x^{2} } — (I)

There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.

For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)

Note 1: the last solution when y=4,
Left-side = 16 + 4 = 20 = 4×5 = 0 (mod 5).
Right-side = 1-1= 0 (mod 5).

Note 2: We call the equation (I) a “Curve over a finite field” since {0, 1,2,.. p-1} is a Field with finite p elements.

Mathematicians for some time have known that if N is a prime number (p), there will be roughly p solutions.

However, the most interesting number is a_{p} = the difference between p and the actual number of solutions.

For N = p = 5, the above equation has actually 4 solutions,
\boxed {a_{5} = 5 - 4 = 1}

Note: a_{p} can be positive or negative.

There is a ‘general rule’ (generating function) to predict a_{p}, and it is inspired from the ubiquitous Fibonacci numbers.

Recall:
Definition of the Fibonacci sequence as a recurrence relation:
\boxed{ F_{n}= \begin{cases} 0, & \text{for }n=0\\ 1, & \text{for }n=1\\ F_{n-2} + F_{n-1} , & \text{for } n \geq { 2} \end{cases} }

Alternatively there is also a generating function for Fibonacci numbers:
q + q(q+q^{2})+ q(q+q^{2})^{2} + q(q+q^{2})^{3} + q(q+q^{2})^{4} + ...

Let’s expand it we get the infinite series:
q + q^{2} + 2q^{3} + 3q^{4 } + 5q^{5} + 8q^{6} + 13q^{7} +...

The above coefficients coincide with
Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13…}

In 1954, the genius German mathematician Martin Eichler took the cue from the above, discovered another generating function:

\boxed {q(1-q^{1})^{2} (1-q^{11})^{2} (1-q^{2})^{2} (1-q^{22})^{2}(1-q^{3})^{2} (1-q^{33})^{2}(1-q^{4})^{2} (1-q^{44})^{2} ... } — (II)

Let’s expand it, we get:
q-2q^{2} -q^{3}+ 2q^{4} + q^{5}+2 q^{6}-2q^{7} -2q^{9} -2q^{10}+ q^{11} -2q^{12}+ 4q^{13}
Let b_{m} denotes the coefficient of the term q^{m}:
b_{1} = 1, b_{2} = -2, b_{3} = -1, b_{4} = 2, b_{5} = 1, ...

Eichler discovered that for any prime p,
\boxed { b_{p} = a_{p}}

Check: b_{5} = 1 = a_{5}

The random numbers of solutions in the elliptic curve equation (I) lies on the generating function (II).

If we view q as a point inside a unit disc on the complex plane, there is a group of symmetries and the function (II) is invariant under this group. The function (II) is called a modular form.

image

The advanced generalisation of the Shimura-Taniyama-Weil Conjecture : we replace each cubic equation by a Representation of the Galois Group; and the modular form generalised by the generating function the “automorphic” function.:

image

Remarks:
1. The Shimura-Taniyama-Weil Conjecture is a special case of Langlands Program.

2. Weil’s “Rosetta stone”:
Number Theory -> Curves over Finite Fields -> Riemann Surfaces

References:

http://en.m.wikipedia.org/wiki/Modularity_theorem

Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

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Click below for more free loan at the Singapore National Library Branches: http://www.nlb.gov.sg/mobile/searches/view_availability/200154975

Our Daily Story #2: The man who cracked FLT

Math Story, Modern Math 1 Comment

Follow up with the story #1 on FLT (Fermat’s Last Theorem),  it was finally cracked 358 years later in 1994 by a British mathematician Professor Andrew Wiles in Cambridge.
The proof of FLT is itself another exciting story, a 7-year lonely task on the attic top of his Cambridge house, nobody in the world knew anything about it, until the very day when Prof Wiles gave a seemingly unrelated lecture which ended with his announcement: FLT is finally proved. The whole world was shocked!

http://en.m.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem

Part 1/5 Andrew Wiles and FLT Proof:

(Part 2 – 5 to follow from YouTube)

Speech at IMO by Andrew Wiles: