Summary of Persistent Homology

Singapore Maths Tuition

We summarize the work so far and relate it to previous results. Our input is a filtered complex $latex K$ and we wish to find its $latex k$th homology $latex H_k$. In each dimension the homology of complex $latex K^i$ becomes a vector space over a field, described fully by its rank $latex beta_k^i$. (Over a field $latex F$, $latex H_k$ is a $latex F$-module which is a vector space.)

We need to choose compatible bases across the filtration (compatible bases for $latex H_k^i$ and $latex H_k^{i+p}$) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module $latex mathscr{M}$ corresponding to $latex K$, which is a direct sum of these vector spaces ($latex alpha(mathscr{M})=bigoplus M^i$). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.

Specifically, each $latex mathcal{P}$-interval $latex (i,j)$ describes a basis element…

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Part 2:   群表示论的基本概念和Abel群的表示 Group Representation (Abel Group)

引言 : Part 1 温习]

群表示是 Group Action ”双面镜” – 了解 群 G 的总信息, 也同时了解 集合 Ω 的性质。

只选择:
1. 子群可逆 线性 变换群  [< “全变换群 S(Ω)”]
2. 集合 Ω = 线性空间

群表示论: 同态 Φ : G -> Ω

第一课: 映射 Mapping (f) 集合 A, B

北大 丘教授说他大学读数学时 (@ video 2:00 mins), 也是”知其然, 不知其所以然”, 即 不知”抽象数学”概念的motivations behind。 当了讲师后, 为了讲课, 重新温故知新, 明白多一些。后来再写书, 才完全透彻了解。这是肺腑之言! 我们以为教授很厉害, 什么都懂, 其实是经过长期的体会, 思考和苦学得来的。这样的名师 才能帮学生少走学习的冤枉路。

学习过程比课本学问更重要! 

这就是为何Nobel Prize winners的老师并没拿Nobel Prize。中国人1300年的状元们, 都是和本身科举失败的老师学习的。


f: A \to B
f: a \mapsto b , a \in A, b \in B
f(A) = \{ f(a) | a \in A \} \subseteq B (f 的值域 : “Im f”)

A : 定义域 domain
B : 陪域 co-domain: 唯一
满射 Surjective, 单射 Injective , 双射 Bijective

Note: 世界各国 (除了法国)的高中数学都只教Functions (函数), 没有 “映射” (Mapping, 法文: l’Application)。丘教授说中国高中数学也不例外。Function 中学教材是 WW1 德国数学教育家 Felix Klein 提倡的。Mapping 映射 是法国Bourbaki 学派 在WW2 改写 大学数学教材的基础 (Set Theory), 推行到法国高中 (Baccalaureate)数学。法国大师Grothendieck 更建议改写 数学的基础(foundation), 用更基础 的 “范畴论” (Category Theory) 代替 “集合论”(Set Theory), 但太先进不被 Bourbaki 同僚接受, 相信有一天他的梦想可以实现。

第二课: 线性空间  Linear Space, 线性变换 Linear Transformation, 同态 Homomorphism

Proposition (命题):  
f : A \to B
\boxed { f \text { reversible} \iff f \text { bijective} }

Projection 投影 P_{U} \implies  线性变换 –– 非常典型的同态例子!

V = U \oplus W , W non-unique

V = U \oplus U^{\perp}

第三课: 群各种同态 (单 / 满 /同构, 核 Ker)

同态 Homomorphism: \sigma : G \to G'
satisfies
\boxed {\sigma (a.b) = \sigma (a) . \sigma (b), \forall a, b \in G}

Homomorphism 同态 (自 ~: Endo-morphism)
Monomorphism 单同态
Epimorphism 满同态

Isomorphism 同构 (自~ : Auto-morphism)

丘教授 Tip: (有些)抽象数学(eg.同构性质) 可以用自觉(intuition)了解 (@ video 11:50 mins)。以上易证。

我的”直觉”例子Durian Kennel 榴莲 核

同态核 Kernel :
\boxed {Ker \: \sigma = | \{ a \in G | \sigma (a) = e' \}}

\boxed {Ker \: \sigma < G }

Proposition: (命题)
Let \: homomorphism \: \sigma : G \to G'
then
\boxed {\sigma \:  injective \iff Ker \: \sigma = \{e\}}

第四课: Normal Subgroup 正规子群 / Quotien Group 商群

H Normal Subgroup of G:
\boxed {H \triangleleft G} \iff \boxed {gH = Hg, \forall g \in G}  \iff
共轭子群 Conjugate Subgroup: \boxed{gHg^{-1} = H}

第五课: 群同态基本定律

商群: {\text {Quotient Group : } G / H} 
[证明]: 商集 G/H 的乘法 有 群 的4个性质 (Closure, Associative, Neutral , Inverse = C.A.N.I.) => G/H 是 商群 

(aH)(bH)
= a(Hb)H [associative]
= a(bH)H [normal subgroup ]
= ab(HH) [associative]
= abH [normal subgroup]
=> G/H closure

群同态基本定律

\boxed { Ker \: \sigma \triangleleft G} \iff  \boxed {G / Ker \: \sigma \cong Im \: \sigma} 同构

继续 Part 3:群的线表示和例子

北大 丘维声的 “群论” List of All Videos:http://www.youtube.com/playlist?list=PLwzFfIxhEkcxvU7-c8rPBbPLHUeacPIpa

Ref:
1. 《抽象代数基础》 (第二版)

习题 1.4 {1 ~ 6}

2. 高等代数学习指导书 (下册) (第二版)

Inspirational Scientist: Dan Shechtman

Singapore Maths Tuition

Source: https://www.theguardian.com/science/2013/jan/06/dan-shechtman-nobel-prize-chemistry-interview

To stand your ground in the face of relentless criticism from a double Nobel prize-winning scientist takes a lot of guts. For engineer and materials scientist Dan Shechtman, however, years of self-belief in the face of the eminent Linus Pauling‘s criticisms led him to the ultimate accolade: his own Nobel prize.

The atoms in a solid material are arranged in an orderly fashion and that order is usually periodic and will have a particular rotational symmetry. A square arrangement, for example, has four-fold rotational symmetry – turn the atoms through 90 degrees and it will look the same. Do this four times and you get back to its start point. Three-fold symmetry means an arrangement can be turned through 120 degrees and it will look the same. There is also one-fold symmetry (turn through 360 degrees), two-fold (turn through 180 degrees) and six-fold symmetry (turn through 60…

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北京大学:数学是什么 ?

丘维声教授 

第1讲 数学的思维方式 

3000 年前 希腊,巴比伦,中国,印度, 10世纪阿拉伯, 16世纪欧洲文艺复兴 数学 => [经典数学 Classical Math]

1830 年 数学的革命 – 法国天才少年 伽罗瓦 (Évariste Galois 1811 – 1832) => [近代数学 Modern Math]

观察 (Observe): 客观现象
\downarrow
抽象 (Abstraction) : 概念, 建立 模型 (Model)
\downarrow
探索 (Explore): 自觉 (Intuition), 解剖 , 类比(Analogy), 归纳 (Induction), 联想, 推理 (Deduction) 等…
\downarrow
猜测 (Conjecture) : eg. Riemann Conjecture (unsolved)
\downarrow
论证 (Prove): 只能用公理 (Axioms)(已知的共识), 定义 (概念), 已经证明的定理 (Theorems), 进行逻辑推理并计算.
\downarrow
揭示 (Reveal): 事物的内在规律 (井然有序)

第二讲: 例子 – 微积分 (Calculus) 的诞生, 演变, 严谨化

思维路程:

15 世纪 天体运动的观察: 哥白尼, 开普勒 三大定律 (天文数据结论, 非数学证明)

17 世纪 理论化: [英]牛顿,[德] Leibniz (非严密的数学)

19 世纪 严密数学: [法] Cauchy 柯西, [德] Wierstrass => “epsilon-delta” 极限 (Limit) => 柯西 数列 (Series).

实数 (R Real Numbers) 的 Complete (完备性 ) : [德国中学数学 老师] Dedekind (戴德金)’s Cut

有理数 (Q Rational Numbers): 稠密 但 不 Complete , 即 有漏洞, 穿插进 无理数 (irrational like \pi, \sqrt{2}  ) 

定理:  如果 数列是 柯西数列 => 一定有极限, 且此 极限一定是 实数

例子: Series S = {1.4 , 1.41, 1.414 … }

S has no limit in \mathbb {Q}, but limit = \sqrt{2} \in  \mathbb{R}

The Map of Mathematics

Show to your schooling children why they need to study Maths – the Queen of all Sciences – which pushes the frontier of human evolution in last 3,000 years. Maths is always invented few centuries or decades before it becomes useful. For examples:  Complex numbers invented accidentally by the 16th century Italian Mathematicians for solving polynomial equation of 3rd degree, became useful in Physics Electrical and Magnetic Fields (19 CE) ; Invention of Analytic Geometry (17 CE) allowed Newton to trace the earth-sun orbit; Calculus propelled Physics and Physical Chemistry; Leibniz’s Binary Math (18 CE) discovery applied in Computing (20 CE)…

Latest Examples

1. Topology was invented in 1900 by French PolyMath Henri Poincaré, today applied in Big Data, AI…

2. His PhD student invented “Derivatives” Partial Differentiation, today applied in Commodity Trading, Stock Trading, Financial Derivatives… with Black-Sholes formula. 1998 USA Sub-Prime Crisis due to the misuse and lack of understanding of its limitation (“fat tail” ).

3. Mathematician SS Chern 陈省身and Nobel Physicist Yang Zhen-Ning 杨振宁were working independently in the USA for 40 years, Chern on Differential Geometry, Yang on Yang-Mill Equation (one the 7 unsolved Math Problems in 21st century). Through a common friend the hedge fund billionaire James Simons – Chern’s former PhD Math student and university colleague of Yang – they realised that the Math “Fiber Bundles” (纤维丛) invented by Chern 30 years earlier could apply in Yang’s Physics (Gauge Theory).