Maths -Topologie

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【区别:代数拓扑 (Algebraic Topology)  微分拓扑 (Differential Topology )  微分几何 ( Differential Geometry ) 代数几何 (Algebraic Geometry ) 交换代数  (Commutative Algebra ) 微分流形 (Differential Manifold )

​【区别:代数拓扑 (Algebraic Topology)  微分拓扑 (Differential Topology )  微分几何 ( Differential Geometry ) 代数几何 (Algebraic Grometry ) 交换代数  (Commutative Algebra ) 微分流形 (Differential Manifold ) ?】月如歌:并不能理解什么叫做楼主所说的配对。我简要谈下我对于上述所列名词的理解。… 

http://www.zhihu.com/question/23848852/answer/26771912 (分享自知乎网)

Sheaves do not belong to algebraic geometry:
https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html

Who cares about topology? (inscribed rectangle problem)

Excellent video for the curious minds! Who cares about Topology such as Torus (aka donut) or Mobius Strip ? They can be used to prove difficult math such as the unsolved problem “Inscribed square/rectangle inside any closed loop”.

To understand the Topology on Loops, please view the lecture  here : Homotopy (同伦) and the Fundamental Group (群) of surface.

Is the Abstract Mathematics of Topology Applicable to the Real World?

1st speaker :
◇ History: Riemann discovered Topology on his papers left behind after death. He told friend Betti.
◇ Betti Number: number of
– ‘scissors’ cut to make a tree (in 2 dim),
– ‘drill’ cut to make a disc (in 3 dim).

2nd speaker:
◇ Evolution (bacteria, viruses) using Topology ‘Barcoding’ technique.

比喻:
病菌(viruses / bacteria)是白骨精多变 (mutate), Topology (拓扑)是孙悟空的火眼金睛, 妖怪不能遁形, 骗得了常人, 骗不过老孙, “三打白骨精”。

3rd speaker:
◇ Liquid Crystal: Homology

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.