【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.

Hairy Ball Theorem

‘Hairy Ball Theorem’ (Topology)
1. “You can’t comb the hairs on a coconut.”
2. “There exists a calm point on earth where there is no wind.”
Both 1 & 2  are ‘Hairy Ball Theorem’.

If we have a ball with hairs sticking out from each point on it, then impossible to comb the hairs flat with a continuous movement without leaving at least one hair sticking up vertically (typically, at the center of a swirl). Not applicable to doughnut.
=> Wind pattern:
Ball = Earth
Combed down hairs = wind
Length of hairs = wind’s force
=> there must be a place on Earth where there is no wind at all.

Topology

Topology (by Poincaré)

Moniker “Rubber-Sheet Geometry“, compared with Geometry’s ‘rigid objects‘.

[Greek]= τοΠοζ(Place) λΟγια(Study)
[Latin]= Analysis Situs (Situation)

1. Remove (invariants) of geometry:

  • a. Euclidean (distance)
  • b. Affine (//, ratio)
  • c. Projective (cross-ratio)

2. Preserve ‘Neighbourhood’ (Nearness)

  • define ‘Continuity’ (Analysis)

3. Elastic deformation (stretch, bend, twist)

  • a line is no longer a line.