Difference between revisions of "Topology and Geometry"

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This is a course that [[Henry Koo|Henry]] and [[Ben Koo|Ben]] are studying during 2021.
This is a course that [[Henry Koo|Henry]] and [[Ben Koo|Ben]] are studying during 2021.


==[[Lecture:Topology and Geometry |Lecture 1]]==
==content==
Starting from Lecture 1 of this course, we have realized that the [[Mobius strip]] is a very powerful mathematical idea. --[[User:Benkoo|Benkoo]] ([[User talk:Benkoo|talk]]) 03:35, 18 July 2021 (UTC)
{{:Lecture content:Topology and Geometry 1}}
{{:Lecture content:Topology and Geometry 2}}
{{:Lecture content:Topology and Geometry 3}}
{{:Lecture content:Topology and Geometry 4}}
{{:Lecture content:Topology and Geometry 5}}
{{:Lecture content:Topology and Geometry 6}}
{{:Lecture content:Topology and Geometry 7}}
{{:Lecture content:Topology and Geometry 8}}
{{:Lecture content:Topology and Geometry 9}}
{{:Lecture content:Topology and Geometry 10}}
{{:Lecture content:Topology and Geometry 11}}
{{:Lecture content:Topology and Geometry 12}}
{{:Lecture content:Topology and Geometry 13}}
{{:Lecture content:Topology and Geometry 14}}
{{:Lecture content:Topology and Geometry 15}}


The [[Mobius strip]] is a strip twisted one or more times. One twist is equal to <math>180^o</math>.
Before the strip becomes a [[Mobius strip]], it can be divided into two sides. We will name them red and blue. Then, you twist the strip to turn it into a [[Mobius strip]]. If the [[Mobius strip]] has an odd twist the blue side will be connected to the red side. If you have an even twist, the blue side will be connected to the blue, and the red will be connected to red. If you start to cut the middle of the blue part and the red part of the Mobius strip you will get two different outcomes:


1. The Mobius strip has an odd twist so you will get a bigger Mobius strip


2. The Mobius strip has an even twist then you will get two Mobius strips. (that are the same length and same number of twists as the Mobius strip before it was cut)




In Topology and Geometry
There are three points to remember.


1. There is so much more to mathematics than numbers and formulas. (For example, there is pictorial thinking)
#[[Lecture:Topology and Geometry 6|Lecture 6]]
##[[Isotopic]]
##The relationship between [[isotopic]] and [[homeomorphic]]
##outside the shape inside the shape
##The positive and negative [[intersection]]
###There is no tangent vector intersection in Topology
#[[Lecture:Topology and Geometry 7|Lecture 7]]
##The positive and negative intersection
###There is no tangent [vector] intersection in Topology
##[[Intersect transversely]]
#[[Lecture:Topology and Geometry 8|Lecture 8]]
##Jordan curve theorem
### If you have a closed curve which does not intersect itself it will divide the plan into two parts.
##Fixed Point Theorem
#[[Lecture:Topology and Geometry 9|Lecture 9]]
#[[Lecture:Topology and Geometry 10|Lecture 10]]
#[[Lecture:Topology and Geometry 11|Lecture 11]]
#[[Lecture:Topology and Geometry 12|Lecture 12]]
#[[Lecture:Topology and Geometry 13|Lecture 13]]
#[[Lecture:Topology and Geometry 14|Lecture 14]]
#[[Lecture:Topology and Geometry 15|Lecture 15]]


2. Always draw pictures whenever you work on mathematics.


3. There is so much more to pictures than photos of objects.


In Topology and Geometry, you should learn to see and draw things that can't be seen physically.
For example take the Mobius strip. When you are doing the experiment of cutting the Mobius strip you will still know what will happen but if you draw it out it will be easier to understand what is happening.
<ref>{{:Lecture:Topology and Geometry}}</ref>
==[[Lecture:Topology and Geometry 2|lecture 2]]==
====This lecture is about====
# Solving problem by deformation
# Understanding by turning it to a higher dimension
# Introduction to Basic Building Blocks of Topology and Geometry
## n-ball <math>B^n</math>
## (n-1)-sphere <math>S^n , n-1</math> (Don't know why I can't write the )
## what is the different between circle and disk
<ref>{{:Lecture:Topology and Geometry 2}}</ref>
==[[Lecture:Topology and Geometry 3|lecture 3]]==
====This Lecture is about====
# The Operation of I:product
## m-cube <math>I^m</math>
# m-torus <math>T^m</math>
# The multiplication of shape in Topology and Geometry
# Quotient in topology
## all kinds of quotient example
## using cut to understanding quotient
<ref>{{:Lecture:Topology and Geometry 3}}</ref>
==[[Lecture:Topology and Geometry 4|lecture 4]]==
====This Lecture is about====
# Quotient in topology
## using cut to understanding quotient
# Introduction to <math> \Sigma g </math> and <math> Ng </math>
# [[Homeomorphism]]
<ref>{{:Lecture:Topology and Geometry 4}}</ref>
==[[Lecture:Topology and Geometry 5|Lecture 5]]==
====This Lecture is about====
#The transformation between  <math> \Sigma g </math> and <math> Ng </math>
#Overflow
#The L dimension object vs K dimension object in M dimension
<ref>{{:Lecture:Topology and Geometry 5}}</ref>
==[[Lecture:Topology and Geometry 6|Lecture 6]]==
====This Lecture is about====
#[[Isotopic]]
#The relationship between [[isotopic]] and [[homeomorphic]]
#outside the shape inside the shape
#The positive and negative [[intersection]]
##There is no tangent vector intersection in Topology
<ref>{{:Lecture:Topology and Geometry 6}}</ref>
==[[Lecture:Topology and Geometry 7|Lecture 7]]==
====This Lecture is about====
#The positive and negative intersection
##There is no tangent [vector] intersection in Topology
#[[Intersect transversely]]
<ref>{{:Lecture:Topology and Geometry 7}}</ref>
==[[Lecture:Topology and Geometry 8|Lecture 8]]==
====This Lecture is about====
#Jordan curve theorem
## If you have a closed curve which does not intersect itself it will divide the plan into two parts.
#Fixed Point Theorem
<ref>{{:Lecture:Topology and Geometry 8}}</ref>
==[[Lecture:Topology and Geometry 9|Lecture 9]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 9}}</ref>
==[[Lecture:Topology and Geometry 10|Lecture 10]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 10}}</ref>
==[[Lecture:Topology and Geometry 11|Lecture 11]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 11}}</ref>
==[[Lecture:Topology and Geometry 12|Lecture 12]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 12}}</ref>
==[[Lecture:Topology and Geometry 13|Lecture 13]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 13}}</ref>
==[[Lecture:Topology and Geometry 14|Lecture 14]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 14}}</ref>
==[[Lecture:Topology and Geometry 15|Lecture 15]]==
====This Lecture is about====
<ref>{{:Lecture:Topology and Geometry 15}}</ref>




Also, we should make proper reference<ref>{{:Lecture:Topology and Geometry 3}}</ref>, and it will show at the Reference section.


Some interesting websites<ref>Gaurish, Gaurish4Math on Topology ,https://gaurish4math.wordpress.com/tag/tadashi-tokieda/, last accessed: July 22, 2021</ref> that referred to this lecture series.


Also, we should make proper reference<ref>{{:Lecture:Topology and Geometry 3}}</ref>, and it will show at the Reference section.
<ref>{{:Lecture:Topology and Geometry 1}}</ref>


=References=
=References=

Latest revision as of 13:49, 26 July 2021

Introduction to Topology and Geometry

This is a course that Henry and Ben are studying during 2021.

content

Lecture 1: Mobius Strip

  1. Mobius strip
  2. In Topology and Geometry There are three points to remember.
    1. There is so much more to mathematics than numbers and formulas. (For example, there is pictorial thinking)
    2. Always draw pictures whenever you work on mathematics.
    3. There is so much more to pictures than photos of objects.


Lecture 2

  1. Solving problem by deformation
  2. Understanding by turning it to a higher dimension
  3. Introduction to Basic Building Blocks of Topology and Geometry
    1. n-ball
    2. (n-1)-sphere (Don't know why I can't write the )
    3. what is the different between circle and disk


Lecture 3

  1. The Operation of I:product
    1. m-cube
  2. m-torus
  3. The multiplication of shape in Topology and Geometry
  4. Quotient in topology
    1. all kinds of quotient example
    2. using cut to understanding quotient


Lecture 4

  1. Quotient in topology
    1. using cut to understanding quotient
  2. Introduction to and
  3. Homeomorphism


Lecture 5

  1. The transformation between and
  2. Overflow
  3. The L dimension object vs K dimension object in M dimension

Lecture content:Topology and Geometry 6 Lecture content:Topology and Geometry 7 Lecture content:Topology and Geometry 8 Lecture content:Topology and Geometry 9 Lecture content:Topology and Geometry 10 Lecture content:Topology and Geometry 11 Lecture content:Topology and Geometry 12 Lecture content:Topology and Geometry 13 Lecture content:Topology and Geometry 14 Lecture content:Topology and Geometry 15




  1. Lecture 6
    1. Isotopic
    2. The relationship between isotopic and homeomorphic
    3. outside the shape inside the shape
    4. The positive and negative intersection
      1. There is no tangent vector intersection in Topology
  2. Lecture 7
    1. The positive and negative intersection
      1. There is no tangent [vector] intersection in Topology
    2. Intersect transversely
  3. Lecture 8
    1. Jordan curve theorem
      1. If you have a closed curve which does not intersect itself it will divide the plan into two parts.
    2. Fixed Point Theorem
  4. Lecture 9
  5. Lecture 10
  6. Lecture 11
  7. Lecture 12
  8. Lecture 13
  9. Lecture 14
  10. Lecture 15



Also, we should make proper reference[1], and it will show at the Reference section.

Some interesting websites[2] that referred to this lecture series.

[3]

References

  1. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 3/15. African Institute of Mathematical Sciences. 
  2. ↑ Gaurish, Gaurish4Math on Topology ,https://gaurish4math.wordpress.com/tag/tadashi-tokieda/, last accessed: July 22, 2021
  3. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 1/15. local page: African Institute of Mathematical Sciences. 
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