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 1==
==[[Lecture:Topology and Geometry |lecture 1]]==
Starting from lecture 1 of this course, we have realized that Mobius strip is a very powerful mathematical ideas. --[[User:Benkoo|Benkoo]] ([[User talk:Benkoo|talk]]) 03:35, 18 July 2021 (UTC)
Starting from lecture 1 of this course, we have realized that Mobius strip is a very powerful mathematical ideas. --[[User:Benkoo|Benkoo]] ([[User talk:Benkoo|talk]]) 03:35, 18 July 2021 (UTC)


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Ex. (For example) Mobius strip, when you are doing the experiment of cutting the Mobius strip yes if you didn't draw it out you still will know what will happen but if you draw it out it will let you more Easier to understand what is happening.
Ex. (For example) Mobius strip, when you are doing the experiment of cutting the Mobius strip yes if you didn't draw it out you still will know what will happen but if you draw it out it will let you more Easier to understand what is happening.


click([[Lecture:Topology and Geometry 1|here]])to learn more about this Lecture.
 
<ref>{{:Lecture:Topology and Geometry}}</ref>  
<ref>{{:Lecture:Topology and Geometry}}</ref>  


==lecture 2==
==[[Lecture:Topology and Geometry 2|lecture 2]]==
====This lecture is about====
====This lecture is about====
# Solving problem by deformation
# Solving problem by deformation
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## what is the different between circle and disk  
## what is the different between circle and disk  


click([[Lecture:Topology and Geometry 2|here]])to learn more about this Lecture.


<ref>{{:Lecture:Topology and Geometry 2}}</ref>
<ref>{{:Lecture:Topology and Geometry 2}}</ref>


==lecture 3==
==[[Lecture:Topology and Geometry 3|lecture 3]]==
====This Lecture is about====
====This Lecture is about====
# The Operation of I:product
# The Operation of I:product
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## using cut to understanding quotient
## using cut to understanding quotient


click([[Lecture:Topology and Geometry 3|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 3}}</ref>
<ref>{{:Lecture:Topology and Geometry 3}}</ref>


==lecture 4==
==[[Lecture:Topology and Geometry 4|lecture 4]]==
====This Lecture is about====
====This Lecture is about====
# Quotient in topology
# Quotient in topology
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# Introduction to <math> \Sigma g </math> and <math> Ng </math>
# Introduction to <math> \Sigma g </math> and <math> Ng </math>
# [[Homeomorphism]]
# [[Homeomorphism]]
click([[Lecture:Topology and Geometry 4|here]])to learn more about this Lecture.
 
<ref>{{:Lecture:Topology and Geometry 4}}</ref>
<ref>{{:Lecture:Topology and Geometry 4}}</ref>


==Lecture 5==
==[[Lecture:Topology and Geometry 5|Lecture 5]]==
====This Lecture is about====
====This Lecture is about====
  1. The transformation between  <math> \Sigma g </math> and <math> Ng </math>
  1. The transformation between  <math> \Sigma g </math> and <math> Ng </math>


click([[Lecture:Topology and Geometry 5|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 5}}</ref>
<ref>{{:Lecture:Topology and Geometry 5}}</ref>


==Lecture 6==
==[[Lecture:Topology and Geometry 6|Lecture 6]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 6|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 6}}</ref>
<ref>{{:Lecture:Topology and Geometry 6}}</ref>


==Lecture 7==
==[[Lecture:Topology and Geometry 7|Lecture 7]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 7|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 7}}</ref>
<ref>{{:Lecture:Topology and Geometry 7}}</ref>


==Lecture 8==
==[[Lecture:Topology and Geometry 8|Lecture 8]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 8|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 8}}</ref>
<ref>{{:Lecture:Topology and Geometry 8}}</ref>


==Lecture 9==
==[[Lecture:Topology and Geometry 9|Lecture 9]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 9|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 9}}</ref>
<ref>{{:Lecture:Topology and Geometry 9}}</ref>


==Lecture 10==
==[[Lecture:Topology and Geometry 10|Lecture 10]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 10|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 10}}</ref>
<ref>{{:Lecture:Topology and Geometry 10}}</ref>


==Lecture 11==
==[[Lecture:Topology and Geometry 11|Lecture 11]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 11|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 11}}</ref>
<ref>{{:Lecture:Topology and Geometry 11}}</ref>


==Lecture 12==
==[[Lecture:Topology and Geometry 12|Lecture 12]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 12|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 12}}</ref>
<ref>{{:Lecture:Topology and Geometry 12}}</ref>


==Lecture 13==
==[[Lecture:Topology and Geometry 13|Lecture 13]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 13|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 13}}</ref>
<ref>{{:Lecture:Topology and Geometry 13}}</ref>


==Lecture 14==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 14|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 14}}</ref>
<ref>{{:Lecture:Topology and Geometry 14}}</ref>


==Lecture 15==
==[[Lecture:Topology and Geometry 15|Lecture 15]]==
====This Lecture is about====
====This Lecture is about====


click([[Lecture:Topology and Geometry 15|here]])to learn more about this Lecture.
<ref>{{:Lecture:Topology and Geometry 15}}</ref>
<ref>{{:Lecture:Topology and Geometry 15}}</ref>


Revision as of 13:49, 20 July 2021

Introduction to Topology and Geometry

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

lecture 1

Starting from lecture 1 of this course, we have realized that Mobius strip is a very powerful mathematical ideas. --Benkoo (talk) 03:35, 18 July 2021 (UTC)

Mobius strip is a strip twist by one or more times. One twist is equal to . Before the strip becomes a Mobius strip, it can be divided into two sides. We will name it red and blue. After you twist the strip and turn it into a Mobius strip. If the Mobius strip has an odd twist the blue part will be connected to the red part. If you have an even twist, the blue part will be connected to the blue, and the red will be connected to red. If you start to cut the middle for the blue part and the red part of the Mobius strip you will get to different kinds of 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 has the same length and same number of twists as the Mobius strip before you cut)


In Topology and Geometry There are three-point 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.

In Topology and geometry, you should learn to see and draw things that can't be seen physically. Ex. (For example) Mobius strip, when you are doing the experiment of cutting the Mobius strip yes if you didn't draw it out you still will know what will happen but if you draw it out it will let you more Easier to understand what is happening.


[1]

lecture 2

This lecture is about

  1. Solving problem by deformation
  2. Understanding by turning it to a higher dimension
  3. Introduction to Basic Building Blocks
    1. n-ball
    2. (n-1)-sphere
    3. what is the different between circle and disk


[2]

lecture 3

This Lecture is about

  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

[3]

lecture 4

This Lecture is about

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

[4]

Lecture 5

This Lecture is about

1. The transformation between   and

[5]

Lecture 6

This Lecture is about

[6]

Lecture 7

This Lecture is about

[7]

Lecture 8

This Lecture is about

[8]

Lecture 9

This Lecture is about

[9]

Lecture 10

This Lecture is about

[10]

Lecture 11

This Lecture is about

[11]

Lecture 12

This Lecture is about

[12]

Lecture 13

This Lecture is about

[13]

This Lecture is about

[14]

Lecture 15

This Lecture is about

[15]



Also, we should make proper reference[16], and it will show at the Reference section.

References

  1. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 1/15. local page: African Institute of Mathematical Sciences. 
  2. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 2/15. African Institute of Mathematical Sciences. 
  3. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 3/15. African Institute of Mathematical Sciences. 
  4. ↑ Tokieda, Tadashi (13 May 2014). Topology and Geometry. 4/15. African Institute of Mathematical Sciences. 
  5. ↑ Tokieda, Tadashi (13 May 2014). Topology and Geometry. 5/15. African Institute of Mathematical Sciences. 
  6. ↑ Tokieda, Tadashi (13 May 2014). Topology and Geometry. 6/15. African Institute of Mathematical Sciences. 
  7. ↑ Tokieda, Tadashi (13 May 2014). Topology and Geometry. 6/15. African Institute of Mathematical Sciences. 
  8. ↑ Tokieda, Tadashi (25 May 2014). Topology and Geometry. 8/15. African Institute of Mathematical Sciences. 
  9. ↑ Tokieda, Tadashi (13 May 2014). Topology and Geometry. 9/15. African Institute of Mathematical Sciences. 
  10. ↑ Tokieda, Tadashi (14 May 2014). Topology and Geometry. 10/15. African Institute of Mathematical Sciences. 
  11. ↑ Lecture:Topology and Geometry 11
  12. ↑ Lecture:Topology and Geometry 12
  13. ↑ Lecture:Topology and Geometry 13
  14. ↑ Lecture:Topology and Geometry 14
  15. ↑ Lecture:Topology and Geometry 15
  16. ↑ Tokieda, Tadashi (12 May 2014). Topology and Geometry. 3/15. African Institute of Mathematical Sciences. 
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