Topology and Geometry

Introduction to Topology and Geometry

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

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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 $B^{n}$
   2. (n-1)-sphere $S^{n-1}$ (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 $I^{m}$
2. m-torus $T^{m}$
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 $\Sigma g$ and $Ng$
3. Homeomorphism

Lecture 5

1. The transformation between $\Sigma g$ and $Ng$
2. Overflow
3. The L dimension object vs K dimension object in M dimension

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, and it will show at the Reference section.

Some interesting websites that referred to this lecture series.