Learn Differential Topology: Point-Set, Algebraic, & Calculus on Manifolds

In summary, the conversation discussed the order in which one typically studies topology, with the general consensus being Point-set Topology, followed by Algebraic Topology, and then Differential Topology. The goal of studying these subjects is to ultimately gain an understanding of Calculus on Manifolds and Morse Theory. While it may be possible to skip certain aspects of Algebraic Topology, it is generally recommended to study it in order to gain a deeper understanding of the subject and its connections to differential forms. The individual in the conversation is considering independent study with a topologist to further their understanding of topology for future studies in calculus of variations.
  • #1
leon1127
486
0
I have just found that topology is very interesting. I just want to know how one studies topology. do they go in the order of Point-set Topology, Algebraic Topology, then Differential Topology? My ultimate goal is to understand Calculus on Manifolds and Morse Theory. Is it possible to jump to Differential Topology with knowledge of basic point-set topology and differential forms?

Thx
 
Physics news on Phys.org
  • #2
This is the order one usually does it in. I can't imagine skipping Algebraic Topology - there are too many key things there and there is a large interplay between it and differential forms.
 
  • #3
slearch said:
This is the order one usually does it in. I can't imagine skipping Algebraic Topology - there are too many key things there and there is a large interplay between it and differential forms.
thx.
but of course when i say skip it does not mean skip it completely, I will also study the key results of it, however if I want to study the part main related to analysis, is it possible to skip the "non-manifold" related part? The possiblility that i have right now is independent study with a topologist, I just wish to learn the most from him that would help me in further study in calculus of variations.
 

Related to Learn Differential Topology: Point-Set, Algebraic, & Calculus on Manifolds

1. What is differential topology?

Differential topology is a branch of mathematics that studies the properties of differentiable manifolds. It uses tools from calculus and algebraic topology to understand the geometric and topological structures of these manifolds.

2. What are some applications of differential topology?

Differential topology has many applications in physics, engineering, and computer science. For example, it is used in the study of fluid dynamics, robotics, and computer graphics. It also has important applications in theoretical physics, particularly in the fields of general relativity and string theory.

3. What are the main topics covered in "Learn Differential Topology: Point-Set, Algebraic, & Calculus on Manifolds"?

This course covers the fundamentals of differential topology, including point-set topology, algebraic topology, and calculus on manifolds. It also covers topics such as vector fields, differential forms, and homotopy theory.

4. Do I need prior knowledge of topology or calculus to take this course?

While some knowledge of topology and calculus may be helpful, this course is designed to be accessible to students with a basic understanding of mathematics. The course starts with the fundamentals and gradually builds upon them, making it suitable for beginners.

5. How can learning differential topology benefit my career?

Learning differential topology can open up opportunities in a variety of fields, including physics, engineering, computer science, and data analysis. It also provides a strong foundation for further study in advanced mathematics, such as differential geometry and algebraic topology. Additionally, the problem-solving and critical thinking skills developed through studying differential topology can be applied to many other areas of study and work.

Similar threads

  • STEM Academic Advising
Replies
4
Views
325
  • STEM Academic Advising
Replies
3
Views
1K
  • STEM Academic Advising
Replies
10
Views
2K
  • STEM Academic Advising
Replies
14
Views
851
  • STEM Academic Advising
Replies
4
Views
1K
  • STEM Academic Advising
Replies
3
Views
1K
  • STEM Academic Advising
Replies
3
Views
1K
  • STEM Academic Advising
Replies
16
Views
760
  • STEM Academic Advising
Replies
3
Views
2K
Replies
8
Views
3K
Back
Top