University Course: Topology - What You Need to Know

[evinda]

Hello! (Wave)

I am thinking about taking the university course Topology.
What do you think about this subject? What is it about?
What knowledges do someone have to have, to take it?
What is the difference from the course Algebraic Topology, which is also offered?

(Thinking)

 

[Fantini]

Topology studies continuity between spaces. It gives a sense of proximity without necessarily using distances.

Analysis courses under your belt will be of service, but if you don't have that you will need just patience to sit through proofs and fight through the abstractness. Nevertheless, it is fun and it starts right from the beginning. Everything is constructed concerning topology is constructed mostly from the ground.

To understand algebraic topology you need to know topology first.

 

[evinda]

Ok... I will think about it... (Thinking) (Thinking)And.. what do you think about Complex Analysis and Harmonic Analysis?

What knowledges are required? (Nerd)

 

[alyafey22]

Ok... I will think about it... (Thinking) (Thinking)And.. what do you think about Complex Analysis and Harmonic Analysis?

What knowledges are required? (Nerd)

Depends of your purpose of taking the course. I took complex analysis because I like that concept and it helps me in solving some complicated integrals. Generally Electric engineers will be interested in such course.

For complex analysis a knowledge of real analysis will be of help but lacking it will not be a great problem.

 

[Fantini]

I don't know Complex Analysis, but I've heard that if you are interested in proving theorems then you need to know topology decently and real analysis. If you're more interested in the calculations part, you can skip over these at first.

 

[evinda]

And..what's about Harmonic Analysis? (Thinking)

 

[Fantini]

Don't know much. All I know is that you get to study the Laplacian on nice spaces, such as the sphere and the torus. My research shows it also has plenty of connections with Fourier Analysis.

 

[mathbalarka]

Complex analysis requires very little knowledge in real analysis, actually. The real analytic part is mostly about Fourier analysis. The interaction with point-set topology comes when you are introduced Jordar curve theorem (technically, I have never studied a whole proof of that, only some basic ideas about JCT for piecewise linear polygonal curves). But in the actual theory, you'll never need that much topology and real analysis.

 

[Euge]

And..what's about Harmonic Analysis? (Thinking)

Harmonic analysis is another big subject that deals with Fourier series and singular integrals in depth. You've probably seen Fourier series before, in which case you already have a head start! Harmonic analysis is actually very useful in the study of PDE, especially in the area of nonlinear dispersive equations.

 

[evinda]

Harmonic analysis is another big subject that deals with Fourier series and singular integrals in depth. You've probably seen Fourier series before, in which case you already have a head start! Harmonic analysis is actually very useful in the study of PDE, especially in the area of nonlinear dispersive equations.

Which of them (Harmonic analysis or Complex Analysis) is easier? (Thinking) Because I have a lot of difficult subjects, this semester.. (Malthe)

 

[Euge]

Which of them (Harmonic analysis or Complex Analysis) is easier? (Thinking) Because I have a lot of difficult subjects, this semester.. (Malthe)

Unless the complex analysis course uses harmonic analysis, the complex course will be easier. I strongly recommend holding off a full harmonic analysis course for graduate school. Now will be a good time to get solid foundation in complex variables, which is what you'll need for the other course anyway (in addition to differential topology).

 

[mathbalarka]

Of course complex analysis is easier (don't listen to me : I am totally biased)

 

[evinda]

I also don't know if I should take the subject Data Structures.
There are 4 sets of theoretical exercises and also a project.. Last year, at the project, they had to implement the operations of a photo-sharing service, like Instagram.

What do you think about this subject? (Thinking)

 

[evinda]

And.. what do you think about the subject Differential Geometry? (Thinking)
What is it about? What knowledge is required?

 

[Euge]

Differential geometry (at least at an introductory level) is a study of extrinsic and intrinsic properties of curves and surfaces. Usually, such a course is divided into two parts: local differential geometry and global differential geometry. I can't say that there are prerequisites that everyone needs for the subject, but it'll be useful to have background knowledge of multivariate calculus, linear algebra, and general topology.

 

[Fantini]

Classical differential geometry of curves and surfaces, in $\mathbb{R}^3$, does not need more than multivariable calculus and linear algebra to get a good handle on the subject and treat very nice questions.

However, differential geometry is one of those areas where there are many flavors of geometry and you can explore as many distinct depths are you wish, requiring ever more abstraction. :) It depends a bit on what kind of differential geometry you wish to tackle.

 

[evinda]

Differential geometry (at least at an introductory level) is a study of extrinsic and intrinsic properties of curves and surfaces. Usually, such a course is divided into two parts: local differential geometry and global differential geometry. I can't say that there are prerequisites that everyone needs for the subject, but it'll be useful to have background knowledge of multivariate calculus, linear algebra, and general topology.

Classical differential geometry of curves and surfaces, in $\mathbb{R}^3$, does not need more than multivariable calculus and linear algebra to get a good handle on the subject and treat very nice questions.

However, differential geometry is one of those areas where there are many flavors of geometry and you can explore as many distinct depths are you wish, requiring ever more abstraction. :) It depends a bit on what kind of differential geometry you wish to tackle.

Sounds interesting.. Maybe I will take this subject! (Mmm)

What do you think about Numerical Linear Algebra and Stochastic Processes?

What are these subjects about? What knowledge is required?