Need help to know the prerequisites of these graduate-level courses

[Arian.D]

I'm an undergraduate student studying my 4th semester of pure mathematics and I already have passed two courses in analysis that covers chapters 1 to 8 of Rudin's mathematical analysis and I've taken a course in abstract algebra that covers basics of group theory (excluding Silow's theorem) and basics of ring and module theory (including Tensor product over modules and injective, projective and flat modules).
I've talked to two of my university professors to join their classes and they've agreed. I got these choices to join:

1- Finite groups theory
2- Functional algebra
3- Manifold I
4- Algebraic varieties and Riemannian manifolds
5- Real Analysis I
6- Banach algebra
7- Operator theory

I'm thinking of joining two of these classes as a guest student but I'm a bit confused. I personally like algebra and geometry very much, but I don't know which one of these courses I could join. Your suggestions will be highly appreciated.

 

[micromass]

Could you perhaps tell more about these courses?? For example, what is the contents and what are the books they are using?? Then we'll be in a better position to advice you.

 

[Arian.D]

well, I know that the professor who teaches Manifold I has her own book which is not famous. I have her lecture notes taught in the previous semester class, she first covers some basic definitions like local charts, coordinate functions, local coordinate system, atlases, maximal atlas, differentiable manifolds, product manifold, differential structure and differentiable functions on a manifold and things of that sort.
The second chapters cover things about tangent space, cotangent space, vector fields, and then she moves on to cover stuff like Torus, Möbius strip, Klein bottle, etc...

I think I could join her manifold I class. I understand the concepts easily and I like the course very much, even though I find it really hard to solve her homework problems and I'm afraid that my low problem solving capability would make me infertile in mathematics :P

About Real Analysis I, I guess they're going to teach from Royden's real analysis I suppose.

and I have no idea about other courses. The professors that teach those courses are on vacations now and I don't have access to them to ask them about the books they're going to use. So please just assume that they are regular graduate courses covering those topics.

 

[micromass]

OK, let me look at the different courses:

1- Finite groups theory

I think you will likely be able to do this course. My guess is that it will cover things like Sylow theorems, solvable groups, nilpotent groups, Jordan-Holder theorem, etc. Maybe you will even go into cohomology and stuff. I think that your prerequisites are likely enough.

2- Functional algebra

I have no idea what this is about.

3- Manifold I

This indeed looks like an introduction to differential geometry. Depending on the lecturer, it can be easy or very hard. I recommend taking things like topology, analysis and linear algebra before taking a manifolds course. Having taken a course of Rudin, that might be enough. But you say that you have troubles with the exercises, which is a bad sign.

4- Algebraic varieties and Riemannian manifolds

Prerequisites here are definitely an algebra course and a complex analysis course. You should be very comfortable with things like maximal ideal, noetherian rings, algebraically closed, etc. for algebra. If you did not yet do complex analysis, then this course is most certainly nothing for you.

5- Real Analysis I

Rudin's real analysis is likely enough for a grad real analysis course. It will probably focus on measure theory and function spaces such as Hilbert and Banach spaces.

6- Banach algebra
7- Operator theory

I don't recommend doing these classes before you did Real Analysis I.

 

[Matrix0]

I understand the importance of having a strong foundation in mathematics for graduate-level courses. It seems like you have a solid understanding of analysis and abstract algebra, which are essential for many advanced courses in mathematics.

Based on your interests in algebra and geometry, I would suggest considering the courses on finite groups theory and algebraic varieties and Riemannian manifolds. These courses will build upon your knowledge of group theory and introduce you to the concepts of algebraic geometry, which combines algebra and geometry.

However, I would also recommend considering taking a course in real analysis, as it is a fundamental subject in mathematics and will provide a strong foundation for other advanced courses. Real analysis is also closely related to geometry, as it deals with the properties of real numbers and functions.

Ultimately, the decision on which courses to take should also depend on your future academic and career goals. If you are interested in pursuing a career in pure mathematics, it may be beneficial to take courses in both algebra and analysis. If you are interested in applied mathematics, then courses in functional algebra, Banach algebra, and operator theory may be more relevant.

I would also recommend speaking with your professors and seeking their advice on which courses would be most beneficial for your academic and career goals. They may also be able to provide more specific information on the prerequisites for each course and help you make a more informed decision.

Best of luck in your studies!