What Advanced Math Topics Should I Focus on Next?

[sammycaps]

Hello everyone, I have a bit of time on my hands, so I'm trying to take an organized approach to learning some more advanced math.

What I have now is,
Real Analysis 1 (First half of Baby Rudin)
Linear Algebra (First half of Friedberg Insel Spence)
Point Set Topology (Munkres almost through Urysohn Metrization)
Some Algebraic Topology (Just some basics about the 1st Fundamental Group)
Basic Ring and Field Theory (Some baby book)
Basic Group Theory (Abstract Algebra-Herstein)
Some ODE (but a while ago)
Some Probability and Statistics (Again a while ago and from weak textbooks)
Some Set Theory (First half of Halmos)

I would like to get a more complete treatment of math at the advanced undergraduate level, and it seems I'm missing Complex Analysis, Differential Geometry, more advanced Algebra, and more advanced Probability and Statistics (I have little measure theory, so maybe I'd need to learn some of that first)

I'll be taking a second analysis course in the Spring so I'll get a treatment of that.

My thoughts were to try to tackle at least some of Dummit and Foote with respect to Algebra, then try Do Carmo's Differential Geometry.

If you have suggestions on books, other topics I should look into, or an order for which I should try to tackle this, that would be much appreciated. I have just about a year and a half.

 

[TonyG]

Hello there!

It's great that you are taking an organized approach to learning advanced math. It can be overwhelming to try and tackle everything at once, so it's good that you have a plan in mind.

Based on your current knowledge and interests, I have a few suggestions for you:

1. Complex Analysis: This is an important area of math that you seem to be missing. I would recommend starting with "Complex Variables" by James Ward Brown and Ruel V. Churchill. It covers the basics of complex numbers, functions, and integration, and also includes some applications. After that, you can move on to "Functions of One Complex Variable" by John B. Conway for a more in-depth treatment.

2. Differential Geometry: Since you already have some knowledge of topology, I think you can start with "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo. It's a good introductory text that covers the basics of curves and surfaces in a clear and concise manner.

3. Advanced Algebra: As you mentioned, Dummit and Foote's "Abstract Algebra" is a great choice for this topic. It's a comprehensive text that covers a wide range of topics in abstract algebra, including group theory, ring theory, field theory, and more. It's a good idea to start with the topics you are most interested in and then move on to the others.

4. Measure Theory: This is an important topic in probability and statistics, so it's good that you are planning on learning it. A good introductory text is "Measure, Integration, and Real Analysis" by Sheldon Axler. It covers the basics of measure theory and integration, and also includes some applications to probability.

5. Other topics: I would also recommend looking into topics such as topology, number theory, and combinatorics. These are important areas of math that have applications in various fields.

In terms of an order to tackle these topics, I would suggest starting with Complex Analysis and Differential Geometry, as they seem to be the ones you are most interested in. Then you can move on to Advanced Algebra and Measure Theory. As you mentioned, you will also be taking a second analysis course in the Spring, so that will also help with your understanding of real analysis.

I hope this helps and best of luck with your studies!