Advanced Topics in Mathematics post DE

[Samit]

Hello,
I am a software professional and 33 years old. I always loved mathematics but never studied/practiced it seriously and forgot whatever I had learned before. One and half years back I started with algebra and then calculus, multivariable calculus and differential equations. All were self-study and now I am fairly comfortable till differential eqns. Next, I need your advice on how to proceed further.

Can you pls. suggest what should I focus on next? Also, can you pls. suggest good reference books? I tried learning Complex Analysis from Advanced Engineering Mathematics by Kreyszig. No doubt it is a great book but I found it difficult. It seems this book is good for revising/practicing for those who are well familiar with those topics. But I am starting from scratch. Generally, I find those books hard where multiple concepts are combined in one sentence and explained very crisply with too many statistical symbols (i.e. less of explanation and more of equations and symbols - for example: I have seen cases where a simple concept that can be expressed simply with a couple of words are explained with only equations and statistical symbols - and too many of them confuses me).

I found the online materials by Prof. Paul Dawkins of Lamar University to be perfect for me. It explains all the concepts beautifully in very simple language. It covers till Diff. Equations. I wish there had been many other advanced math materials from the same author.

Eagerly awaiting your response. Basically, what I am looking for is to curve out a plan for the next 1 or 2 years with your help. Pls. also include other suggestions that you might think should be useful to me.
Rgds,

 

[HuskyNamedNala]

Probability is pretty good. I love probability and statistics - especially from the way it makes you think about the world. You can combined your probability knowledge and study stochastic calculus. That is a fledgling field - about where CFD was 30 years ago.

 

[HuskyNamedNala]

Or, you can expand on your diff-eq and study nonlinear differential equations, which will lead you down the path of fractal mathematics. But it is also important to determine what your goal is first.

 

[Samit]

nonlinear differential equations

Thanks a lot for your response. For now, goal is to gain knowledge - you see, I have just started. Any interesting related topic will be fine. I shall seek new goal once I reach a certain level.
nonlinear differential equations - can you pls. suggest a book to start with?

 

[axmls]

Perhaps Calculus by Spivak? It's an excellent introduction to analysis. Or you could go with a full analysis book. Would you rather expand to different areas of math (i.e. start learning number theory, probability and statistics, etc.) or develop a deeper knowledge of the things you already know?

 

[Samit]

Perhaps Calculus by Spivak? It's an excellent introduction to analysis. Or you could go with a full analysis book. Would you rather expand to different areas of math (i.e. start learning number theory, probability and statistics, etc.) or develop a deeper knowledge of the things you already know?

Hi, Thanks a lot for your response. I would prefer to develop a deeper knowledge of the things that I already know before moving to new topics. Can you suggest a good way to start this?

 

[HuskyNamedNala]

Nonlinear Dynamics and Chaos focuses on qualitative analysis of Non-linear DEs. The last chapter gets into fractal mathematics.

 

[axmls]

A good analysis textbook is what I'd recommend personally. I used Spivak's Calculus after my regular calculus courses to develop a better theoretical understanding of calculus. It was really my introduction to proof-based mathematics. Has very interesting (and some quite difficult) questions, but they're important. He has solutions and a solutions manual, though, so it's not impossible to check your work.

Or as stated above, you could go with partial differential equations or some other area of DEs. Maybe even, instead of analysis, an introductory abstract algebra textbook.

I personally enjoyed Spivak. Terrence Tao also has a free analysis textbook (2 volumes) available on his website. You may want to skim through the first few pages of those and see if that's what you'd like to learn.