It may not suit your goals, but one approach is to enroll in a masters program before entering a doctoral program. This could help you get back into the groove of academic life,
and also give you a chance to meet new professors who could write letters for your application to a more high-powered doctoral program. (I once advised a student who had spent quite a long time, maybe 8 years, in the software industry before returning to academia, and
this is the route she took. I think it served her well; because of the masters, which involved a mixture of coursework and a small thesis, she was very solidly prepared for her doctoral work, and was one of the strongest students in her cohort.)
Maybe I am reading too much into your pseudonym and your partly apologetic and partly condescending comments about the course you are going to take, but please,
Don't disparage the "rules" and computational aspects of differential equations.
Firstly, it is a beautiful subject with direct scientific origin and arguably most applications (save only calculus, perhaps) of all the courses you'd ever take. Secondly, these scientific connections continue to motivate and shape the development of the subject. Thirdly, rigor and abstraction are not substitutes for the actual mathematical content. Bourbaki never wrote a volume on differential equations, and the reason, I think, is that the subject is too content-rich to be amenable to axiomatic treatment. Finally, I've taught students who were gung-ho about rigorous real analysis, Rudin style, but couldn't compute the Taylor expansion of $\sqrt{1+x^3}.$ Knowing that the Riemann-Hilbert correspondence is an equivalence of triangulated categories may feel empowering, but as a matter of technique, it is mere stardust compared with the power of being able to compute the monodromy of a Fuchsian differential equation by hand.
Having forewarned you, here are my favorite introductory books on differential equations, all eminently suitable for self-study:
- Piskunov, Differential and integral calculus
- Filippov, Problems in differential equations
- Arnold, Ordinary differential equations
- Poincaré, On curves defined by differential equations
- Arnold, Geometric theory of differential equations
- Arnold, Mathematical methods of classical mechanics
You will find a lot of geometry, including an excellent exposition of calculus on manifolds, in the right context, in Arnold's Mathematical methods.
Best Answer
I believe that Jech's book is a solid part of any graduate student's preparation for independent research in set theory. He covers most or even all of the main topics of set-theoretic research, and he does so at quite a high level, including some extremely advanced material. For further study of large cardinals, however, the book should probably be supplemented by Kanamori's book The Higher Infinite (see my review), and for learning forcing, I always encourage my graduate students to read both Jech and also Kunen's book, as well as some others, especially Bell's book on forcing via Boolean-valued models (and my own article on the Boolean ultrapower), and to play all these texts off of one another, as each has some strengths the others lack.
Jech's book is extremely thorough, and I suppose that if you mastered every last bit of it, then indeed I think it would position you for independent research in set theory. But of course, the more typical pattern is to read at first only the easier parts of it, while also learning from other books, and gradually bring oneself to the research level that way.
During and after your study of Jech, you will need someone to guide you to research topics and problems, to suggest problems or areas that might be fruitful for your independent work or which may interest you.