Let me start off with a description of my background. I am an undergraduate student, with some background in real analysis (Rudin, Principles of Mathematical Analysis), measure theory (Royden, Real Analysis: Parts I and III) and a little functional analysis (first five chapters of Rudin's 'Real and Complex Analysis'). I know no geometry, beyond Loring Tu's 'Introduction to Manifolds'. Additionally, just to give you the entire picture, I have an algebra background from Dummit's and Foote's 'Abstract Algebra', until but not including the section on Galois Theory. I also know a little spectral theory from William Arveson's 'A Short Course in Spectral Theory'.
I recently took a course on ordinary differential equations, where we followed a combination of Arnold's 'Ordinary Differential Equations' and George Simmons's 'Differential Equations with Applications and Historical Notes'. I liked the course, and decided to study partial differential equations from Evans's 'Partial Differential Equations'.
I would like a roadmap that answers the following questions:
What related areas of math would I need to know to understand research-level papers in the subject? Here, I understand that there could be many frontiers of the subject, each requiring its own set of prerequisites. I am looking for an answer that explains what exactly the area of research aims at doing and what the prerequisite content to work in that area would be. If certain things are required only in parts as prerequisites, I would very much appreciate it if you could name the theorems or the sections of books I would have to read.
What could I read immediately after reading Evans? I've heard that Michael Taylor's three volume series is good, and plan on taking a look at that, but more well-directed (as in, towards a specific sub-area of research) recommendations would be appreciated.
If you could estimate the time it would take (on an average) to study the material you suggest, especially if it isn't a book that mentions that, I would appreciate it, since it helps to know whether I am rushing through things (in which case, I often remember nothing later) or am going too slowly (in which case, I will try seeking help from some more advanced students or a professor).