(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math and physics getting ready to apply to grad school. So far, I know that I enjoy PDEs (and most kinds of analysis, generally), mathematical physics, and especially dynamical systems. But this is way too broad for choosing a Ph.D. program. My background in these subjects is somewhere between the undergraduate and graduate level, but certainly not 'up to date' or research-level.
I want to get a clearer picture of these fields as they exist today. If I'm going to try to contribute to these topics in the next 5-10 years, I'd like to know what I'll be getting into.
If you work in these fields, or you have colleagues who do, how would you describe the current state of affairs?
Specifically:
- What are people trying to accomplish?
- How do those things fit into a larger picture?
- What are the obstructions?
- What are some of the major recent advances?
- What would you like to see happen over the next few decades?
I'm looking for technical descriptions, preferably with references to actual papers. Pretend you were describing your work to a fellow mathematician from an entirely different specialty. Don't worry if it's over my head (I'm sure it will be); the point is to get a taste, to help narrow my interests, and to have a guide to come back to over the years.
Thanks!
Best Answer
I am afraid that the task you describe as "obtaining a clear picture" of PDE, Mathematical physics and dynamical systems is impossible. I doubt that there are people who have a "clear picture" of all these three fields. Each of them is enormous. My advise is to find an adviser who works in the broad area of analysis, and to rely on his directions for the study of some specific area. Of course you can read about the broader area, and perhaps after 40 years of work and reading you will obtain some general picture:-)
For a general introduction to dynamical systems, I recommend the book of Katok and Hasselblatt, Introduction to modern theory of dynamical systems, or another book of the same authors, A first course in dynamics, with a panorama of recent developments.
I don't think there exists a modern survey of the whole mathematical physics. But Reed and Simon 4 volume course, Methods of Mathematical Physics gives an introduction to some parts of it.