My specialty is Partial Differential Equations (PDE) and especially those of hyperbolic type. These arise in mathematical physics as equations of motion in acoustics, electromagnetic theory, elasticity, fluid dynamics, and other areas. Consequently there are a wealth of applications and natural questions.
In broadest terms the subject of PDE addresses the question: what features are forced upon a function because it is the solution of a partial differential equation from some family? As very few problems are explicitly solvable, the analysis is almost always qualitative, the key ingredients being the derivation of estimates for solutions and the construction of approximate solutions. There is a vast variety of technique that has evolved for both purposes and a student usually approaches research by learning techniques developed for a small class of problems and then broadening with time.
The entry level course is Math 656 which requires real analysis. Baby functional analysis is good background too, as the basic estimates are in terms of the norms of the classical Banach spaces. The texts of F. John, L.C. Evans and J. Rauch give the flavor of the course.
The subject is appealing because;
1. It uses the full arsenal of tools in analysis.
2. Problems arise in a wide variety of settings: several complex variables, differential geometry, mathematical physics, topology,...
3. Problems arise in context which makes them more interesting and suggest approaches. On the other hand, this demands that one appreciate the context. Interest from science and engineering means that there are opportunities for contacts with specialists in other areas.
1. You can't really start till the alpha analysis courses are complete.
2. The breadth of the area in subject matter and methods means that it takes time and hard work before one feels at home with the basics.
My recent work has included work on the short wavelength asymptotic analysis of solutions of linear and nonlinear hyperbolic problems.
The analysis of non reflective and/or absorbing layers in numerical methods.
Precise finite speed of propagation estimates for linear problems.
The quasilinear versions are outstanding open problems.
The mathematics of high powered lasers.