In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about the classification of finite simple groups, and gains some slight sense of what a group theorist might wonder about.
In a topology class, one learns about topological spaces, and conceives of an algebraic topologist as someone who studies topological spaces, their algebraic invariants, and wonders about ways of classifying such spaces. Similarly, a differential geometer might be described as someone who studies manifolds and their invariants, and an algebraic geometer as someone who studies varieties and schemes and their invariants.
Now, obviously these sorts of one-sentence descriptions are rather simplistic, especially since many (most?) mathematicians work at the interface of a variety of different areas.
That being said, I feel that I have absolutely no conception of what contemporary analysts actually do. My sense is that contemporary analysis does not, for example, resemble the material found in (say) Folland's text.
To be slightly more concrete, my question boils down to these:
- What areas of analysis are at the center of active research?
- What sort of questions are analysts concerned with? What are some major themes that each subject is concerned with? What are the big-picture goals of each subject?
My sense is that current areas of research include:
- Harmonic analysis (and Fourier analysis)
- Operator theory
- Partial differential equations (PDE)
- Several complex variables (SCV)
- Geometric measure theory (GMT)
My sense is that analysts care about things like regularity, growth, and oscillations, and might be concerned with:
- Approximation problems
- Interpolation problems
- Optimization problems
- Boundary-value problems
However, all of this is really the extent of my understanding.
Note on motivation: Just to be clear, I am someone who really likes analysis. Part of my motivation for asking (other than curiosity) is that I seem to meet very few American undergraduates or first-year graduate students who are interested in pursuing analysis, and sometimes wonder if this is because few of us seem to have any idea what analysts actually do.
Note also: Saying that analysts are mathematicians who really like estimates does not count :-)
Apologies if this question is too vague or too broad.