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.
Best Answer
Given your preamble about algebra, topology, and geometry it sounds like your question is: "what are the basic objects of study in analysis?" I think there is an answer which is just as satisfying (while being just as much of an oversimplification) as the answers to the corresponding questions in other areas of mathematics: the basic objects of study in analysis are functions on Euclidean space or possibly linear spaces of continuous functions on Euclidean space.
There are a number of caveats. For instance, it is quite often useful to consider functions on more exotic spaces than just Euclidean space, just as it is often useful to consider schemes in algebraic geometry even though the main objects of study are generally varieties. Also, while abstract constructions like products or cohomology theories are fairly central to each of algebra, topology, and geometry they are not as important in analysis because it is not so useful to think of a function as an object in some category (though it can be useful to think of a space of functions as such).
Once you have this perspective in mind, a lot of analysis falls into place fairly naturally.
Similarly, much research in analysis can ultimately be traced back to a basic question about functions, such as:
How well can a general function be approximated by simpler sorts of functions, such as functions with rich algebraic structure like polynomials or functions with rich symmetry like trigonometric series? What can be said about functions which are particularly well approximated by simpler functions?
How is the structure of the domain of a function reflected in its analytical properties and vice-versa?
How can geometric techniques help locate a specific function with desirable properties among an ocean of possibilities?
To what extent are the properties of a function determined by an equation to which it is a solution?
What are the useful notions of distance between functions, and what properties do nearby functions necessarily share?
All that said, I want to disagree with your claim that Folland's textbook is an inadequate guide to current research in analysis. Like any good textbook in an area as old as analysis it lacks the breadth and depth necessary to make serious contact with current research, but it still has in its pages some of the basic results and first hints of many active research areas (with the notable exception of spectral theory / operator algebras).