What do modern-day analysts actually do?

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.

• Harmonic analysis is the study of functions which have rich symmetries
• Functional analysis is the use of geometric techniques to study large spaces of functions
• PDE theory is the study of functions which arise naturally as solutions to equations
• Operator algebras are generalizations of rings of continuous functions (C*-algebras) or rings of measurable functions (Von Neumann algebras)
• Complex analysis is the study of functions which can be approximated in a very strong sense by polynomial functions
• Measure theory is the theory of functions which arise as limits (in a weak sense) of continuous functions

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).

Well, one area has become quite prominent since I was in college, which is applications of PDE to differential geometry. The Ricci flow, investigated for years and years by Hamilton, led eventually to the proof of the Poincare conjecture and Geometrization Conjectures in three dimensions, POINCARE Later the differentiable sphere theorem, uncertain in dimensions 7 and above, was established with these techniques, SCHOEN

Prior to this, manifolds were investigated by the behavior of geodesics, see Comparison Theorems in Riemannian Geometry by Cheeger and Ebin. The new question is often, here is a PDE that gives some geometric/topological information that does have solutions on small neighborhoods. Can we extend the solution to an entire manifold? The case that may be familiar is that of oriented compact surfaces, all of which have Riemannian metrics with constant curvature.

Since you mention GMT, still undecided is the Willmore conjecture, about the oriented closed torus in $\mathbb R^3$ that achieves the minimum integral of the square of the mean curvature. Leon Simon proved that a minimizer exists.

Here's one I tried, a conjecture of Meeks: let us be given two convex curves in parallel planes, close enough together such that there is at least one minimal surface with the two curves being the boundary of the surface. Does it follow that the surface is topologically an annulus? No idea.

I'm a graduate student in operator algebras, so I'll try to say a little bit about operator algebras as I see it. That said, one need only look at math.OA to see that this is far from exhaustive.

Operator algebraists study algebras of operators on topological vector spaces, i.e. von Neumann algebras or $C^*$-algebras. I'm less familiar with $C^*$-algebras than I would like to be, but much of the study of von Neumann algebras seems to be in classifying particular von Neumann algebras or showing that they have certain properties.

One of the basic questions we can ask about a von Neumann algebra is whether it has a trivial center. A von Neumann algebra that has a trivial center is called a factor, and can be classified into certain types (known as types $I$, $II_1$, $II_\infty$ and $III$) based on their lattices of projections.

Other properties that people sometimes study are solidity, rigidity, and injectivity. Another question that one might take interest in is whether von Neumann algebras are isomorphic. Numerous people, notably Vaughan Jones, have looked at questions pertaining to subfactors (which are pretty much what you'd expect).

One class of example of von Neumann algebras is the matrix algebras (these, and tensor products of these, are the type $I$ von Neumann algebras), but these are less interesting. Given a discrete group $G$, one can construct a von Neumann algebra $L(G)$ using the left regular representation of $G$ on $L^2(G)$.

The example of $G=F_n$, the free group on $n$ generators has been studied extensively, but there is still much that isn't known. For instance, it is not known whether $L(F_m)$ and $L(F_n)$ are isomorphic for distinct $m,n\ge 2$. This problem gave rise to the study of free probability, a noncommutative analog of probability theory, which has since developed into a field of study in its own right.