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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:

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.

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Often wondered this myself, good show. – Alex Youcis Dec 28 '11 at 7:00
+1! Eagerly looking forward to answers – mathmath8128 Dec 28 '11 at 7:12
I think that this is best answered by looking at the relevant categories of the Arxiv. – Michael Greinecker Dec 28 '11 at 10:36
Seeing a lot of great answers. However I would like to point towards real life applications such as image denoising and materials technology. Dunno if you had this in mind for an answer. I have used things such as nonlinear diffusions on Lie-Groups to enhance images. Further the numerical analysis involved for showing stability involves things like spectral analysis of operators. So a large number of analysts actually use techniques mentioned by you to model and simplify such real-life applications. – UPS Mar 21 '13 at 15:27

5 Answers 5

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.

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Great example to begin with!A terrific introduction to this area can be found in Jurgen Jost's RIEMANNIAN GEOMETRY AND GEOMETRIC ANALYSIS. I can't think of a better textbook for a second course in differential geometry. But be warned-not easy going and you'll need your analysis chops on. – Mathemagician1234 Dec 28 '11 at 7:28
Willmore Conjecture: Proof announced on Feb 27, 2012. – Jesse Madnick May 11 '12 at 5:34
@JesseMadnick, yes, Rob Kusner is taking it seriously, – Will Jagy May 11 '12 at 19:57

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.

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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).

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This is a very interesting perspective. Most surprising to me is the emphasis on euclidean space. Were I to have guessed at the "objects of analysis," I would have said topological vector spaces (over $\mathbb{R}$ or $\mathbb{C}$), but this makes sense, too. – Jesse Madnick Jun 7 '12 at 23:31
I'd be interested in hearing about particular results in Folland's book that are good "first hints of ... active research areas." (For the record, I never meant to seriously claim that Folland's text is an "inadequate guide to current research" -- just that from my perspective as a beginner, it's hard to see the connections.) – Jesse Madnick Jun 7 '12 at 23:43
When I was first thinking about this I almost wrote that the basic objects in analysis are functions on the real line - this alone accounts for a huge portion of the subject. But these days there is a great deal of interest in various sorts of interactions between geometry and analysis (with information flowing both ways), so it's really better to say that analysis is about functions of several variables. Of course p-adic analysis is a rapidly growing field, so maybe it's better to say that analysis is about functions on finite dimensional vector spaces over complete fields. – Paul Siegel Jun 8 '12 at 15:49
To see my point about Folland's book, look at some of the later chapters. Between the chapter on Fourier analysis and the section on Haar measure, you've got the rudiments of harmonic analysis. The chapter on distributions provides an excellent basic introduction to PDE theory, and combined with an earlier chapter you get the flavor of modern functional analysis as well. The last chapter might as well be the first chapter of a book on geometric measure theory, and similarly the chapter on probability is an excellent self-contained introduction to modern analytic techniques in that subject. – Paul Siegel Jun 8 '12 at 15:57

Besides the geometric analysis described in Will Jagy's answer:

PDE theory is certainly one of the biggest areas of modern research in analysis. For example, currently tons of work is being produced about various aspects of the Navier-Stokes equation. If you do an arxiv search for papers with "Navier-Stokes" in the title you get a pretty good sampling. The big fish would be the Navier-Stokes conjecture of the millennium problems, but there are numerous papers coming out nowadays giving results on existence, uniqueness, decay of solutions, stability of solutions, statistical properties of solutions, you name it.

Another really popular area of PDE theory nowadays would be Schrodinger equations. Again, they are interested in existence and uniquess for initial data in various function spaces, decay of solutions, the effects of potentials in various function spaces, solitons, expansion formulas for solutions, the list goes on.

Many other PDEs are popular as well. In general the most popular ones are those coming out of physical applications.

There is still a good amount of classical harmonic analysis going on, but a lot of harmonic analysis people combine their field with the PDE problems mentioned above. The non-PDE areas include issues involving convergence of Fourier series and integrals, connections with combinatorics, and generalizing classical harmonic analysis to weighted function spaces or multilinear analogues.

Operator algebras/functional analysis is big nowadays too but I'm less familiar with it. They too have many connections to physics. I think several complex variables is less active.

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Also, much of what probabilists study is essentially analysis. Concepts such as convergence and concentration of measures (in all kinds of exotic spaces), mixing of random walks, understanding spectra of random matrices can be thought to be analysis questions.

Also, don't forget analytic number theory. A key question here is understanding the distribution of primes by proving analytical properties about the Riemann zeta function and its relatives. Recent breakthroughs on the (weak) twin prime conjecture (Zhang 2013) are essentially results about analysis.

Perhaps the foremost analyst today is Terence Tao, so you may want to see his work to get a sense of what the big questions currently are.

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